CAMBRIDGE MONOGRAPHS ONAPPLIED AND COMPUTATIONALMATHEMATICS
Series Editors
m ablowitz s davis j hincha iserles j ockendon p olver
28 Multiscale Methods for FredholmIntegral Equations
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The Cambridge Monographs on Applied and Computational Mathematics seriesreflects the crucial role of mathematical and computational techniques incontemporary science The series publishes expositions on all aspects of applicableand numerical mathematics with an emphasis on new developments in thisfast-moving area of research
State-of-the-art methods and algorithms as well as modern mathematicaldescriptions of physical and mechanical ideas are presented in a manner suited tograduate research students and professionals alike Sound pedagogical presentation isa prerequisite It is intended that books in the series will serve to inform a newgeneration of researchers
A complete list of books in the series can be found atwwwcambridgeorgmathematicsRecent titles include the following
14 Simulating Hamiltonian dynamics Benedict Leimkuhler amp Sebastian Reich15 Collocation methods for Volterra integral and related functional differential
equations Hermann Brunner16 Topology for computing Afra J Zomorodian17 Scattered data approximation Holger Wendland18 Modern computer arithmetic Richard Brent amp Paul Zimmermann19 Matrix preconditioning techniques and applications Ke Chen20 Greedy approximation Vladimir Temlyakov21 Spectral methods for time-dependent problems Jan Hesthaven Sigal Gottlieb amp
David Gottlieb22 The mathematical foundations of mixing Rob Sturman Julio M Ottino amp
Stephen Wiggins23 Curve and surface reconstruction Tamal K Dey24 Learning theory Felipe Cucker amp Ding Xuan Zhou25 Algebraic geometry and statistical learning theory Sumio Watanabe26 A practical guide to the invariant calculus Elizabeth Louise Mansfield27 Difference equations by differential equation methods Peter E Hydon28 Multiscale methods for Fredholm integral equations Zhongying Chen Charles A
Micchelli amp Yuesheng Xu29 Partial differential equation methods for image inpainting Carola-Bibiane
Schonlieb
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Multiscale Methods forFredholm Integral Equations
ZHONGYING CHENSun Yat-Sen University Guangzhou China
CHARLES A MICCHELLIState University of New York Albany
YUESHENG XUSun Yat-Sen University Guangzhou China
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University Printing House Cambridge CB2 8BS United Kingdom
Cambridge University Press is part of the University of Cambridge
It furthers the Universityrsquos mission by disseminating knowledge in the pursuit ofeducation learning and research at the highest international levels of excellence
wwwcambridgeorgInformation on this title wwwcambridgeorg9781107103474
copy Zhongying Chen Charles A Micchelli and Yuesheng Xu 2015
This publication is in copyright Subject to statutory exceptionand to the provisions of relevant collective licensing agreementsno reproduction of any part may take place without the written
permission of Cambridge University Press
First published 2015
A catalogue record for this publication is available from the British Library
Library of Congress Cataloguing in Publication dataChen Zhongying 1946ndash
Multiscale methods for Fredholm integral equations Zhongying Chen Sun Yat-SenUniversity Guangzhou China Charles A Micchelli State University of New York
Albany Yuesheng Xu Sun Yat-Sen University Guangzhou Chinapages cm ndash (The Cambridge monographs on applied and computational
mathematics series)Includes bibliographical references and index
ISBN 978-1-107-10347-4 (Hardback)1 Fredholm equations 2 Integral equations I Micchelli Charles A
II Xu Yuesheng III TitleQA431C4634 2015
515prime45ndashdc23 2014050239
ISBN 978-1-107-10347-4 Hardback
Cambridge University Press has no responsibility for the persistence or accuracy ofURLs for external or third-party internet websites referred to in this publication
and does not guarantee that any content on such websites is or will remainaccurate or appropriate
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Contents
Preface page ixList of symbols xi
Introduction 1
1 A review of the Fredholm approach 511 Introduction 512 Second-kind matrix Fredholm equations 713 Fredholm functions 1114 Resolvent kernels 1715 Fredholm determinants 2016 Eigenvalue estimates and a trace formula 2417 Bibliographical remarks 31
2 Fredholm equations and projection theory 3221 Fredholm integral equations 3222 General theory of projection methods 5323 Bibliographical remarks 78
3 Conventional numerical methods 8031 Degenerate kernel methods 8032 Quadrature methods 8633 Galerkin methods 9434 Collocation methods 10535 PetrovndashGalerkin methods 11236 Bibliographical remarks 142
4 Multiscale basis functions 14441 Multiscale functions on the unit interval 14542 Multiscale partitions 153
v
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vi Contents
43 Multiscale orthogonal bases 16644 Refinable sets and set wavelets 16945 Multiscale interpolating bases 18446 Bibliographical remarks 197
5 Multiscale Galerkin methods 19951 The multiscale Galerkin method 20052 The fast multiscale Galerkin method 20553 Theoretical analysis 20954 Bibliographical remarks 221
6 Multiscale PetrovndashGalerkin methods 22361 Fast multiscale PetrovndashGalerkin methods 22362 Discrete multiscale PetrovndashGalerkin methods 23163 Bibliographical remarks 263
7 Multiscale collocation methods 26571 Multiscale basis functions and collocation functionals 26672 Multiscale collocation methods 28173 Analysis of the truncation scheme 28874 Bibliographical remarks 298
8 Numerical integrations and error control 30081 Discrete systems of the multiscale collocation method 30082 Quadrature rules with polynomial order of accuracy 30283 Quadrature rules with exponential order of accuracy 31484 Numerical experiments 31885 Bibliographical remarks 321
9 Fast solvers for discrete systems 32291 Multilevel augmentation methods 32292 Multilevel iteration methods 34793 Bibliographical remarks 354
10 Multiscale methods for nonlinear integral equations 356101 Critical issues in solving nonlinear equations 356102 Multiscale methods for the Hammerstein equation 359103 Multiscale methods for nonlinear boundary
integral equations 377104 Numerical experiments 402105 Bibliographical remarks 413
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Contents vii
11 Multiscale methods for ill-posed integral equations 416111 Numerical solutions of regularization problems 416112 Multiscale Galerkin methods via the Lavrentiev
regularization 420113 Multiscale collocation methods via the Tikhonov
regularization 438114 Numerical experiments 456115 Bibliographical remarks 463
12 Eigen-problems of weakly singular integral operators 465121 Introduction 465122 An abstract framework 466123 A multiscale collocation method 474124 Analysis of the fast algorithm 478125 A power iteration algorithm 483126 A numerical example 484127 Bibliographical remarks 487
Appendix Basic results from functional analysis 488A1 Metric spaces 488A2 Linear operator theory 494A3 Invariant sets 502
References 519Index 534
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Preface
Fredholm equations arise in many areas of science and engineeringConsequently they occupy a central topic in applied mathematics Traditionalnumerical methods developed during the period prior to the mid-1980s includemainly quadrature collocation and Galerkin methods Unfortunately all ofthese approaches suffer from the fact that the resulting discretization matricesare dense That is they have a large number of nonzero entries This bottleneckleads to significant computational costs for the solution of the correspondingintegral equations
The recent appearance of wavelets as a new computational tool in appliedmathematics has given a new direction to the area of the numerical solution ofFredholm integral equations Shortly after their introduction it was discoveredthat using a wavelet basis for a singular integral equation led to numericallysparse matrix discretization This observation combined with a truncationstrategy then led to a fast numerical solution of this class of integral equations
Approximately 20 years ago the authors of this book began a systematicstudy of the construction of wavelet bases suitable for solving Fredholmintegral equations and explored their usefulness for developing fast multiscaleGalerkin PetrovndashGalerkin and collocation methods The purpose of this bookis to provide a self-contained account of these ideas as well as some traditionalmaterial on Fredholm equations to make this book accessible to as large anaudience as possible
The goal of this book is twofold It can be used as a reference text forpractitioners who need to solve integral equations numerically and wish touse the new techniques presented here At the same time portions of thisbook can be used as a modern text treating the subject of the numericalsolution of integral equations which is suitable for upper-level undergraduatestudents as well as graduate students Specifically the first five chapters of thisbook are designed for a one-semester course which provides students with a
ix
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x Preface
solid background in integral equations and fast multiscale methods for theirnumerical solutions
An early version of this book was used in a summer school on appliedmathematics sponsored by the Ministry of Education of the Peoplersquos Republicof China Subsequently the authors used revised versions of this book forcourses on integral equations at our respective institutions These teachingexperiences led us to make many changes in presentation resulting from ourinteractions with our many students
We are indebted to our many colleagues who gave freely of their time andadvice concerning the material in this book and whose expertise on the subjectof the numerical solution of Fredholm equations collectively far exceedsours We mention here that a preliminary version of the book was provided toKendall Atkinson Uday Banerjee Hermann Brunner Yanzhao Cao WolfgangDahmen Leslie Greengard Weimin Han Geroge Hsiao Hideaki KanekoRainer Kress Wayne Lawton Qun Lin Paul Martin Richard Noren SergeiPereverzyev Reinhold Schneider Johannes Tausch Ezio Venturino and AihuiZhou We are grateful to them all for their constructive comments whichimproved our presentation
Our special thanks go to Kendall Atkinson for his encouragement andsupport in writing this book We would also like to thank our colleagues at SunYat-Sen University including Bin Wu Sirui Cheng Xianglin Chen as well asthe graduate student Jieyang Chen for their assistance in preparing this book
Finally we are deeply indebted to our families for their understandingpatience and continued support throughout our efforts to complete this project
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Symbols
ae almost everywhere sect11Alowast adjoint operator of A sect211A[i j] minor of matrix A with lattice vectors i and j sect12B(XY) normed linear space of all bounded linear operators from
X into Y sect211C set of complex numbers sect11C(D) linear space of all real-valued continuous functions on D sect21Cm(D) linear space of all real-valued m-times continuously
differentiable functions on D sect21Cinfin(D) linear space of all real-valued infinitely differentiable functions
on D sect21C0(D) subspace of C(D) consisting of functions with support contained
inside D sectA1Cinfin0 (D) subspace of Cinfin(D) consisting of functions with support
contained inside D and bounded sectA1cσ (D) positive constant defined in sect212card T cardinality of T sect222cond(A) condition number of A sect223D(λ) complex-valued function at λ defined by (118)det(A) determinant of matrix A sect12diag(middot) diagonal matrix sect12diam(S) diameter of set S sect11Hm(D) Sobolev space sectA1Hm
0 (D) Sobolev space sectA1Lp(D) linear space of all real-valued pth power integrable functions
(1 le p ltinfin) sect21
xi
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xii List of symbols
Linfin(D) linear space of all real-valued essentially bounded measurablefunctions sect21
m(D) positive constant defined in sect212N set of positive integers 1 2 3 sect11N0 set of integers 0 1 2 sect11Nn set of positive integers 1 2 n for n isin N sect11PA characteristic polynomial of matrix A sect12R set of real numbers sect11Rd d-dimensional Euclidean space sect11Re f real part of f sect16rq(A) minor equation of A (14)Rλ resolvent kernel sect14rank A rank of matrix A sect335s(n) dimension of space Xn sect33span S span of set S sect331U index set (i j) i isin N0 j isin Zw(i) sect41Un index set (i j) i isin Zn+1 j isin Zw(i) sect451vol(S) volume of set S sect11Wmp(D) Sobolev space sectA1Wmp
0 (D) Sobolev space sectA1w(n) dimension of space Wn sect41Z set of integers 0plusmn1plusmn2 sect11Zn set of integers 0 1 2 nminus 1 for n isin N sect11
(middot) gamma function sect212nabla gradient operator sect213 Laplace operator sect213ρ(T ) resolvent set of operator T sect112σ(T ) spectrum of operator T sect112ωdminus1 surface area of unit sphere in Rd sect213ω(K h) modules of continuity of K sect13
factorial for example (14)⋃perp union of orthogonal sets sect41oplusdirect sum of spaces sect41otimestensor product (direct product) sect13
functional composition sect331|α| sum of components of lattice vector α sect21|sminus t| Euclidean distance between s and t sect212
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List of symbols xiii
A norm of operator A sect211 middot minfin norm of Cm(D) sect21 middot p norm of Lp(D) (1 le P le infin) sect21(middot middot) inner product sect21〈middot middot〉 value of a linear functional at a function sect211sim same order sect511
sminusrarr pointwise converge sect211uminusrarr uniformly converge sect211
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Introduction
The equations we consider in this book are primarily Fredholm integralequations of the second kind on bounded domains in the Euclidean spaceThese equations are used as mathematical models for a multitude of physicalproblems and cover many important applications such as radiosity equa-tions for realistic image synthesis [18 85 244] and especially boundaryintegral equations [12 177 203] which themselves occur as reformulationsof other problems typically originating as partial differential equations Inpractice Fredholm integral equations are solved numerically using piecewisepolynomial collocation or Galerkin methods and when the order of thecoefficient matrix (which is typically full) is large the computational costof generating the matrix as well as solving the corresponding linear systemis large Therefore to enhance the range of applicability of the Fredholmequation methodology it is critical to provide alternate algorithms whichare fast efficient and accurate This book is concerned with this challengedesigning fast multiscale methods for the numerical solution of Fredholmintegral equations
The development and use of multiscale methods for solving integral equa-tions is a subject of recent intense study The history of fast multiscale solutionsof integral equations began with the introduction of multiscale Galerkin(PetrovndashGalerkin) methods for solving integral equations as presented in[28 64 68 88 94 95 202 260 261] and the references cited thereinMost noteworthy is the discovery in [28] that the representation of a singularintegral operator by compactly supported orthonormal wavelets producesnumerically sparse matrices In other words most of their entries are so smallin absolute value that to some degree of precision they can be neglectedwithout affecting the overall accuracy of the approximation Later the papers[94 95] studied PetrovndashGalerkin methods using periodic multiscale basesconstructed from refinement equations for periodic elliptic pseudodifferential
1
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2 Introduction
equations and in this restricted environment stability convergence and matrixcompression were investigated For a first-kind boundary integral equationa truncation strategy for the Galerkin method using spline-based multiscalebasis functions of low degree was proposed in [260] Also in [261] forelliptic pseudodifferential equations of order zero on a three-dimensionalmanifold a Galerkin method using discontinuous piecewise linear multiscalebasis functions on triangles was studied
In another direction a general construction of multidimensional discontin-uous orthogonal and bi-orthogonal wavelets on invariant sets was presentedin [200 201] Invariant sets include among others the important cases ofsimplices and cubes and in the two-dimensional case L-shaped domainsA similar recursive structure was explored in [65] for multiscale functionrepresentation and approximation constructed by interpolation on invariantsets In this regard an essential advantage of this approach is the existenceof efficient schemes for generating recursively multilevel partitions of invariantsets and their associated multiscale functions All of these methods even extendto domains which are a finite union of invariant sets thereby significantlyexpanding the range of their applicability Therefore the constructions givenin [65 200 201] led to a wide variety of multiscale basis functions which onthe one hand have desirable simple recursive structure and on the other handcan be used in diverse areas in which the Fredholm methodology is applied
Subsequently the papers [64 68 202] developed multiscale piecewise poly-nomial Galerkin PetrovndashGalerkin and discrete multiscale PetrovndashGalerkinmethods An important advantage of multiscale piecewise polynomials is thattheir closed-form expressions are very convenient for computation Moreoverthey can easily be related to standard bases used in the conventional numericalmethod thereby providing an advantage for theoretical analysis as wellAmong conventional numerical methods for solving integral equations thecollocation method has received the most favorable attention in the engineeringcommunity due to its lower computational cost in generating the coefficientmatrix of the corresponding discrete equations In comparison the implemen-tation of the Galerkin method requires much more computational effort forthe evaluation of integrals (see for example [19 77] for a discussion of thispoint) Motivated by this issue [69] proposed and analyzed a fast collocationalgorithm for solving general multidimensional integral equations Moreovera matrix truncation strategy was introduced there by making a careful choice ofbasis functions and collocation functionals the end result being fast multiscalealgorithms for solving the integral equations
The development of stable efficient and fast numerical algorithms for solv-ing operator equations including differential equations and integral equations
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Introduction 3
is a main focus of research in numerical analysis and scientific computationsince such algorithms are particularly important for large-scale computationWe review the three main steps in solving an operator equation The firstis at the level of approximation theory Here we must choose appropriatesubspaces and suitable bases for them The second step is to discretize theoperator equations using these bases and to analyze convergence properties ofthe approximate solutions The end result of this step of processing is a discretelinear system and its construction is considered as a main task for the numericalsolution of operator equations The third step employs methods of numericallinear algebra to design an efficient solver for the discrete linear system Theultimate goal is of course to solve the discrete linear system efficiently andobtain an accurate approximate solution to the original operator equationTheoretical considerations and practical implementations in the numericalsolution of operator equations show that these three steps of processing areclosely related Therefore designing efficient algorithms for the discrete linearsystem should take into consideration the choice of subspaces and their basesthe methodologies of discretization of the operator equations and the specificcharacteristics and advantages of the numerical solvers used to solve theresulting discrete linear system In this book we describe how these threesteps are integrated in a multiscale environment and thereby achieve our goalof providing a wide selection of fast and accurate algorithms for the second-kind integral equations We also describe work in progress addressing relatedissues of eigenvalue and eigenfunction computation as well as the solution ofFredholm equations of the first kind
This book is organized into 12 chapters plus an appendix Chapter 1 isdevoted to a review of the Fredholm approach to solving an integral equation ofthe second kind In Chapter 2 we introduce essential concepts from Fredholmintegral equations of the second kind and describe a general setup of projectionmethods for solving operator equations which will be used in later chaptersThe purpose of Chapter 3 is to describe conventional numerical methodsfor solving Fredholm integral equations of the second kind including thedegenerate kernel method the quadrature method the Galerkin method thePetrovndashGalerkin method and the collocation method In Chapter 4 a generalconstruction of multiscale bases of piecewise polynomial spaces includingmultiscale orthogonal and interpolating bases is presented Chapters 5 6and 7 use the material from Chapter 4 to construct multiscale GalerkinPetrovndashGalerkin and collocation methods We study the discretization schemesresulting from these methods propose truncation strategies for building fastand accurate algorithms and give a complete analysis for the order of con-vergence computational complexity stability and condition numbers for the
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4 Introduction
truncated schemes In Chapter 8 two types of quadrature rule for the numericalintegration required to generate the coefficient matrix are introduced and errorcontrol strategies are designed so that the quadrature errors will neither ruin theoverall convergence order nor increase the overall computational complexityof the original multiscale methods The goal of Chapter 9 is to investigate fastsolvers for the discrete linear systems resulting from multiscale methods Weintroduce multilevel augmentation methods and multilevel iteration methodsbased on direct sum decompositions of the range and domain of the operatorequation In Chapters 10 11 and 12 the fast algorithms are applied to solvingnonlinear integral equations of the second kind ill-posed integral equations ofthe first kind and eigen-problems of compact integral operators respectivelyWe summarize in the Appendix some of the standard concepts and resultsfrom functional analysis in a form which is used throughout the book Theappendix provides the reader with a convenient source of the backgroundmaterial needed to follow the ideas and arguments presented in other chaptersof this book
Most of the material in this book can only be found in research papers Thisis the first time that it has been assembled into a book Although this book ispronouncedly a research monograph selected material from the initial chapterscan be used in a semester course on numerical methods for integral equationswhich presents the multiscale point of view
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1
A review of the Fredholm approach
In this chapter we pay homage to Ivar Fredholm (April 7 1866ndashAugust 171927) and review his approach to solving an integral equation of the secondkind The methods employed in this chapter are classical and differ from theapproach taken in the rest of the book We include it here because those readersinexperienced in integral equations should be familiar with these importantideas The basic tools of matrix theory and some complex analysis are neededand we shall provide a reasonably self-contained discussion of the requiredmaterial
11 Introduction
We start by introducing the notation that will be used throughout this bookLet C R Z and N denote respectively the set of complex numbers the setof real numbers the set of integers and the set of positive integers We alsolet N0 = 0 cup N For the purpose of enumerating a nonempty finite set ofobjects we use the sets Nd = 1 2 d and Zd = 0 1 dminus1 both ofwhich consist of d distinct integers For d isin N let Rd denote the d-dimensionalEuclidean space and a subset of Rd By C() we mean the linear space ofall continuous real-valued functions defined on We usually denote matricesor vectors over R in boldface for example A = [Aij i j isin Nd] isin Rdtimesd andu = [uj j isin Nd] isin Rd When the vector has all integer coordinates thatis u isin Zd we sometimes call it a lattice vector Moreover we usually denoteintegral operators by calligraphic letters Especially the integral operator witha kernel K will be denoted by K that is for the kernel K defined on times andthe function u defined on we define
(Ku)(s) =int
K(s t)u(t)dt s isin
5
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6 A review of the Fredholm approach
The most direct approach to solving a second-kind integral equation merelyreplaces integrals by sums and thereby obtains a linear system of equationswhose solution approximates the solution of the original equation The studyof the resulting linear system of equations leads naturally to the importantnotion of the Fredholm function and determinant which remain a central toolin the theory of second-kind integral equations see for example [183 253]We consider this direct approach when we are given a continuous kernelK isin C( times ) on a compact subset of Rd with positive Borel measurea continuous function f isin C() and a nonzero complex number λ isin C Thetask is to find a function u isin C() such that for s isin
u(s)minus λ
int
K(s t)u(t)dt = f (s) (11)
To this end for each positive h gt 0 we partition into nonempty compactsubsets i i isin Nn
=⋃iisinNn
i
such that different subsets have no overlapping interior and
diam i = max|xminus y| x y isin i le h
where |x| is the 2-norm of the vector x isin Rd This partition can be constructedby first putting a large ldquoboxrdquo around the set and then decomposing this boxinto cubes each of which has diameter less than or equal to h The sets i arethen formed by intersecting the set with the cubes where we discard sets ofzero Borel measure Therefore the partition of constructed in this manneris done ae Next we choose any finite set of points T = ti i isin Nn suchthat for any i isin Nn we have that ti isin i With these points we now replaceour integral equation (11) with a linear system of equations Specifically wechoose the number ρ = minusλ and the ntimes n matrix A defined by
A = [vol(j)K(ti tj) i j isin Nn]where vol(j) denotes the volume of the set j and replace (11) with thesystem of linear equations
(I+ ρA)u = f (12)
Here f is the vector obtained by evaluating the function f on the set T Of course the point of view we take here is that the vector u isin Rn
which solves equation (12) is an approximation to the function u on theset T Therefore the problem of determining the function u is replaced by thesimpler one of numerically solving for the vector u when h is small Certainly
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12 Second-kind matrix Fredholm equations 7
an important role is played by the determinant of the coefficient matrix ofthe linear system (12) Its properties especially as h rarr 0+ will be ourmain concern for a significant part of this chapter We start by studying thedeterminant of the coefficient matrix of the linear system (12) and then derivea formula for the entries of the inverse of the matrix I + ρA in terms of thematrix A
12 Second-kind matrix Fredholm equations
We define the minor of an n times n matrix If A = [Aij i j isin Nn] is an n times nmatrix q is a non-negative integer in Zn+1 i = [il l isin Nq] j = [jl l isin Nq]are lattice vectors in N
qn we define the corresponding minor by
A[i j] = det[Air js r s isin Nq]Sometimes the more elaborate notation
A
(i1 i2 iqj1 j2 jq
)(13)
is used for A[i j] When i = j that is for a principal minor of A we use thesimplified notation A[i] in place of A[i i] For a positive integer q isin Nn we set
rq(A) = 1
qsumiisinNq
n
A[i] (14)
and also choose r0(A) = 1
Lemma 11 If A is an ntimes n matrix and ρ isin C then
det(I+ ρA) =sum
qisinZn+1
rq(A)ρq (15)
Before proving this lemma we make two remarks
Remark 12 Using the extended notation for a minor as indicated in (13)we see that equation (14) is equivalent to the formula
rq(A) = 1
qsum
[illisinNq]isinNqn
A
(i1 iqi1 iq
) (16)
Certainly if any two components of the vector i = [il l isin Nq] are equalthen the corresponding minor has a repeated row (and column) and so is zeroThese terms may be neglected Moreover any permutation of the componentsof the vector i affects both a row and a column exchange of the determinant
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8 A review of the Fredholm approach
appearing in (16) and so does not affect the value of the determinant Sincethere are q such permutations we get that
rq(A) =sum
1lei1lti2ltmiddotmiddotmiddotltiqlen
A
(i1 iqi1 iq
) (17)
Remark 13 If the characteristic values of the matrix A are denoted by λj j isin Nn then
rq(A) =sum
1lei1lti2ltmiddotmiddotmiddotltiqlen
λi1λi2 middot middot middot λiq (18)
The right-hand side of this equation is an elementary symmetric function ofthe eigenvalues of A which is invariant under a similarity transformation ofthe matrix A This fact will be the basis of the proof of Lemma 11 presentedbelow
We next present the proof of Lemma 11
Proof By Schurrsquos upper-triangular factorization theorem (see for example[142] p 79) the matrix A can be factored in the form
A = Pminus1TP (19)
where P is an orthogonal matrix and T is an upper-triangular matrix whosediagonal entries are the eigenvalues of A (chosen in any prescribed order) Foran upper-triangular matrix T we observe from (18) that
det(I+ ρT) =prodjisinNn
(1+ ρλj)
=sum
qisinZn+1
ρqsum
1le j1ltj2ltmiddotmiddotmiddotltjqle n
λj1λj2 middot middot middot λjq
=sum
qisinZn+1
rq(T)ρq
thereby verifying that equation (15) is valid at least for an upper-triangularmatrix For a general matrix we use the reduction to the upper-triangular caseby an orthogonal similarity (19) Since all determinants in equations (15) and(16) are unchanged under an orthogonal similarity this comment proves thegeneral case
We now add a more difficult computation which provides an expansion ofthe elements of the matrix (I+ ρA)minus1 as a rational function of ρ To this end
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12 Second-kind matrix Fredholm equations 9
we introduce some needed constants We define for any k l isin Nn and q isin Zn+1
the constants
ulkq = 1
qsumiisinNq
n
A
[l ik i
] (110)
In (110) there are (not necessarily principal) minors of order q+ 1 Moreoverwhen q = n minus 1 and k l isin Nn with k = l we have that ulkq = 0 since Nn hasonly n distinct elements Also for the same reason ulkn = 0 for all k l isin Nn
We shall relate these constants But first we introduce for k l isin Nn
polynomials Ukl defined at ρ to be
Ulk(ρ) =sum
qisinZn+1
ulkqρq
Now to relate all of these quantities we start with the minor
A
(l i1 iqk i1 iq
)(111)
where l k isin Nn and expand it by the Laplace expansion by minors across itsfirst row to obtain the formula
A
(l i1 iqk i1 iq
)= AlkA
(i1 iqi1 iq
)+sum
misinNq
(minus1)mAlimA
(i1 i2 im+1 iqk i1 im middot middot middot iq
)
(112)
The symbol im appearing in the minor above is to be interpreted to mean thatthe imth column of A is deleted in that minor We now sum both sides of thisformula over all integers i1 i2 iq in Nn and interchange the summationsof the second term on the right-hand side of the resulting equation to yield theformulasum[ippisinNq]isinNq
nA
(l i1 iqk i1 iq
)=sum[ippisinNq]isinNq
nAlkA
(i1 iqi1 iq
)+summisinNq
sum[ippisinNq]isinNq
n(minus1)mAlim A
(i1 i2 im+1 iqk i1 im middot middot middot iq
)
(113)
The first term on the right-hand side of equation (113) is clearly Alkqrq(A)which follows from our definition (16) The value for the second term requiresmore explanation
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10 A review of the Fredholm approach
For the second term we point out that there are two sums over im isin NnThe outer sum already appears in the right-hand side of equation (113)and the inner one appears in the sum over all indices i1 i2 iq in NnIn the inner sum we first fix im and then sum over all the other indicesi1 imminus1 im+1 iq isin Nn This leads us to the expression
sum[i1 imminus1 im+1 iq]isinNqminus1
n
(minus1)mA
(i1 iq
k i1 imminus1 im+1 iq
)
We now locate the imth row in the minor of A and move it forward to the firstrow This requires mminus 1 row exchanges and gives us the expression
minussum
[i1 imminus1 im+1 iq]isinNqminus1n
A
(im i1 imminus1 im+1 iqk i1 imminus1 im+1 iq
)
Next we multiply this expression by Alim and compute the (first) sum of it overim isin Nn This yields the quantity
minussumrisinNn
Alr
sum[ippisinNqminus1]isinNqminus1
n
A
(r i1 iqminus1
k i1 iqminus1
)
But this quantity is independent of im and so appears q times in the other(second) sum over im So in summary we get the equationsum[ippisinNq]isinNq
n
A
(l i1 iqk i1 iq
)= Alk
sum[ippisinNq]isinNq
n
A
(i1 iqi1 iq
)
minusqsumrisinNn
Alr
sum[ippisinNqminus1]isinNqminus1
n
A
(r i1 iqminus1
k i1 iqminus1
)
which is equivalent to the formulasumiisinNq
n
A
[l ik i
]= Alk
sumiisinNq
n
A[i] minus qsumrisinNn
Alr
sumjisinNqminus1
n
A
[r jk j
]
When q = 0 the second term on the right is zero while the first sum on theright is set to one Likewise the expression on the left is set to Alk and so thisformula is still true when q = 0 We now multiply both sides by ρq
q and sumover q isin Zn+1 Upon simplification we conclude for l k isin Nn that
Ulk(ρ) = Alk det(I+ ρA)minus ρsumrisinNn
AlrUrk(ρ) (114)
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13 Fredholm functions 11
Let U be the ntimes n matrix defined by
U = (Ulk(ρ))lkisinNn
Using this matrix we re-express equation (114) in an equivalent matrix formas follows
Uminus A det(I+ ρA)+ ρAU = 0 (115)
Likewise by performing the Laplace expansion on (111) by minors across itscolumns we get the equation
Uminus A det(I+ ρA)+ ρUA = 0 (116)
From these formulas we get that
(I+ ρA)minus1 = Iminus ρUdet(I+ ρA)
(117)
Indeed this can be obtained by checking the equations
(I+ ρA)(
Iminus ρUdet(I+ρA)
)=(
Iminus ρUdet(I+ρA)
)(I+ ρA) = I
which may be confirmed by employing equations (115) and (116)
13 Fredholm functions
We now apply Lemma 11 to Fredholmrsquos discretization given in equation (12)For this purpose we introduce the notation for the Fredholm minors of acontinuous kernel K isin C(times)
Definition 14 For any two vectors x = [xl l isin Nq] and y = [yl l isin Nq] inq we define the corresponding qth-order minor of a continuous kernel K as
K
[xy
]= det[K(xl yr) l r isin Nq]
If x = y we use the simplified notation K[x]Sometimes to avoid confusion we may use the extended notation
K
(x1 xq
y1 yq
)for the minor K
[xy
]and similarly we might use
K
(x1 xq
x1 xq
)for K[x]
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12 A review of the Fredholm approach
Now returning to equation (12) we recall in that case the matrix A is givenby the formula
A = [vol(j)K(ti tj) i j isin Nn]For each lattice vector i = [il l isin Nq] in N
qn we form the vector si=[til
lisinNq] in q from the points in the set T = ti i isin Nn the subset
i =otimeslisinNq
il
of q and obtain a formula for the determinant of the coefficient matrix of thelinear system (12) namely
Dh(λ) =sum
qisinZn+1
(minus1)q
q λqsumiisinNq
n
vol(i)K[si]
Note that the vector si = [til l isin Nq] is in i and that
q =⋃iisinNq
n
i
thereby forming a partition of the set q ae In our expanded notation thedeterminant becomes
Dh(λ) =sum
qisinZn+1
(minus1)qλqsum
1lei1lti2ltmiddotmiddotmiddotltiqlen
vol(i)K
(ti1 tiqti1 tiq
)
We now begin the task of identifying the limit of the polynomials Dh ashrarr 0+
Definition 15 If K is a continuous kernel on times and λ isin C we definethe complex-valued function D at λ isin C by
D(λ) =sumqisinN0
(minus1)qλq
qintq
K[x]dx (118)
Next we point out that D is an entire function of λ We shall refer to thefunction D as the Fredholm function To prove that the Fredholm function D isan entire function we recall the Hadamard inequality for the determinant of anntimes n matrix (see for example [142]) When we use the Hadamard inequalityit is convenient to express an ntimes n matrix A in terms of its columns Thus wewrite A = [ai i isin Nn] in which each ai isin Rn is the ith column of A
Lemma 16 If A = [ai i isin Nn] is an ntimes n matrix then
| det A| leprodiisinNn
|ai|
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13 Fredholm functions 13
where |ai| denotes the 2-norm of ai
We now set
Kinfin = max|K(x y)| x y isin and derive the next lemma as an immediate application of the Hadamardinequality
Lemma 17 If K isin C(times) and x y isin q then∣∣∣∣K [xy
]∣∣∣∣ le Kqinfinqq2
Proof Since for each r isin Nq and xr y isin we have thatsumrisinNq
K2(xr y) le qK2infin
the result of this lemma follows immediately from the Hadamard inequality
Lemma 18 The function D defined by equation (118) is an entire function
Proof For q isin N0 and λ isin C we have by Lemma 17 that∣∣∣∣ (minus1)qλq
qintq
K[x]dx
∣∣∣∣ le vol()q|λ|qq K
qinfinqq2
By using the inequality
q lt eq (119)
valid for all q isin N we obtain that
vol()q|λ|qq K
qinfinqq2 =(
vol()|λ|Kinfinradicq
)q qq
q le(
vol()|λ|Kinfineradicq
)q
Hence when q is chosen to satisfy the condition
q gt (2vol()|λ| middot Kinfine)2
we conclude that ∣∣∣∣ (minus1)qλq
qintq
K[x]dx
∣∣∣∣ lt 2minusq
Thus the series on the right-hand side of equation (118) converges absolutely
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14 A review of the Fredholm approach
Theorem 19 For λ isin C
limhrarr0+
Dh(λ) = D(λ)
uniformly and absolutely on any bounded subset of C
For the proof of Theorem 19 we need some ancillary lemmas The first isabout ntimes n matrices A = [ai i isin Nn] and B = [bi i isin Nn]Lemma 110 If there exists a constant θ such that for all i isin Nn |ai| le θ
and |bi| le θ then
| det Aminus det B| le θnminus1sumiisinNn
|ai minus bi|
Proof The proof follows from the Hadamard inequality applied to theformula
det Aminus det B =sumiisinNn
det[b1 biminus1 ai minus bi ai+1 an]
For the next fact we need to recall the definition of the modulus of continuityof function K namely for h gt 0
ω(K h) = max|K(x y)minus K(xprime yprime)| |xminus xprime| le h |yminus yprime| le hLemma 111 For any x y isin q with |xminus y| le h there holds the estimate
|K[x] minus K[y]| le q
Kinfin (Kinfinradicq)qω(K h)
Proof We apply Lemma 110 to obtain the inequalities
|K[x] minus K[y]| le (Kinfinradicq)qminus1sumiisinNq
⎛⎝sumrisinNq
(K(xr xi)minus K(yr yi))2
⎞⎠12
le (Kinfinradicq)qminus1
q32ω(K h)
which gives the desired estimate
The next lemma estimates the difference between a multivariate integral anda finite sum which approximates it
Lemma 112 If f isin C(q) and q is partitioned as q =⋃iisinNqni ae and
si isin i for i isin Nqn then∣∣∣∣∣∣
intq
f (x)dxminussumiisinNq
n
vol(i)f (si)
∣∣∣∣∣∣ le (vol())qω(f h)
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13 Fredholm functions 15
where h is chosen so that diam i le h for all i isin Nqn
Proof We have thatsumiisinNq
n
vol(i)f (si)minusintq
f (x)dx =sumiisinNq
n
vol(i)f (si)minusinti
f (x)dx
=sumiisinNq
n
inti
( f (si)minus f (x))dx
Hence taking absolute values of both sides of the equation gives the inequality∣∣∣∣∣∣sumiisinNq
n
vol(i)f (si)minusintq
f (x)dx
∣∣∣∣∣∣ lesumiisinNq
n
inti
| f (si)minus f (x)|dx
le ω( f h)sumiisinNq
n
vol(i) le (vol())qω( f h)
We are now ready to prove Theorem 19
Proof Note that
|Dh(λ)minus D(λ)| le 1 +2
where
1 =∣∣∣∣∣∣Dh(λ)minus
sumqisinZn+1
(minus1)qλq
qintq
K[x]dx
∣∣∣∣∣∣and
2 =∣∣∣∣∣∣sum
qisinN0Zn+1
(minus1)qλq
qintq
K[x] dx
∣∣∣∣∣∣ We estimate 1 and 2 separately
We apply the above two lemmas to the function f q rarr R defined forx isin q as f (x) = K[x] and obtain the inequality∣∣∣∣∣∣
sumiisinNq
n
vol(i)K[si] minusintq
K[x]dx
∣∣∣∣∣∣ le q
Kinfin (vol()Kinfinradicq)qω(K h)
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16 A review of the Fredholm approach
Therefore we obtain that
1 lesum
qisinZn+1
|λ|qq
∣∣∣∣∣∣sumiisinNq
n
vol(i)K[si] minusintq
K[x]dx
∣∣∣∣∣∣le⎧⎨⎩ sum
qisinZn+1
|λ|qq
q
Kinfin (vol()Kinfinradicq)q
⎫⎬⎭ω(K h)
le sum
qisinN0
q
(vol()|λ| middot Kinfineradic
q
)qω(K h)
Kinfin
where we have again used inequality (119) from the proof of Lemma 18 andwe demand that q is sufficiently large so that the inequality
vol()|λ| middot Kinfineradicq
lt1
2
holds Therefore the series above converges uniformly and absolutely on anybounded set In fact if we introduce the constant
c =sumqisinN0
q
(vol()|λ| middot Kinfineradic
q
)qKinfin
we can write the above observation as 1 le cω(K h)For 2 we also have that
2 lesum
qisinN0Zn+1
|λ|qqintq|K[x]| dx
lesum
qisinN0Zn+1
|λ|qq (Kinfin
radicq)qvol(q)
lesum
qisinN0Zn+1
(vol()|λ|Kinfineradic
q
)q
lesum
qisinN0Zn+1
1
2q
Since n ge vol()2dhd we see that h rarr 0+ implies both n rarr infin and q rarr infin
so that the right-hand side of the above inequality goes to zero uniformly in λ
in any bounded subset of the complex plane These two inequalities prove thedesired result Indeed when n is sufficiently large we have
|Dh(λ)minus D(λ)| le cω(K h)+sum
qisinN0Zn+1
1
2q
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14 Resolvent kernels 17
Since ω(K h) rarr 0 as h rarr 0+ the right-hand side of the above inequalitytends to zero uniformly
14 Resolvent kernels
We now explain the central role that the Fredholm determinant plays in solvinga second-kind integral equation with a continuous kernel
To this end for each λ isin C we introduce the resolvent kernel Rλ defined fors t isin as
Rλ(s t) =sumqisinN0
(minus1)q+1
q λq+1intq
K
[s xt x
]dx
Here of course the expression K[
sxtx
]stands for the Fredholm minor
K
(s x1 xq
t x1 xq
)
Alternatively we may write the resolvent kernel in the form
Rλ(s t) = minusλ⎧⎨⎩K(s t)+
sumqisinN
(minus1)q
q λqintq
K
[s xt x
]dx
⎫⎬⎭
As in the proof of Lemma 18 we see for fixed s t isin that Rλ(s t) is an entirefunction of λ isin C and for a fixed λ isin C it is continuous in s t isin The nextobservation gives an indication of the role of the resolvent kernel in solving asecond-kind integral equation
Proposition 113 For any λ isin C we have that
Rλ minus λKRλ + λD(λ)K = 0 (120)
and
Rλ minus λRλK + λD(λ)K = 0 (121)
Proof We first prove equation (120) For x isin q s t isin we expand the
determinant K[
s xt x
]along its first row to obtain the formula
K
(s x1 xq
t x1 xq
)= K(s t)K
(x1 xq
x1 xq
)+sumjisinNq
(minus1) jK
(x1 x2 xj xj+1 xq
t x1 xjminus1 xj+1 xq
)K(s xj)
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18 A review of the Fredholm approach
Now we integrate both sides of this equation over x isin q and note that theintegral of each summand is independent of the summation index j Hence weconclude thatintq
K
[s xt x
]dx = K(s t)
intq
K[x]dxminus qint
K(s r)
(intqminus1
K
[r xt x
]dx)
dr
and this equation is also by definition valid for q = 0 We now substitute thisequation into the power series for the resolvent kernel to obtain the equation
Rλ(s t) = λ
int
K(s r)Rλ(r t)dr minus λD(λ)K(s t) (122)
Next we choose any u isin C() and multiply both sides of equation (122) byu(t) and integrate both sides of the resulting equation over t isin to obtainequation (120) The second equation (121) is proved in an analogous fashion
by expanding the minor K[
s xt x
]along its first column
The next result is concerned with the unique solvability of equation (11)
Theorem 114 If D(λ) = 0 then (I minus λK)minus1 is given by the formula
(I minus λK)minus1 = I minus D(λ)minus1Rλ
Proof We compute
(I minus λK)(I minus D(λ)minus1Rλ) = I minus D(λ)minus1[Rλ + λD(λ)K minus λKRλ]and now use equation (120) to conclude that the term inside the brackets iszero Similarly we verify that
(I minus D(λ)minus1Rλ)(I minus λK) = I
by using equation (121)
It follows immediately from Theorem 114 that if λ is not a zero of thefunction D then equation (11) has a unique solution in C() for any f isin C()We now prove the converse of Theorem 114 namely if D(λ) = 0 then IminusλKis not invertible To this end we note the following ancillary fact
Lemma 115 For any λ isin C there holds the equationint
Rλ(s s)ds = λDprime(λ) (123)
The proof of this formula follows directly from a term-by-term integrationand differentiation of the relevant power series
Corollary 116 If D has a zero of order m at some μ = 0 then there is ay isin for which Rμ(y y) has a zero of order strictly less than m
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14 Resolvent kernels 19
Proof From Lemma 115 above and our hypothesis we have thatint
partmminus1
partλmminus1 (Rλ(s s)) |λ=μds = μD(m)(μ) = 0
Theorem 117 If K is a continuous kernel such that D(λ) = 0 then IminusλK isnot invertible
Proof Choose a y isin and consider the function uy defined to be Rλ(middot y)Since D(λ) = 0 by hypothesis equation (120) implies that uy minus λKuy = 0We would be finished with the proof if there was a y isin for which uy = 0Otherwise we must argue differently
Suppose that D vanishes at μ and the multiplicity of this zero is m (note thatD(0) = 1 and so each zero of D must be of finite multiplicity) Let l be thesmallest non-negative integer such that for all x y isin the function Rλ(x y)has a zero of order l at μ Corollary 116 says that l lt m We now expandRλ(x y) near λ = μ and have that
Rλ = (λminus μ)lg + O(λminus μ)l+1
where the function g is continuous on times and never vanishes thereinSubstituting this equation into equation (122) we obtain that
(λminusμ)lg(s t)+O(λminusμ)l+1 = λ
int
K(s tprime)[(λminusμ)lg(tprime t)+O(λminusμ)l+1
]dtprime
minus λD(λ)K(s t)
Dividing both sides of the above equation by (λminus μ)l we get that
g(s t)+ O(λminus μ) = λ
int
K(s tprime)[g(tprime t)+ O(λminus μ)]dtprime minus λD(λ)
(λminus μ)lK(s t)
Letting λrarr μ we have that
g(s t) = μ
int
K(s tprime)g(tprime t)dtprime (124)
However we may rewrite equation (124) in the equivalent form
(I minus μK)g = 0
which together with the fact that g is not equal to zero establishes that I minusμKis not invertible thereby proving the theorem
We see from the above theorems the importance of the zeros of D in thestudy of second-kind integral equations Since D(0) = 1 the zeros of D are
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20 A review of the Fredholm approach
nonzero countable and only have infinity as a possible accumulation pointThe reciprocals of the zeros of D correspond to the nonzero eigenvalues ofthe operator K These observations prove many of the general statements wehave made about compact operators in the Appendix see Section A27 Weshall say more about the zeros of D later First we want to prove Fredholmrsquosremarkable formula for the determinant of a product of two operators
15 Fredholm determinants
In this section we consider the Fredholm determinant and provide a result onthe Fredholm determinant product
Definition 118 Let K isin C( times ) The linear operator I + K C() rarrC() defined for s isin and u isin C() by
((I +K)u)(s) = u(s)+int
K(s t)u(t)dt
has a Fredholm determinant which is given by
det(I +K) =sumqisinN0
1
qintq
K[x]dx
Alternatively we may express the Fredholm determinant directly in termsof the Fredholm function namely
det(I +K) = D(minus1)
Note that the Fredholm determinant of the operator K is the same as theFredholm determinant of the operator Klowast corresponding to the adjoint kernelwhich is defined for s t isin by Klowast(s t) = K(t s) That is for any continuouskernel K on the compact set we have the equation
det(I +K) = det(I +Klowast) (125)
The main result of this section is the Fredholm determinant product formula
Theorem 119 If K H isin C(times) then
det(I +H)(I +K) = det(I +H) det(I +K)
We start our discussion of this product formula by introducing a continuouskernel L associated with two prescribed continuous kernels K and H
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15 Fredholm determinants 21
Definition 120 If K and H are continuous kernels we define L to be thekernel of the operator
L = K +H+HK
Remark 121 The kernel L has the characteristic property that
I + L = (I +H)(I +K) (126)
There are some special cases of Theorem 119 that are readily proven Forexample if det(I + K) = 0 then I + K is not one to one Therefore itfollows that I + L is also not one to one Consequently we conclude thatdet(I + L) = 0 Similarly if det(I + H) = 0 then det(I + Hlowast) = 0 fromwhich it follows that I +H is not onto Hence the operator I + L is also notonto and so det(I+L) = 0 In other words in the proof of the above theoremwe may assume that the operators I + L I +H and I +K are all invertible
To proceed further we need the following differentiation formula To thisend we define for τ isin R and the continuous kernel G the one-parameterfamily of kernels given as Kτ = K + τG
Proposition 122 If K and G are continuous kernels then
d
dτdet(I +Kτ )|τ=0 = det(I +K)
int
G(s s)dsminusint
(int
Rminus1(s t)G(t s)dt
)ds
Proof For each x isin q a straightforward differentiation of the requisitedeterminant and a Laplace expansion yield the formula
d
dτ(Kτ [x]) |τ=0 =
sumlkisinNq
(minus1)l+kK
(x1 xkminus1 xk+1 xq
x1 xlminus1 xl+1 xq
)G(xk xl)
(127)
We now integrate both sides of this equation over x isin q The integral on theleft-hand side of the equation is clear As for the right-hand side we distinguishthe summands corresponding to k = l from the remaining summands All thesummands corresponding to k = l have the same integral given byint
qminus1K[x]dx middot
int
G(s s)ds (128)
There are q(qminus 1) remaining terms corresponding to the case k = l They alsohave the same integral which is independent of k and l and is computed to be
minusq(qminus 1)int2
(intqminus2
K
[s xt x
]dx)
G(t s)dsdt
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22 A review of the Fredholm approach
Therefore in summary we obtain thatintq
d
dτ(Kτ [x]) |τ=0 dx = q
intqminus1
K[x]dx middotint
G(s s)ds
minus q(qminus 1)int
[int
(intqminus2
K
[s xt x
]dx)
G(t s)
dt
]ds
(129)
Substituting equation (129) into the series expansion
det(I +Kτ ) =sumqisinN0
1
qintq
Kτ [x]dx
and rearranging terms yields the desired formula
d
dτdet(I +Kτ )|τ=0 = D(minus1)
int
G(s s)dsminusint
(int
Rminus1(s t)G(t s)dt
)ds
Remark 123 When det(I +K) = 0 we may use equation (120) to expressthe formula in Proposition 122 in the equivalent form
d
dτlog det(I +Kτ )|τ=0 =
int
W(s s)ds (130)
where W is the continuous kernel corresponding to the integral operator(I minus Dminus1(minus1)Rminus1
)G
Indeed by Proposition 122 we have that
d
dτlog det(I +Kτ )|τ=0 =
d
dτdet(I +Kτ )|τ=0
det(I +K)
=int
G(s s) dsminus Dminus1(minus1)int
(int
Rminus1(s t)G(t s)dt
)ds
=int
W(s s) ds
where
W(s s) = G(s s)minus Dminus1(minus1)int
Rminus1(s t)G(t s) dt
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15 Fredholm determinants 23
Moreover for any u isin C() s isin we have thatint
W(s t)u(t) dt =int
G(s t)minus Dminus1(minus1)
int
Rminus1(s tprime)G(tprime t) dtprime
u(t) dt
=int
G(s t)u(t) dt minus Dminus1(minus1)int
int
Rminus1(s tprime)G(tprime t)u(t) dtprime dt
= (((I minus Dminus1(minus1)Rminus1)G)u)(s)
which ensures the desired result By Theorem 114 we can rewrite this integraloperator in the equivalent simpler form (I + K)minus1G In order to make it easyto recall the association of the kernel W with this integral operator we expressequation (130) in the notationally suggestive form
d
dτlog det(I +Kτ )|τ=0 =
int
(I +K)minus1G
(s s)ds (131)
As we already pointed out in equation (125) the Fredholm determinants ofK and Klowast are the same Hence since Kτ
lowast = Klowast + τGlowast we also have that
d
dτlog det(I +Kτ )|τ=0 =
int
(I +Klowast)minus1Glowast
(s s)ds (132)
We are now ready to prove Theorem 119
Proof We consider two one-parameter perturbations Kτ = K+τK0 and Hτ =H + τH0 and set
I + Lτ = (I +Hτ ) (I +Kτ )
Note that
Lτ = L+ τL0 + o(τ 2) τ rarr 0+ (133)
where L is defined in Definition 120 and L0 is given by the formula
L0 = (I +H)K0 +H0 (I +K) (134)
Combining equations (131) (132) (133) and (134) we obtain that
d
dτlog det(I + Lτ )|τ=0 =
int
(I + L)minus1 (I +H)K0
(s s)ds
+int
(I + Llowast)minus1 (I +Klowast
)Hlowast0(s s)ds
But we also have that
(I + L)minus1 = (I +K)minus1 (I +H)minus1
and
(I + Llowast)minus1 = (I +Hlowast)minus1 (I +Klowast
)minus1
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24 A review of the Fredholm approach
Therefore we get that
d
dτlog det(I + Lτ )|τ=0
=int
(I +Hlowast)minus1Hlowast0
(s s)ds+
int
(I +K)minus1K0
(s s)ds
= d
dτlog det(I +Hτ )|τ=0 + d
dτlog det(I +Kτ )|τ=0
Let γ [0 1] rarr R be the function defined at τ isin [0 1] as
γ (τ) = log det(I + Lτ )minus logdet(I +Hτ ) det(I +Kτ )The above formula means that γ prime(0) = 0 Clearly we can derive the sameconclusion along any continuous path H(τ ) and K(τ ) as long as H(0) = Hand K(0) = K Moreover we can also show that the derivative of γ is zero atany point μ in (0 1) provided that both the integral operators I + H(μ) andI +K(μ) have inverses
Next we choose any continuously differentiable path σ [0 1] rarr [0 1]such that σ(0) = 0 σ(1) = 1 and then for any τ isin [0 1] which satisfiesdet(I + τH) = 0 and det(I + τK) = 0 we introduce the operatorsHτ = σ(τ)H and Kτ = σ(τ)K Consequently we conclude by our previousremarks that the function γ defined above must be constant in the interval[0 1] This together with the fact that γ (0) = 0 and γ (1) = log det(I + L)minuslog det(I +H)) det(I +K) completes the proof
16 Eigenvalue estimates and a trace formula
We now turn our attention to a new topic and provide an estimate of the growthof the zero of D in the complex plane We accomplish this only when =[0 1] and the kernel K has the property that there are positive constants ρ gt 0and α isin (0 1] such that for all u s t isin [0 1]
|K(u s)minus K(u t)| le ρ|sminus t|α
When this is the case we say that K is Holder continuous of order α withconstant ρ (with respect to the second variable) For Holder continuous kernelsof order α we can provide a better estimate for the Fredholm minor than thatgiven in Lemma 17
Lemma 124 If K is Holder continuous of order α with constant ρ then forany x y isin [0 1]q ∣∣∣∣K [x
y
]∣∣∣∣ le 4α(qq)12minusαρq
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16 Eigenvalue estimates and a trace formula 25
Proof First we reorder the columns of the minor appearing on the left-handside of the inequality so that the components of the vector y = [yi i isin Nn]are increasing Next for each i isin Nqminus1 we subtract the (i + 1)th columnfrom the ith column in the above minor (which does not alter its value) usethe Hadamard inequality and then the hypothesis on the kernel to obtain theinequality ∣∣∣∣K [x
y
]∣∣∣∣ le (ρradic
q)qprod
iisinNqminus1
|yi+1 minus yi|α
We now apply the arithmeticndashgeometric inequality to obtain the inequalities∣∣∣∣K [xy
]∣∣∣∣ le (ρradic
q)q
1
qminus 1
sumiisinNqminus1
(yi+1 minus yi)
(qminus1)α
le 4α(qq)12minusαρq
thereby confirming the desired bound
Remark 125 The above lemma holds if K is a Holder continuous kernel oforder α with respect to the first variable rather than the second variable
Note that for the next result we introduce the constant
μ = 2
1+ 2α
For α isin ( 12 1] we have that μ isin [ 23 1)
Theorem 126 If K(s t) is a Holder continuous kernel of order α isin ( 12 1]
with respect to either s or t in [0 1] then for every ε gt 0 there is a constanta gt 0 such that for all λ isin C
|D(λ)| le ae|λ|μ+ε
Proof To prove this theorem we first note that any polynomial appearingin the proof of this result can be neglected because any polynomial is surelydominated in the complex plane by an exponential
For every λ isin C we have that
|D(λ)| le 4αsum
misinN0
am|λ|m
where we define the constant
am = ρm(mm)12minusα
m
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26 A review of the Fredholm approach
This constant satisfies the inequality
am le(
ρe
mε
μ+ε
)m 1
mm
μ+ε (135)
Now for m large enough the expression in parentheses on the right-hand sideof inequality (135) can be made less than one Hence as we can ignorepolynomials of a fixed degree in |λ| we assume without loss of generalitythat it suffices to bound from above the power seriessum
misinN
|λ|mm
mμ+ε
Certainly the series above is bounded for |λ| le 1 and so we only have toconsider it for |λ| gt 1 In that case we break the sum above into two parts Forthe first part we sum over positive integers m lt (2|λ|)μ+ε and for the secondpart over the remaining integers For the first sum we have thatsum
mlt(2|λ|)μ+ε
|λ|mm
mμ+εle c|λ|(2|λ|)μ+ε
where
c =summisinN
mminusm
μ+ε ltinfin
Since
lim|λ|rarrinfin |λ|minusε log |λ| = 0
we conclude that there is a constant a gt 0 such that
|λ|(2|λ|)μ+ε le ae|λ|μ+2ε
For the other summands corresponding to m ge (2|λ|)μ+ε we have that
|λ|mm
mμ+εle 2minusm
and so we conclude that summge(2|λ|)μ+ε
|λ|mm
mμ+εlesummisinN
2minusm ltinfin
This theorem says under the hypothesis on the kernel K that D is an entirefunction of order less than or equal to μ
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16 Eigenvalue estimates and a trace formula 27
Corollary 127 If λn n isin N are the zeros of D in C ordered so that0 lt |λ1| le |λ2| le middot middot middot le |λn| le middot middot middot and K(s t) is a Holder continuous kernelin either s or t on [0 1] with exponent α isin ( 1
2 1] thensumnisinN|λn|minus1 ltinfin
The proof of this corollary relies on standard techniques in the study ofentire functions We briefly review some of the details
The first step is to recall Jensenrsquos formula (see for example [2] p 208) Ifρ gt 0 f is a function analytic in the disc ρ = z |z| lt ρ and continuouson the boundary with a finite number of zeros aj j isin Nm in the closed discρ then
log | f (0)| +sumjisinNm
logρ
|aj| =1
2π
int 2π
0log | f (ρeiθ )|dθ (136)
We apply Jensenrsquos formula to the function D in the following way We letν(ρ) be the number of zeros of D (counting multiplicities) in the disc ρ Recall that |λ1| le |λ2| le middot middot middot and so it follows that
n le ν(|λn|) (137)
Now according to Jensenrsquos formula applied to the function D on the disc 2ρ
we have that sumjisinNk
log2ρ
|λj| =1
2π
int 2π
0log |D(2ρeiθ )|dθ (138)
where λj j isin Nk are the zeros of D in 2ρ Since |λj| le 2ρ for j isin Nkwe have that each summand on the left-hand side of equation (138) is non-negative We neglect the terms corresponding to the zeros of D in 2ρ ρ thereby obtaining the inequality
ν(ρ) log 2 lt1
2π
int 2π
0log |D(2ρeiθ )|dθ (139)
Now choose ε gt 0 so that μ+ ε lt 1 (recall that when α isin ( 12 1] we have
that μ lt 1) Next we use the estimate for D(λ) in Theorem 126 and get that
ν(ρ) log 2 le log a+ (2ρ)μ+ε (140)
Consequently for sufficiently large m there is a positive constant c such thatfor n ge m we have |λn| ge 1 and
ν(|λn|) le c|λn|μ+ε
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28 A review of the Fredholm approach
According to the inequality (137) this inequality implies for n ge m that
n1
μ+ε le c|λn|In other words for some other positive constant b gt 0 we have thatsum
nisinN|λn|minus1 le b
sumnisinN
nminus1
μ+ε ltinfin
Theorem 128 If K(s t) is a Holder continuous kernel on [0 1] in either sor t with exponent α isin ( 1
2 1] then
D(λ) =prodnisinN
(1minus λ
λn
)
Proof According to the above corollary we see that the right-hand side of theabove equation is an entire function of λ isin C Moreover the function h definedat λ as
h(λ) = D(λ)prodnisinN
(1minus λ
λn
)is free of zeros in C Therefore by the Weierstrass factorization theorem thereis an entire function g such that for all λ isin C
D(λ) = eg(λ)prodnisinN
(1minus λ
λn
) (141)
We now show that g is a constant But since D(0) = 1 the result will followTherefore the last remaining task is to show that g is constant To this end weneed the following version of the PoissonndashJensen formula (see for example[2] p 208) Specifically if f is a function analytic in ρ with only zeros aj j isin Nm in ρ and z isin ρ then
log | f (z)| = minussumjisinNm
log
∣∣∣∣∣ ρ2 minus ajz
ρ(zminus aj)
∣∣∣∣∣+ 1
2π
int 2π
0Re
ρeiθ + z
ρeiθ minus zlog | f (ρeiθ )|dθ
(142)
We shall first differentiate both sides of this equation with respect to zappropriately and then examine the behavior of each resulting term on theright-hand side when f = D and ρ rarrinfin For the process of differentiation werecall the following lemma
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16 Eigenvalue estimates and a trace formula 29
Lemma 129 If f is analytic in a neighborhood of z then
part
partzlog | f (z)| = f prime(z)
f (z)
and
part
partzRef (z) = f prime(z)
To prove this lemma we write f (z) = u+ iv and then by a direct applicationof the CauchyndashRiemann equations the result may be verified
We now fix z choose ρ gt 2|z| and apply the derivative operator part
partz to bothsides of equation (142) for the choice f = D to get the equation
Dprime(z)D(z)
=sumjisinNn
λj
ρ2 minus λjz+sumjisinNn
1
zminus λj+ 1
π
int 2π
0ρeiθ (ρeiθ minus z)minus2 log |D(ρeiθ )|dθ
(143)
We first estimate the integral by using Corollary 127 Since ρ gt 2|z| we knowthat |ρeiθ minus z| ge ρ2 which together with Theorem 126 yields that∣∣∣∣ρeiθ (ρeiθ minus z)minus2 log |D(ρeiθ )|
∣∣∣∣ le ρ|ρeiθ minus z|minus2(log a+ |ρeiθ |μ+ε)
le 4
ρ(log a+ ρμ+ε)
and thus the absolute value of the third term on the right-hand side ofequation (143) is bounded by 8
ρ
(log a+ ρμ+ε
) which tends to zero as
ρ rarr infin For the first sum on the right-hand side of equation (143) we notethat
|ρ2 minus λjz| ge ρ2 minus ρ|z| ge ρ2
2
Therefore we see that the first sum on the right-hand side of equation (143) isbounded by 2ρminus1ν(ρ) which according to inequality (140) also goes to zeroas ρ rarrinfin Consequently equation (143) yields as ρ rarrinfin the equation
Dprime(z)D(z)
=sumjisinN
1
zminus λj (144)
However we also have that
D(z) = eg(z)prodjisinN
(1minus z
λj
) (145)
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30 A review of the Fredholm approach
Now we differentiate both sides of equation (145) to obtain that
Dprime(z)D(z)
= gprime(z)+sumjisinN
1
zminus λj (146)
Comparing equations (144) and (146) leads to the desired conclusion thatg is constant thereby completing the proof of the theorem
Our discussion in this chapter indicates that a Holder continuous kernelK(s t) on [0 1] in either s or t with exponent α isin ( 1
2 1] has eigenvalues whichare l1-summable We let μn n isin N be the nonzero eigenvalues of K so thatλn = 1μn We then have the following result
Theorem 130 If K(s t) is a Holder continuous kernel on [0 1] in either s ort with exponent α isin (12 1] thenint
K(s s)ds =sumnisinN
μn
and
det(I +K) =prodnisinN
(1+ μn)
Proof First we prove the second equality Indeed we have that
det(I +K) = D(minus1) =prodnisinN
(1+ λminus1n ) =
prodnisinN
(1+ μn)
Next we show the first equality On the one hand we know that
Dprime(z)D(z)
=sumjisinN
1
zminus λj
which in turn yields
Dprime(0) = D(0)sumjisinNminus 1
λj= minus
sumjisinN
μn
On the other hand by the power series expansion of D we have that
Dprime(λ) =sumqisinN0
(minus1)q+1 λq
qintq+1
K[x] dx
and hence it follows that
Dprime(0) = minusint
K(s s)ds
Combining the above two aspects we obtain the desired result
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17 Bibliographical remarks 31
This completes our brief discussion of the classical theory of Fredholmintegral equations We next turn our attention to further essential backgroundmaterial on integral equations and postpone until Chapter 4 the main subject ofthis book namely multiscale methods for the numerical solution of Fredholmintegral equations
17 Bibliographical remarks
Most of the material presented in this chapter is taken from the book [183] Forthe important notion of the Fredholm function and the Fredholm determinantreaders are referred to [183 253] Readers may find a discussion of thedistribution of the eigenvalues of the Fredholm integral operator in [141] Inaddition [249] is a nice reference for the topic of integral equations wherethree fundamental papers on integral equations written by three outstandingmathematicians [Ivar Fredholm (1903) David Hilbert (1904) and ErhardSchmidt (1907)] published in the first decade of the twentieth century weretranslated into English with commentaries
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2
Fredholm equations and projection theory
In this chapter we provide concepts and results useful for the study ofthe Fredholm integral equation of the second kind especially the theory ofprojection methods needed for the development of the multiscale methods
21 Fredholm integral equations
The main concern of this book is the Fredholm integral equation of the secondkind In this section we review concepts relevant to the study of this class ofintegral equations The general form for a Fredholm integral equation of thesecond kind is
uminusKu = f (21)
where the linear operator K is defined on a normed linear space with valuesin another such space the function f is given and u is a solution to bedetermined Typically these spaces consist of real or complex-valued functionson a measurable subset in the d-dimensional Euclidean space Rd Thefunction Ku is determined by a kernel K(s t) s t isin and the Fredholmintegral operator is defined by the formula
(Ku)(s) =int
K(s t)u(t)dt s isin (22)
or in short by
(Ku)(s) =int
K(s middot)u(middot) s isin
32
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21 Fredholm integral equations 33
Properties of the operator K are inherited from those of the kernel K We shallrestrict ourselves to kernels for which K is a compact operator that is it mapsbounded subsets to relatively compact ones Let us postpone the discussion ofoperators of this type until we describe our preferred notation and terminology
For d isin N let Rd be the d-dimensional Euclidean space and a domain(open set) in Rd By Cm() m isin N0 we mean the linear space of all real-valued functions defined on which are m-times continuously differentiablethere That is all derivatives up to and including all those of total order mare continuous on Therefore the space of infinitely differentiable functionson is given by Cinfin() = ⋂
misinN0Cm() We use for the closure of
and denote by Cm() the subspace of all functions together with all theirderivatives up to order m that are bounded and uniformly continuous on theclosure of The simplified notational convention C() and C() for thespaces C0() and C0() respectively will be used throughout the book Fora lattice vector α isin Nd
0 with non-negative coordinates we use the notation|α| = sumiisinNd
αi Corresponding to any vector t = [tj j isin Nd] isin Rd wedenote the |α|-derivative of a function u at t (when it exists) by
Dαu(t) = part |α|u(t)parttα1
1 middot middot middot parttαdd
When the set is compact the linear space Cm() m isin N0 is a Banach spacewith the norm
uminfin = max|Dαu(t)| t isin α isin Zdm+1 |α| le m
and for m = 0 we simply denote it by uinfinFor a Lebesgue measurable subset of Rd the linear space of all real-
valued functions defined on for which their pth powers 1 le p lt infin areintegrable is denoted by Lp() Unless stated otherwise all integrals are takenin the sense of the Lebesgue integration Likewise we use Linfin() for thelinear space of all real-valued essentially bounded (that is bounded except ona zero measure set) measurable functions Moreover Lp() is a Banach spacewith the norm
up =⎧⎨⎩(int
|u(t)|pdt
)1p
1 le p ltinfin
inf sup|u(t)| t isin E meas(E) = 0 p = infin
In the special case p = 2 L2() is a Hilbert space equipped with the innerproduct
(u v) =int
u(t)v(t)dt u v isin L2()
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34 Fredholm equations and projection theory
We remark that the integral of an integrable function f over a measurableset isin Rd with respect to the Lebesgue measure will be denoted by
int
f (t)dtor in short by
int
f when there is no independent variable indicated and noconfusion can be expected Additional details about the spaces can be found instandard texts for example [1 183 236 276]
211 Compact integral operators
Compact operators play a central role in the theory of integral equationsand hence are frequently discussed in that context We review some of theirimportant properties here which from time to time we shall refer to in thetext Readers are referred to [10 15 47 177 183 203 236] for additionaldetails on the material reviewed here and to the Appendix of this book forbasic elements of functional analysis
We use the symbol B(XY) for the normed linear space of all bounded linearoperators from a normed linear space X into a normed linear space Y withoperator norm A = supAx x isin X x le 1 When Y is a Banach spacethen so is B(XY) and in the case that X = Y we denote it simply by B(X)Convergence of a sequence of operators An n isin N sube B(XY) to an operatorA isin B(XY) relative to the operator norm is said to be uniform convergenceand will be denoted by An
uminusrarr A This notion of convergence differs from theweaker requirement that for all x isin X we have that limnrarrinfinAnx = Ax Thisis called pointwise convergence and denoted by An
sminusrarr AThe normed linear space B(XR) is called the dual space of X and is denoted
by Xlowast The dual space of a normed linear space is always a Banach space andconsists of all continuous linear functionals on X For every x isin X isin Xlowast weuse the familiar bracket notation 〈 x〉 (or 〈x 〉) = (x) and associated withany operator A isin B(XY) its adjoint operator Alowast isin B(YlowastXlowast) is defined forall x isin X isin Ylowast by
〈Ax〉 = langAlowast xrang
An operator and its adjoint have the same norm that is A = Alowast WhenX is a Hilbert space the Riesz representation theorem (see Theorem A31 inthe Appendix) can be used to identify X with its dual space In that case theadjoint of any A isin B(X) is likewise in B(X) and A is said to be self-adjointwhenever A = Alowast
Definition 21 A linear operator K from a normed linear space X to a normedlinear space Y is said to be compact if it maps each bounded set in X to arelatively compact set in Y
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21 Fredholm integral equations 35
It follows from the definition of a compact operator that A isin B(XY) iscompact if there is a sequence of compact operators An n isin N sube B(XY)which uniformly converges to it that is An
uminusrarr AWe now describe some examples of compact operators which are relevant
To do this we first recall the ArzelandashAscoli theorem which states that a setS is relatively compact in C() when is compact if and only if S isuniformly bounded and equicontinuous (for more details see Theorem A43in the Appendix) This classical result leads us to our first example
1 The Fredholm operator defined by a continuous kernel
Proposition 22 If sube Rd is a compact set and K isin C( times ) thenthe corresponding integral operator K given in (22) is a compact operator inB(C())
Proof First note by the Lebesgue dominated convergence theorem (cf [236])that K isin B(C()) since its operator norm satisfies the inequality
K le max int
|K(s t)|dt s isin
Next let S sube C() be a bounded subset Choose a constant c gt 0 such that forany v isin S we have that vinfin le c Therefore for any v isin S and s s1 s2 isin we conclude that
|(Kv)(s)| le c meas() max|K(s t)| (s t) isin and∣∣(Kv
)(s1)minus
(Kv)(s2)∣∣ le c
int
∣∣K(s1 t)minus K(s2 t)∣∣dt
le c meas() max∣∣K(s1 t)minus K(s2 t)∣∣ s1 s2 t isin
where meas() denotes the Lebesgue measure of the domain Since the ker-nel K is bounded and uniformly continuous ontimes the right-hand side of thefirst inequality is finite and that of the second inequality can be made as smallas desired provided that |s1 minus s2| is small enough Thus not only is the imageof the set S under K namely K(S) uniformly bounded and equicontinuousTherefore an appeal to the ArzelandashAscoli theorem completes the proof
2 The Fredholm operator defined by a Schmidt kernelRecall that a kernel K is called a Schmidt kernel provided that K isin L2(times)Proposition 23 If sube Rd is a measurable set and K is a Schmidt kernel thenthe integral operator K defined by (22) is a compact operator in B(L2())
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36 Fredholm equations and projection theory
Proof We first show that this linear operator K is in B(L2()) Indeed forany v isin L2() and any compact subset 0 of by the Fubini theorem(Theorem A14) and the CauchyndashSchwarz inequality (Section A13) we havethatint
0
|(Kv)(s)|ds leint0
int
|K(s t)v(t)|dsdt
le (meas(0))12(int
0
int
|K(s t)|2dsdt
)12
vL2()
Therefore again by the Fubini theorem we obtain that(Kv)(s) exists almost
everywhere for s isin 0 and is measurable hence it will be so on the entire Once again using the Fubini theorem and the CauchyndashSchwarz inequality weobtain that
KvL2() =(int
∣∣∣∣ int
K(s t)v(t)dt
∣∣∣∣2ds
)12
le(int
int
|K(s t)|2dsdt
)12(int
|v(t)|2dt
)12
(23)
= KL2(times)vL2()
which proves that K isin B(L2())We next show that K is actually a compact operator First we consider the
case that K is a degenerate kernel That is there exist K1j K2j isin L2() j isin Nn
such that the kernel K for any s t isin has the form
K(s t) =sumjisinNn
K1j(s)K2j(t)
In this case we have for v isin L2() that
Kv =sumjisinNn
K1j
int
K2j(t)v(t)dt
which implies that the range of K is finite-dimensional and so K is indeed acompact operator Next we show that for any K isin L2(times) there is alwaysa sequence of degenerate kernels Kj j isin N sube L2(times) such that
limjrarrinfinKj minus KL2(times) = 0 (24)
From this fact it will follow that K is a compact operator Indeed let Kj be thecompact operator corresponding to the kernel Kj thus Kj is compact Frominequality (23) we obtain that
Kj minusK le Kj minus KL2(times)
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21 Fredholm integral equations 37
and consequently we conclude that KjuminusrarrK Therefore by Proposition
A47 part (v) we see that K is compactNow the existence of a sequence Kj j isin N of degenerate kernels in
L2( times ) which satisfy (24) follows from Theorem A31 and CorollaryA30 Specifically we use the Fubini theorem and conclude that the onlyfunction h isin L2(times) with the property thatint
int
f (s)g(t)h(s t)dsdt = 0
for all f g isin L2() is the zero function Therefore the set of degeneratekernels forms a dense subset of L2(times)
We remark that a constructive approximation argument can also be usedto establish the existence of a sequence of degenerate kernels Kj j isin Nwhich satisfy (24) when is compact For example first we approximatethe Schmidt kernel K by a kernel in C( times ) and then approximate thecontinuous kernel uniformly on by bivariate polynomials In particularwhen K isin C(times) the degenerate kernel Kj j isin N can also be chosen inC(times) so that
limjrarrinfinKj minus KC(times) = 0
thereby giving an alternate proof of Proposition 23
212 Weakly singular integral operators
We now turn our attention to a class of weakly singular integral operatorswhich is a kind of the most important compact integral operators
Definition 24 Let be a bounded and measurable subset of Rd If thereexists a bounded and measurable function M defined on times such that fors t isin but s = t
K(s t) = M(s t) log |sminus t| (25)
or
K(s t) = M(s t)
|sminus t|σ (26)
where σ is a constant in the interval (0 d) and |s minus t| is the Euclideandistance between s t isin then the function K is called a kernel with a weaksingularity and the operator K defined by (22) is called a weakly singularintegral operator The case that the kernel K has a logarithmic singularity (25)is sometimes referred to merely by saying that σ = 0
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38 Fredholm equations and projection theory
We introduce several constants which are convenient for our discussion ofweakly singular integral operators namely
cσ () = sup int
dt
|sminus t|σ s isin
m() = sup|M(s t)| s t isin and
diam() = sup|sminus t| s t isin
Also we let Sd be the unit sphere in Rd and recall that
vol(Sd) = dπd2
(d2+ 1)
where is the gamma function In the next lemma we estimate an upper boundof cσ ()
Lemma 25 If sube Rd is a bounded and measurable set and σ isin [0 d) then
cσ () le vol(Sd)diam()dminusσ
d minus σ (27)
Proof Fix a choice of s isin We use spherical coordinates with center at sto estimate the integral in the definition of cσ () Specifically for any t isin we have that dt = rdminus1drdωd where r isin [0 diam()] and ωd is the Lebesguemeasure on the unit sphere Sd Consequently we obtain the estimate
int
dt
|sminus t|σ leint diam()
0
dt
rσ=int
Sd
(int diam()
0rdminus1minusσdr
)dωd
= vol(Sd)diam()dminusσ
d minus σ (28)
Note that from inequality (28) it follows that every weakly singular kernelK is in L1(times) and when σ isin [0 d2) K is likewise a Schmidt kernel Weconsider the general weakly singular integral operator in the next result
Proposition 26 The integral operator K defined by (22) with a weaksingular kernel (26) is in B(L2()) with the norm satisfying the inequalityK le m()cσ ()
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21 Fredholm integral equations 39
Proof We first observe by the Fubini theorem for any u isin L2() thatint
int
u2(t)
|sminus t|σ dsdt =int
u2(t)
(int
ds
|sminus t|σ)
dt le cσ ()u2L2() (29)
Therefore the function v defined for all s isin as
v(s) =int
u2(t)
|sminus t|σ dt
exists at almost every s isin and is integrable Next we point out for eachs t isin that
|K(s t)u(t)| le m()1
|sminus t|σ2
|u(t)||sminus t|σ2
le m()
2
1
|sminus t|σ +m()
2
u2(t)
|sminus t|σ
For any sisin both terms on the right-hand side of this inequality are integrablewith respect to t isin and so we conclude that |K(s t)u(t)| is finite for almostevery t isin Moreover by the CauchyndashSchwarz inequality we have that
[(Ku)(s)]2 =[int
K(s t)u(t)dt
]2
le m2()
int
dt
|sminus t|σint
u2(t)
|sminus t|σ dt
le m2()cσ ()int
u2(t)
|sminus t|σ dt
which implies that Ku isin L2() is square-integrable that is K is definedon L2() and maps L2() to L2() Moreover integrating both sides of thelast inequality with respect to s isin and employing estimate (29) yields theinequality
Ku2L2()le m2()cσ ()
2u2L2()
which completes the proof
We next establish the compactness of a weakly singular integral operator onL2()
Theorem 27 The integral operator K with a weakly singular kernel (26) isa compact operator in B(L2())
Proof For ε gt 0 let Kε and Kprimeε be the integral operators whose kernelsKε Kprimeε are defined respectively for s t isin by the equations
Kε(s t) =
K(s t) |sminus t| ge ε
0 |sminus t| lt ε
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40 Fredholm equations and projection theory
and
Kprimeε(s t) =
0 |sminus t| ge ε
K(s t) |sminus t| lt ε
These kernels were chosen to provide the decomposition
K = Kε +Kprimeε (210)
Since for s tisin |Kε(s t)| lem()εσ and is bounded it follows thatKε isin L2(times) Consequently we conclude from Proposition 23 that Kε is acompact operator in B(L2()) Moreover setting sε =t t isin |sminust|ltεfor each s isin the CauchyndashSchwarz inequality yields the inequality
|(Kprimeε(u)(s)| =∣∣∣∣∣intsε
M(s t)
|sminus t|σ u(t)dt
∣∣∣∣∣le m()
(intsε
1
|sminus t|σ dt
)12 (int
u2(t)
|sminus t|σ dt
)12
We bound the first integral on the right-hand side of this inequality by themethod of proof used for Lemma 25 and then integrate both sides of theresulting inequality over t isin to obtain the inequality
Kprimeε(u)L2() le[
m2()vol(Sd)(2ε)dminusσ
d minus σ
int
int
u2(t)
|sminus t|σ dsdt
]12
This inequality combined with (29) and Lemma 25 leads to the estimate
KprimeεuL2() lem()vol(Sd)[2εdiam()](dminusσ)2
d minus σuL2()
In other words we have proved that limεrarr0 Kprimeε = 0 and so being theuniform limit of the compact operators Kε ε gt 0 K is compact
The next result establishes a similar fact for K to be a compact operator inB(C()) under a modified hypothesis
Proposition 28 If sube Rd is a compact set of positive measure and thefunction M in (26) is in C(times) then the integral operator K with a weaklysingular kernel (26) is a compact operator in B(C())
Proof Let S be a bounded subset of C() By the ArzelandashAscoli theorem itsuffices to prove that the set K(S) is uniformly bounded and equicontinuous
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21 Fredholm integral equations 41
We choose a positive constant c which ensures for any u isin S that uinfin le cand so we have for any u isin S that
|(Ku)(s)| =∣∣∣∣int
M(s t)
|sminus t|σ u(t)dt
∣∣∣∣ le c m()cσ () s isin
Therefore using Lemma 25 we conclude that the set K(S) is uniformlybounded
Next we shall show not only that Ku isin C() but also that the set K(S) isequicontinous To this end we choose ε gt 0 points s + h s isin and obtainthe equation
(Ku)(s+ h)minus (Ku)(s) =int
[M(s+ h t)
|s+ hminus t|σ minusM(s t)
|sminus t|σ]
u(t)dt
Let B(s 2ε) be the sphere with center at s and radius 2ε and set (s) = B(s 2ε) We have that
|(Ku)(s+ h)minus (Ku)(s)| le c m()
(intB(s2ε)
dt
|s+ hminus t|σ +int
B(s2ε)
dt
|sminus t|σ)
+ cint(s)
∣∣∣∣ M(s+ h t)
|s+ hminus t|σ minusM(s t)
|sminus t|σ∣∣∣∣ dt
(211)
For every s isin it follows from the method used to prove Lemma 25 thatintB(s2ε)
dt
|sminus t|σ levol(Sd)(4ε)dminusσ
d minus σ
When |h| lt 2ε we are assured that B(s 2ε) sube B(s+ h 4ε) and consequentlywe obtain the next inequalityint
B(s2ε)
dt
|s+ hminus t|σ leint
B(s+h4ε)
dt
|s+ hminus t|σ levol(Sd)(16ε)dminusσ
d minus σ
where in the last step we again employ the method of proof for Lemma 25These two inequalities demonstrate that the first two quantities on the right-hand side of (211) do not exceed
2c m()vol(Sd)(16ε)dminusσ
d minus σ
We now estimate the third integral appearing on the right-hand side of (211)To this end we observe that on the compact set W =(s t) s tisin |sminust| ge εthe function defined as H(s t) = M(s t)|s minus t|σ is uniformly continuous
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42 Fredholm equations and projection theory
Hence there exists a δ gt 0 such that whenever (sprime tprime) (s t) isin W with |sprime minus s| leδ |tprime minus t| le δ we have that∣∣∣∣ M(sprime tprime)
|sprime minus tprime|σ minusM(s t)
|sminus t|σ∣∣∣∣ le ε
Now assume that |h| le min(ε δ) and t isin (s) Therefore we have that(s + h t) (s t) isin W and so the third term on the right-hand side of (211) isbounded by c meas()ε for any s isin and |h| le min(ε δ) Therefore not onlyis the function Ku continuous on but also the set K(S) is equicontinuousAn application of the ArzelandashAscoli theorem establishes that the set K(S) iscompact and so K is a compact operator
213 Boundary integral equations
Some important boundary value problems of partial differential equations overa prescribed domain can be reformulated as equivalent integral equationsover the boundary of the domain The resulting integral equations on theboundary are called boundary integral equations (BIEs) The superiority ofthe BIE methodology for solving boundary value problems rests on the factthat the dimension of the domain of functions appearing in the BIE will beone lower than in the original partial differential equation This means thatthe computational effort required to solve the partial differential equations canbe reduced significantly by using an efficient numerical method to solve theassociated BIE In this subsection we briefly review the BIE reformulation ofboundary value problems for the Laplace equation This material will providethe reader with some concrete integral equations that supplement the generaltheory described throughout the book
We begin with a discussion of the Green identities and integral representa-tions of harmonic functions To this end we let sube Rd be a bounded opendomain with piecewise smooth boundary part Throughout our discussion we
use the standard notation nabla =[
partpartxl
l isin Nd
]for the gradient and the Laplace
operator is defined by u = (nablanablau) Let A = [aij i j isin Nd] be a d times dsymmetric matrix with entries in C2() b = [bi i isin Nd] a vector field withcoordinates in C1() and c a scalar-valued function in C()
The proof of the lemma below is straightforward
Lemma 29 If u Rd rarr R and a Rd rarr Rd are in C1() then
nabla middot (ua) = unabla middot a+ a middot nablau
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21 Fredholm integral equations 43
For the next lemma we introduce the vector field P Rd rarr Rd
P = A(vnablauminus unablav)+ buv
the second-order elliptic partial differential operator
Mu = nabla middot Anablau+ b middot nablau+ cu
and its formal adjoint operator
Mlowastv = nabla middot Anablavminusnabla middot bv+ cv
Lemma 210 If u v Rd rarr R are in C2() and A b c as above then
vMuminus uMlowastv = nabla middot P
Proof By direct computation using Lemma 29 we have that
vMuminus uMlowastv = vnabla middot (Anablau)minus unabla middot (Anablav)+ vb middot nablau+ unabla middot (bv)
while the definition of P proves that the right-hand side of this equation equalsnabla middot P
The formula appearing in the above lemma is often referred to as the adjointidentity
Next we write the adjoint identity in an alternate form For this purpose weintroduce the notation
Pu = Anablau middot n and Qv = Anablav middot nminus vb middot n (212)
where n denotes the unit outer normal along part It follows from the Gaussformula and Lemma 210 thatint
(vMuminus uMlowastv) =intpart
(vPuminus uQv) (213)
We are interested in getting a representation for the solution u of thehomogeneous elliptic equation Mu = 0 in terms of its values on the boundaryof the domain The standard method for doing this employs the fundamentalsolution to the inhomogeneous problem corresponding to the adjoint operatorWe briefly describe this process Recall that the fundamental solution of thelinear partial differential operator M is a function U defined on times suchthat for each x isin the solution u of the equation Mu = f is given by theintegral representation
u(x) =intRd
U(x y)f (y)dy (214)
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44 Fredholm equations and projection theory
We assume that the fundamental solution of the adjoint operator Mlowast isavailable and is denoted by Glowast Therefore we are ensured that the solutionv of the adjoint equation Mlowastv = f is given for each x isin as
v(x) =intRd
Glowast(x y)f (y)dy (215)
The function Glowast leads us to the following basic result
Proposition 211 If u is the solution of the homogeneous equation
Mu = 0 (216)
then for each x isin
u(x) =intpart
[u(y)(QGlowast(x middot))(y)minus Glowast(x y)Pu(y)
]dy (217)
where P Q are defined by (212) Glowast is the fundamental solution of theoperator Mlowast
Proof It follows from (213) thatint
[v(y)(Mu)(y)minus u(y)(Mlowastv)(y)]dy =intpart
[v(y)(Pu)(y)minus u(y)(Qv)(y)]dy
We choose v = Glowast(x middot) in this formula and use the definition of Glowast to get thedesired conclusion
Note that (217) expresses u over in terms of the boundary values of uand its normal derivative on part Let us specialize this result to the Laplaceoperator Thus we choose M to be the Laplace operator and observe in thiscase that M = Mlowast = and Pu = Qu = nablau middot n Therefore we get from(213) the following theorem
Theorem 212 (Green theorem) If u v isin C2() thenint
(vuminus uv) =intpart
(vpartu
partnminus u
partv
partn
) (218)
The fundamental solution of the Laplace operator is given by the formula
G(x y) =minus 1
2π log |xminus y| d = 2minus 1
(dminus2)ωdminus1
1|xminusy|dminus2 d ge 3
(219)
where ωdminus1 denotes the surface area of the unit sphere in Rd (cf [7 177])
Definition 213 If u isin C2() satisfies the Laplace equation u = 0 on then u is called harmonic on
It follows from direct computation that u = G(middot y) is harmonic on Rd y
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21 Fredholm integral equations 45
Corollary 214 If u is harmonic on thenintpart
partu
partn= 0
Proof This result follows from Theorem 212 by choosing v = 1
In what follows we review the techniques of finding solutions to theLaplace equation under various circumstances Especially we discuss thedirect method for solving the Dirichlet and Neumann problems in boththe interior and exterior of the domain Moreover we shall also reviewboth single and double-layer representations for the solution of these problemsAll of our remarks pertain to the practically important cases of two andthree dimensions The two-dimensional case will be presented in some detailwhile the corresponding three-dimensional case will be stated without thebenefit of a detailed explanation since it follows the pattern of proof of thetwo-dimensional case We begin with a proposition which shows that harmonicfunctions have boundary integral representations
Proposition 215 If sube R2 is a bounded open domain with smoothboundary part prime = R2 and u is a harmonic function on then
u(x) = 1
2π
intpart
(u(y)
part
partnlog |xminus y| minus log |xminus y|partu(y)
partn
)dy x isin (220)
u(x) = 1
π
intpart
[u(y)
part
partnlog |xminus y| minus log |xminus y|partu(y)
partn
]dy x isin part (221)
0 =intpart
[u(y)
part
partnlog |xminus y| minus log |xminus y|partu(y)
partn
]dy x isin prime (222)
Proof The first equation (220) follows from Proposition 211 We now turnto the case that x isin part To deal with the singularity on x = y of the integrandon the right-hand side of equation (221) we choose a positive ε and denote byε the domain obtained from after removing the small disc B(x ε) = y |x minus y| le ε On this punctured domain we can use the Green identity (218)to obtain the equationint
ε
(v(y)u(y)minus u(y)v(y))dy =intpartε
(v(y)
partu(y)
partnminus u(y)
partv(y)
partn
)dy
(223)
Let v be the fundamental solution (219) of the Laplace operator Both of thefunctions u and v are harmonic functions on ε Therefore the above equation
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46 Fredholm equations and projection theory
can be written as
1
2π
intpartε
(log |xminus y|partu(y)
partnminus u(y)
part
partnlog |xminus y|
)dy = 0 (224)
To evaluate the limit as ε rarr 0 of the integral on the left-hand side of theequation above we split it into a sum of the following two integrals
I1ε = 1
2π
intε
(log |xminus y|partu(y)
partnminus u(y)
part
partnlog |xminus y|
)dy
and
I2ε = 1
2π
intpartεε
(log |xminus y|partu(y)
partnminus u(y)
part
partnlog |xminus y|
)dy
with ε = partB(x ε)⋂ It follows that
I1ε = 1
2π
intε
(log ε
partu(y)
partn+ εminus1u(y)
)dy (225)
from which we obtain that
limεrarr0
I1ε = 1
2u(x) (226)
Moreover we have that
limεrarr0
I2ε = 1
2π
intpart
(log |xminus y|partu(y)
partnminus u(y)
part
partnlog |xminus y|
)dy (227)
Combining equations (224)ndash(227) yields (221) Finally we note that whenx isin prime both functions u and v = 1
2π log |x minus middot| are harmonic functions on Thus (222) follows from the Green identity (218)
We state the corresponding result for the three-dimensional case The prooffollows the pattern of the proof for Proposition 215
Proposition 216 If sube R3 is a bounded open domain with smoothboundary part prime = R3 and u is a harmonic function on then
u(x) = minus 1
4π
intpart
[u(y)
part
partn
1
|xminus y| minus1
|xminus y|partu(y)
partn
]dy x isin (228)
u(x) = minus 1
2π
intpart
[u(y)
part
partn
1
|xminus y| minus1
|xminus y|partu(y)
partn
]dy x isin part (229)
0 =intpart
[u(y)
part
partn
1
|xminus y| minus1
|xminus y|partu(y)
partn
]dy x isin prime (230)
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21 Fredholm integral equations 47
Next we make use of these boundary value formulas for harmonic functionsto rewrite several boundary value problems as integral equations We startwith a description of methods for obtaining boundary integral equations ofthe direct type Later we turn our attention to the indirect methods of singleand double-layer potentials First we consider the following interior boundaryvalue problems
The interior Dirichlet problem Find u isin C()⋂
C2() such thatu(x) = 0 x isin u(x) = u0(x) x isin part
(231)
where u0 isin C(part) is a given boundary function
The interior Neumann problem Find u isin C()⋂
C2() such thatu(x) = 0 x isin partu(x)partn = u1(x) x isin part
(232)
where u1 isin C(part) is a given boundary function satisfyingintpart
u1(x)dx = 0
It is known that both of these problems have unique solutions (see forexample Chapter 6 of [177]) We now use (221) and (229) with the boundarycondition u = u0 and reformulate the interior Dirichlet problem for (231)when d = 2 as the BIE of the first kind
1
π
intpart
log |xminus y|ρ(y)dy = f (x) x isin part (233)
where
ρ = partu
partnand f = minusu0 + 1
π
intpart
u0(y)part
partnlog | middot minusy|dy
For the case d = 3 we choose
ρ = partu
partnand f = u0 + 1
2π
intpart
u0(y)part
partn
1
| middot minusy|dy
and obtain the reformulation of the interior Dirichlet problem for (231) withd = 3 as the BIE of the first kind
1
2π
intpart
1
|xminus y|ρ(y)dy = f (x) x isin part (234)
In a similar manner we treat the interior Neumann problem First for d = 2we use (221) and (229) with the boundary condition partu
partn = u1 and convert the
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48 Fredholm equations and projection theory
interior Neumann problem (232) to the equivalent BIE of the second kind
u(x)minus 1
π
intpart
u(y)part
partnlog |xminus y|dy = g(x) x isin part (235)
where
g = minus 1
π
intpart
u1(y) log | middot minusy|dy
For the corresponding three-dimensional case d = 3 we set
g = 1
2π
intpart
u1(y)
| middot minusy|dy
and obtain the BIE of the second kind
u(x)+ 1
2π
intpart
u(y)part
partn
1
|xminus y|dy = g(x) x isin part (236)
Let us now consider the Dirichlet and Neumann problems in the exteriordomain prime = Rd for d = 2 3 Specifically we reformulate the followingtwo problems as BIE
The exterior Dirichlet problem Find u isin C(prime)⋂
C2(prime) such thatu(x) = 0 x isin primeu(x) = u0(x) x isin part
(237)
where u0 isin C(part) is a given boundary function
The exterior Neumann problem Find u isin C(prime)⋂
C2(prime) such thatu(x) = 0 x isin primepartu(x)partnprime = u1(x) x isin part
(238)
where u1 isin C(part) is a given boundary function satisfyingintpart
u1(x)dx = 0and nprime is the outer unit normal to part(= partprime) with respect to prime
It is known (see for example [177]) that both problems have uniquesolutions under the condition that
u(x) = O(|x|minus1
) |x| rarr infin
|nablau(x)| = O(|x|minus2
) |x| rarr infin
(239)
Below we state the analog of both Proposition 215 and Proposition 216 forthe exterior domain prime We begin with the case d = 2
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21 Fredholm integral equations 49
Proposition 217 If sube R2 is a bounded open domain with smoothboundary part prime = R2 and u is a harmonic function on prime then thereholds
u(x) = 1
2π
intpart
[u(y)
part
partnprimelog |xminus y| minus log |xminus y|partu(y)
partnprime
]dy x isin prime (240)
u(x) = 1
π
intpart
[u(y)
part
partnprimelog |xminus y| minus log |xminus y|partu(y)
partnprime
]dy x isin part (241)
0 =intpart
[u(y)
part
partnprimelog |xminus y| minus log |xminus y|partu(y)
partnprime
]dy x isin (242)
Proof We only prove (240) The other two equations are similarly obtainedLet the ball BR = x |x| lt R be chosen such that sub BR and let primeR =prime⋂
BR Consequently it follows from (220) with = primeR that
u(x) = 1
2π
intpart
[u(y)
part
partnprimelog |xminus y| minus log |xminus y|partu(y)
partnprime
]dy+ IR x isin primeR
(243)
where
IR = 1
2π
intpartBR
[u(y)
part
partnlog |xminus y| minus log |xminus y|partu(y)
partn
]dy x isin primeR
and n is the outer unit normal to partBR Using the condition (239) we have thatthere exists a positive constant c such that
|IR| le 1
2π
intpartBR
[|u(y)|
∣∣∣∣ partpartnlog |xminus y|
∣∣∣∣+ |log |xminus y||∣∣∣∣partu(y)
partn
∣∣∣∣] dy
le c log R
2πRrarr 0
Note that the upper bound tends to zero as R tends to infinity Therefore thisestimate combined with (243) yields (240)
The three-dimensional version of Proposition 216 for the exterior domainis described next The proof is similar to that of Proposition 217 and so isomitted
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50 Fredholm equations and projection theory
Proposition 218 If sube R3 is a bounded open domain with smoothboundary part prime = R3 and u is a harmonic function on prime then
u(x) = minus1
4π
intpart
[u(y)
part
partnprime1
|xminus y| minus1
|xminus y|partu(y)
partnprime
]dy x isin prime (244)
u(x) = minus 1
2π
intpart
[u(y)
part
partnprime1
|xminus y| minus1
|xminus y|partu(y)
partnprime
]dy x isin part (245)
0 =intpart
[u(y)
part
partnprime1
|xminus y| minus1
|xminus y|partu(y)
partnprime
]dy x isin (246)
We now make use of (241) and (245) to rewrite the exterior Dirichletproblem (237) for d = 2 as the BIE
1
π
intpart
log |xminus y|ρ(y)dy = f (x) x isin part (247)
where
ρ = partu
partnprimeand f = minusu0 + 1
π
intpart
u0(y)part
partnprimelog | middot minusy|dy
while for d = 3 we have the equation
1
2π
intpart
1
|xminus y|ρ(y)dy = f (x) x isin part (248)
where
ρ = partu
partnprimeand f = u0 + 1
2π
intpart
u0(y)part
partnprime1
| middot minusy|dy
For the exterior Neumann problem (238) the BIE is of the second kind andit is explicitly given for d = 2 as
u(x)minus 1
π
intpart
u(y)part
partnprimelog |xminus y|dy = g(x) x isin part (249)
where
g(x) = minus 1
π
intpart
u1(y) log |xminus y|dy
The case d = 3 is covered by the following BIE
u(x)+ 1
2π
intpart
u(y)part
partnprime1
|xminus y|dy = g(x) x isin part (250)
where
g(x) = 1
2π
intpart
u1(y)
|xminus y|dy
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21 Fredholm integral equations 51
In the remaining part of this section our goal is to describe the BIE forthe Laplace equation of the indirect type First we consider representing theunknown harmonic function u as a single-layer potential
u(x) =intpart
ρ(y)G(x y)dy x isin Rd part (251)
and then later as a double-layer potential
u(x) =intpart
ρ(y)partG(x y)
partndy x isin Rd part (252)
where G is the fundamental solution of the Laplace operator and ρ isin C(part)is a function to be determined depending on the nature of the boundaryconditions We show that the single and double-layer potentials can be usedto solve interior and exterior problems for both the Dirichlet and Neumannproblems
Let us start with the single-layer method For the interior or exteriorDirichlet problem the boundary condition and the continuity of u on part leadto the demand that the function ρ satisfies the first-kind Fredholm integralequation int
part
ρ(y)G(x y)dy = u0(x) x isin part
In particular for d = 2 the solution u of the interior (resp exterior) Dirichletproblem has the single-layer representation given by the equation
u(x) = 1
2π
intpart
ρ(y) log |xminus y|dy x isin (resp prime) (253)
with ρ satisfying the requirement that
1
2π
intpart
ρ(y) log |xminus y|dy = u0(x) x isin part (254)
In the three-dimensional case we get the equation
u(x) = minus 1
4π
intpart
ρ(y)
|xminus y|dy x isin (respprime) (255)
where ρ satisfies the equation
minus 1
4π
intpart
ρ(y)
|xminus y|dy = u0(x) x isin part (256)
For the two-dimensional interior Neumann problem we consider the equa-tionintpartε
ρ(y) log |xminus y|dy =intε
ρ(y) log |xminus y|dy+intpartε
ρ(y) log |xminus y|dy
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52 Fredholm equations and projection theory
where partε is the boundary of the domain ε = B(x ε) and ε =partB(x ε)
⋂ By taking the directional derivative in the normal direction n
of both sides of this equation letting εrarr 0 using the boundary conditionpartupartn = u1 and arguments similar to that used in the proof of Proposition 215we conclude that u is represented as the single-layer potential (253) with ρ
satisfying the second-kind Fredholm integral equation
minusρ(x)2+ 1
2π
intpart
ρ(y)part
partnlog |xminus y|dy = u1(x) x isin part (257)
In a similar manner the solution u to the three-dimensional interior Neumannproblem is represented as the single-layer potential (255) with ρ satisfying thesecond-kind Fredholm integral equation
minusρ(x)2minus 1
4π
intpart
ρ(y)part
partn
1
|xminus y|dy = u1(x) x isin part (258)
For the exterior Neumann problem a similar argument leads to the resultthat the solution u in the two and three-dimensional cases is representedas (253) and (255) with ρ satisfying the second-kind Fredholm integralequations respectively
ρ(x)
2+ 1
2π
intpart
ρ(y)part
partnlog |xminus y|dy = u1(x) x isin part (259)
and
ρ(x)
2minus 1
4π
intpart
ρ(y)part
partn
1
|xminus y|dy = u1(x) x isin part (260)
We close this section with a review of the double-layer potentials forharmonic functions We start with the two-dimensional interior Dirichletproblem Suppose that u= u+ isin C() is a harmonic function in Let uminusbe the solution of the exterior Neumann problem with
partuminus(y)partn
= partu+(y)partn
y isin part
which satisfies (239) It follows from Propositions 215 and 217 that
u(x) = 1
2π
intpart
(u+(y)minus uminus(y))part
partnlog |xminus y|dy x isin
This equation can be written as a double-layer potential
u(x) = 1
2π
intpart
ρ(y)part
partnlog |xminus y|dy x isin (261)
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22 General theory of projection methods 53
with ρ = u+ minus uminus According to the proof of Proposition 215 we have forx isin part that
limxrarrx
intpart
ρ(y)part
partnlog |xminus y|dy = minusπρ(x)+
intpart
ρ(y)part
partnlog |xminus y|dy
This with the boundary condition of the Dirichlet problem concludes that ρsatisfies the second-kind Fredholm integral equation
minusρ(x)2+ 1
2π
intpart
ρ(y)part
partnlog |xminus y|dy = u0(x) x isin part (262)
Similarly the solution u to the three-dimensional interior Dirichlet problem isrepresented as the double-layer potential
u(x) = minus 1
4π
intpart
ρ(y)part
partn
1
|xminus y|dy x isin with ρ satisfying the second-kind Fredholm integral equation
ρ(x)
2minus 1
4π
intpart
ρ(y)part
partn
1
|xminus y|dy = u0(x) x isin part
For the exterior Dirichlet problem the corresponding integral equations in thetwo and three-dimensional cases are
ρ(x)
2+ 1
2π
intpart
ρ(y)part
partnlog |xminus y|dy = u0(x) x isin part
and
minusρ(x)2minus 1
4π
intpart
ρ(y)part
partn
1
|xminus y|dy = u0(x) x isin part
respectively
22 General theory of projection methods
The main concern in the present section is the general theory of projectionmethods for approximate solutions of operator equations of the form
Au = f (263)
where AisinB(XY) f isinY are given and u isin X is the solution to be determinedThe case of central importance to us takes the form A= I minusK where I is theidentity operator in B(X) and K is a compact operator in B(X) In this case(263) becomes
(I minusK) u = f (264)
a Fredholm equation of the second kind
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54 Fredholm equations and projection theory
221 Projection operators
We begin with a description of various projections an essential tool for thedevelopment of approximation schemes for (263) The following notation willbe used throughout the book For a linear operator A not defined on all of thelinear space X we denote by D(A) its domain and by N(A)=x xisinX Ax =0 its null space For the range of A we use R(A) = Ax x isin D(A)Alternatively we may sometimes write A(U) for the range of A where U isthe domain of A We start with the following definition
Definition 219 Let X be a normed linear space and V a closed linearsubspace of X A bounded linear operator P X rarr V is called a projectionfrom X onto V if for all v isin V
Pv = v (265)
Note that a projection P X rarr V necessarily has the property that V =R(P) For later use we make the following remark
Proposition 220 Let X be a normed linear space and P isin B(X) Then P isa projection on X if and only if P2 = P Moreover in this case if P = 0 thenP ge 1
Proof If P X rarr R(P) is a projection then for all x isin X we have thatP2x = P(Px) = Px Conversely if P2 = P then any v isin R(P) written asv = Px for some x isin X satisfies the equation Pv = P2x = Px = v Finally itfollows from the equation P2 = P that P2 ge P2 = P which impliesthat P ge 1 when P = 0
Now we describe the three kinds of projection which are most importantfor the practical development of approximation schemes to solve operatorequations We have in mind the well-known orthogonal and interpolationprojections and perhaps the less familiar concept of a projection defined bya generalized best approximation
1 Orthogonal projectionsWe recall the standard definition of the orthogonal projection on a Hilbertspace
Definition 221 Let X be a Hilbert space with inner product (middot middot) Two vectorsx y isin X are said to be orthogonal provided that (x y) = 0 If V is a nontrivialclosed linear subspace of X then a linear operator P from X onto V is calledthe orthogonal projection if for all x isin X y isin V it satisfies the equation
(Px y) = (x y) (266)
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22 General theory of projection methods 55
In other words the orthogonal projection onto V has the property that xminusPxis orthogonal to all y isin V The orthogonal projection satisfies P= 1 and isself-adjoint that is Plowast = P Moreover we have the following well-knownextremal characterization of the orthogonal projection
Proposition 222 If X is a Hilbert space and V a nontrivial closed linearsubspace of X then there exists an orthogonal projection P from X onto V
and for all x isin X
xminus Px = minxminus v v isin VMoreover the last equation uniquely characterizes Px isin V
Proof The existence of Px follows from the completeness of X and theparallelogram law The remaining claim follows from the definition of theorthogonal projection which gives for x isin X and v isin V that
xminus v2 = xminus Px2 + Pxminus v2
2 Interpolating projectionsWe next introduce the concept of interpolating projections
Definition 223 Let X be a Banach space and V a finite-dimensional subspaceof X A subset j j isin Nm of the dual space Xlowast is V-unisolvent if for anycj jisinNm sube R there exists a unique element v isin V satisfying for all jisinNmthe equation
j(v) = cj (267)
To emphasize the pairing between X and Xlowast the value of a linear functional isin Xlowast at x isin X will often be denoted by 〈x 〉 That is we define〈x 〉 = (x) This convenient notation is standard and the next propositionis elementary To state it we use the Kronecker symbol δij i j isin Nm that isδij = 0 except for i = j in which case δij = 1
Proposition 224 If X is a Banach space and V is an m-dimensional subspaceof X then j j isin Nm is V-unisolvent if and only if there exists a linearlyindependent set xj j isin Nm sube V which satisfies for all i j isin Nm the equation
j(xi) = δij (268)
In this case the operator P Xrarr V defined for each x isin X and j isin Nm bylangPx j
rang = langx jrang
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56 Fredholm equations and projection theory
is a projection from X onto V and is given by the formula
Px =sumjisinNm
j(x)xj (269)
According to the above result any m-dimensional subspace V of X and anyset of linear functionals j j isin Nm in X which is V-unisolvent determinethe projection (269) An important special case occurs when the Banach spaceX consists of real-valued functions on a compact set of Rd In this case ifthere is a subset of points tj j isin Nm in such that the linear functionalsj j isin Nm are defined for each x isin X by the equation
j(x) = x(tj) (270)
that is j is the point evaluation functional at tj then the operator P Xrarr V
defined by (269) is called the Lagrange interpolation If for some j isin Nmj(x) is determined not only by the value of the function x at some point of but also by derivatives of x P is called the Hermite interpolation In the casethat P isin B(C()) is a Lagrange interpolation its operator norm is given by
P = max
⎧⎨⎩sumjisinNm
|xj(t)| t isin ⎫⎬⎭ (271)
3 Generalized best approximation projectionsAs our final example we describe the generalized best approximation pro-jections which were introduced in [77] Let X be a Banach space and Xlowast itsdual space For n isin N we assume that Xn and Yn are two finite-dimensionalsubspaces of X and Xlowast respectively with the same dimension
Definition 225 For x isin X an element Pnx isin Xn is called a generalized bestapproximation to x from Xn with respect to Yn if for all isin Yn it satisfies theequation
〈xminus Pnx 〉 = 0 (272)
Similarly given isin Xlowast an element P primen isin Yn is called a generalized bestapproximation from Yn to with respect to Xn if for all x isin Xn it satisfies theequation lang
x minus P primenrang = 0
Figure 21 displays schematically the generalized best approximationprojection Pnx to x isin X from Xn with respect to Yn in a Hilbert space Inthis case equation (272) means (xminus Pnx)perpYn
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22 General theory of projection methods 57
Yn
Xn
x
Pnx
0
Figure 21 Generalized best approximation projections
For a further explanation we provide an example as follows Let I = [0 1]and X = L2(I) We subdivide the interval I into n subintervals by pointstj = jh j isin Nnminus1 and set tjminus 1
2= (jminus 1
2 )h jisinNn where h = 1n We let Xn be
the space of continuous piecewise linear polynomials with knots at tj j isin Nnminus1and Yn be the space of piecewise constant functions with knots at tjminus 1
2 j isin Nn
Clearly dimXn= dimYn = n + 1 For xisinX we define the generalized bestapproximation Pnx to x from Xn with respect to Yn by the equation
〈xminus Pnx y〉 = 0 for all y isin Yn
Let
φj(t) =⎧⎨⎩
1minus (tj minus t)h tjminus1 le t le tj1minus (t minus tj)h tj lt t le tj+10 elsewhere
j isin Zn+1
and
ψj(t) =
1 tjminus 12le t le tj+ 1
2
0 elsewherej isin Zn+1
Two groups of functions φj j isin Zn+1 and ψj j isin Zn+1 form thebases for Xn and Yn respectively Thus Pnx can be written in the formPnx = sum
jisinZn+1cjφj where the vector un = [cj j isin Zn+1] satisfies the
linear equation
Anun = fn
in which An = [langφjψirang
i j isin Zn+1] and fn = [〈xψi〉 i isin Zn+1]We now present a necessary and sufficient condition for which each x isin X
has a unique generalized best approximation from Xn with respect to Yn In
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58 Fredholm equations and projection theory
what follows we denote by Xperpn the set of all linear functionals in Xlowast whichvanish on the subspace Xn that is the annihilator of Xn in Xlowast
Proposition 226 For each x isin X the generalized best approximation Pnx tox from Xn with respect to Yn exists and is unique if and only if
Yn cap Xperpn = 0 (273)
When this is the case and Pnx is the generalized best approximation of x Pn Xrarr Xn is a projection
Proof Let x isin X be given and assume that spaces Xn and Yn have basesxj j isin Nm and j j isin Nm respectively The existence and uniqueness of
Pnx =sumiisinNm
cixi isin Xn
satisfying equation (272) means that the linear systemsumiisinNm
cilangxi j
rang = langx jrang j isin Nm (274)
has a unique solution c = [cj j isin Nm] isin Rm for any x isin X This is equivalentto the fact that the mtimes m matrix
A = [langxj irang
i j isin Nm]is nonsingular Moreover a vector b = [bj j isin Nm] isin Rm is in the null spaceof A if and only if the linear functional = sumjisinNm
bjj is in the subspace
Yn cap Xperpn This proves the first assertion For the remaining claim when Yn capXperpn = 0 we solve (274) and define for x isin X
Pnx =sumjisinNm
cjxj
Clearly Pn Xrarr Xn is a linear operator that by construction satisfies (272)Let us show that Pn is also a projection For any x isin X P2
n x isin Xn isa generalized best approximation to Pnx from Xn with respect to Yn ByDefinition 225 we conclude for all isin Yn thatlang
Pnxminus P2n x
rang= 0
This together with (272) implies for all isin Yn thatlangxminus P2
n x rang= 0
By the uniqueness of the solution to equation (274) we obtain that P2n x = Pnx
and so Pn is indeed a projection
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22 General theory of projection methods 59
We state the corresponding result for the generalized best approximation to from Yn with respect to Xn The proof is similar to that of Proposition 226where in this case the transpose of the matrix A is used
Proposition 227 For each isin Y the generalized best approximation to isin Y from Yn with respect to Xn exists and is unique if and only if
Xn cap Yperpn = 0 (275)
When this is the case and P primen is the generalized best approximation of fromYn with respect to Xn then P primen Yrarr Yn is a projection
In view of Propositions 226 and 227 we shall always assume that con-dition (273) (resp (275)) holds whenever we refer to Pn (resp P primen) as thegeneralized best approximation projection from Xn with respect to Yn (respthe generalized best approximation projection from Yn with respect to Xn) Wealso remark that condition (273) or (275) implies that dimYn = dimXn
Proposition 226 allows us to connect the concept of the generalized bestapproximation to the familiar concept of dual bases Indeed by the HahnndashBanach theorem every finite-dimensional subspace Xn of a normed linearspace X has a dual basis (see Theorem A32 in the Appendix) Specificallyif xj j isin Nm is a basis for Xn there is a subset j j isin Nm sube Xlowast suchthat for i j isin Nm j(xi) = δij According to Proposition 226 the generalizedbest approximation to x isin X from Xn with respect to Yn = spanj j isin Nmexists and is given by
Pnx =sumjisinNm
j(x)xj
Conversely if (273) holds we can find a dual basis for Xn in Yn
Proposition 228 If Yn cap Xperpn = 0 then P primen = Plowastn and Yn = PlowastnXlowast
Proof For all x isin X and isin Xlowast we have thatlangxP primen
rang = langPnxP primenrang = 〈Pnx 〉 = langxPlowastn
rang
from which the desired result follows
The next proposition gives an alternative sufficient condition to ensure thatevery xisinX has a unique generalized best approximation in Xn with respect toYn
Proposition 229 If there is a constant c gt 0 and a linear operator Tn Xn rarr Yn with TnXn = Yn such that for all x isin Xn
x2 le c 〈x Tnx〉 then (273) holds
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60 Fredholm equations and projection theory
Proof Our hypothesis implies that for any isin Yn cap Xperpn there exists x isin Xn
such that Tnx = and so
x le radicc 〈x Tnx〉12 = radicc 〈x 〉12 = 0
Therefore we obtain that x = 0 and consequently we also have that =Tnx= 0
The next issue that concerns us is the conditions which guarantee that asequence of projections Pn n isin N converges pointwise to the identityoperator in X that is Pn
sminusrarr I As we shall see later this property iscrucial for the analysis of projection methods Generally condition (273) isnot sufficient to ensure that this is the case Therefore we need to introducethe concept of a regular pair
Definition 230 A pair of sequences of subspaces Xn sube X n isin N and Yn subeY n isin N is called a regular pair if there is a positive constant c such that forall n isin N there are linear operators Tn Xn rarr Yn with TnXn = Yn satisfyingthe conditions that for all x isin Xn
(i) x le c 〈x Tnx〉12(ii) Tnx le cx
In this definition it is important to realize that the constant c appearing aboveis independent of n isin N
If X is a Hilbert space so that Xlowast can be identified by the Riesz representationtheorem (see Theorem A31 in the Appendix) with X itself and we also havefor all n isin N that Xn = Yn then conditions (i) and (ii) are satisfied withTn = I n isin N Thus in this case we have a regular pair On the contrary ifXnYn is a regular pair then from Proposition 229 we conclude that (273)holds and so Pn is well defined
For the next proposition we find it appropriate to introduce the quantity
dist(xXn) = minxminus u u isin XnProposition 231 If Xn n isin N and Yn n isin N form a regular pair andPn Xrarr Xn is the corresponding generalized best approximation projectionthen for each x isin X and n isin N
(i) Pn le c3(ii) dist(xXn) le xminus Pnx le (1+ c3)dist(xXn)
Consequently if for each x isin X limnrarrinfin dist(xXn)= 0 then limnrarrinfin Pnx= x
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22 General theory of projection methods 61
Proof For each x isin X and u isin Xn we have that
uminus Pnx2 le c2 〈uminus Pnx Tn(uminus Pnx)〉= c2 〈uminus x Tn(uminus Pnx)〉le c2uminus xTn(uminus Pnx)le c3uminus xuminus Pnx
Therefore we conclude that
uminus Pnx le c3uminus xThe choice u = 0 in the above inequality establishes (i) As for (ii) the lowerbound is obvious while the upper bound is obtained by using the inequalitythat for any u isin Xn
xminus Pnx le xminus u + uminus Pnx le (1+ c3)uminus x
222 Projection methods for operator equations
In this subsection we describe projection methods for solving operator equa-tions which include the PetrovndashGalerkin method the Galerkin method theleast-squares method and the collocation method
Let X and Y be Banach spaces A X rarr Y be a bounded linear operatorand f isin Y We wish to find an u isin X such that
Au = f
if it exists Projection methods have the common feature of specifying asequence Xn n isin N of subspaces and choosing a un isin Xn for which theresidual error
rn = Aun minus f
is ldquosmallrdquo so that un is a good approximation to the desired u How this is donedepends on the method used We shall review some of the principal strategiesfor making rn small
We begin with a description of the PetrovndashGalerkin method The idea behindthis method is to make the residual rn isinY small by choosing finite-dimensionalsubspaces Ln nisinN in Ylowast with dim Ln = dim Xn and attempting to findun isinXn so that for all isin Ln we have
〈Aun 〉 = 〈f 〉 (276)
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62 Fredholm equations and projection theory
Specifically we choose bases Xn= spanxj jisinNm and Ln= spanj jisinNmand write un in the form
un =sumjisinNm
cjxj
where the vector un = [cj j isin Nm] isin Rm must satisfy the linear equation
Anun = fn
where An = [langAxj irang
i j isin Nm] and fn = [〈f i〉 i isin Nm]For the purpose of theoretical analysis it is also useful to express un isinXn
as a solution of an operator equation This can be done by specifying anysequence Yn nisinN of subspaces for which there are generalized bestapproximation projections Pn Y rarr Yn with respect to Ln This means thatfor all isinLn not only 〈Aun minus f 〉 = 0 but also 〈Aun minus PnAun 〉 = 0 and〈f minus Pnf 〉 = 0 From these three equations we conclude that PnAunminusPnf isinYn cap Lperpn = 0 and so we obtain that
PnAun = Pn f
Therefore we conclude that equation (276) is equivalent to the operatorequation
Anun = Pn f
where An = PnA|Xn Here the symbol A|Xn stands for the operator Arestricted to the subspace Xn and so An isin B(XnYn) This means that theoperator An can be realized as a square matrix because dim Xn = dim Ln =dim Yn
We remark that the commonly used Galerkin method and also the least-squares method are special cases of the PetrovndashGalerkin method Specificallyif X=Y is a Hilbert space we identify Xlowast with X and choose Xn=Yn n isin Nthen the projection Pn above is the orthogonal projection of X onto Xn andequation (276) means that Aunminus f isinXperpn with un isinXn Alternatively the least-squares method chooses Yn = A(Xn) instead of the choice Xn=Yn specifiedfor the Galerkin method which yields the requirement Aun minus f isinA(Xn) withun isinXn This means that un satisfies the equation
f minusAun = dist (f A(Xn))
Equivalently un has the property that Alowast(Aunminus f ) isin Xperpn So in particular wesee that the least-squares method is equivalent to the Galerkin method appliedto the operator equation
AlowastAu = Alowast f
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22 General theory of projection methods 63
Our final example is the collocation method This approaches the task ofmaking the residual rn small by making it zero on some finite set of pointsSpecifically the setup requires that Y = C() where is a compact subset ofRd Choose a finite set T sube and demand that rn|T = 0 where rn|T denotesthe restriction of the function rn to the finite set T Again to solve for un isin X
we restrict our search to un isin Xn where dim Xn = card T and card T denotesthe number of distinct elements in T that is the cardinality of T In termsof a basis for Xn we have as before un = sumjisinNm
cjxj un = [cj j isin Nm]fn = [f (tj) j isin Nm]where T = tj j isin Nm and An = [(Axi)(tj) i j isin Nm]These quantities are joined by the linear system of equations
Anun = fn
An operator version of this linear system follows by choosing a subspace Yn subeC() with dim Yn = card T which admits an interpolation projection Pn Yrarr Yn corresponding to the family of linear functionals t t isin T wherethe linear functional t is defined to be the ldquodelta functionalrdquo at t that is foreach f isin C() we have that t(f ) = f (t) Therefore (Pnf )|T = 0 if and onlyif f |T = 0 and so we get that
Anun = Pnf
where An = PnA|Xn
223 Convergence and stability
In this subsection we discuss the convergence and stability of projectionmethods for operator equations The setup is as before namely X and Y arenormal linear spaces A isin B(XY) Xn n isin N and Yn n isin N aretwo sequences of finite-dimensional subspaces of X and Y respectively withdim Xn = dim Yn and Pn Yrarr Yn is a projection The projection equationfor our approximate solution un isin Xn for a solution u isin X to the equationAu = f is
Anun = Pn f (277)
where as before
An = PnA|Xn (278)
and An isin B(XnYn) Our goal is to clarify to what extent the sequenceun n isin N if it exists approximates u isin X We start with a definition
Definition 232 The projection method above is said to be convergent if thereexists an integer q isin N such that the operator An isin B(XnYn) is invertible for
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64 Fredholm equations and projection theory
n ge q and for each f isin A(X) the unique solution of (277) which we callun = Aminus1
n Pnf converges as n rarr infin to a u isin X that satisfies the operatorequation Au = f
We remark that convergence of the approximate solutions un n isin N tou as defined above does not require that the operator equation Au = f has aunique solution although this is often the case in applications
We first describe a consequence of convergence
Theorem 233 If the projection method is convergent then there is an integerq isin N and a constant c gt 0 such that for all n ge q
Aminus1n PnA le c (279)
If in addition A is onto that is A(X) = Y then the q and c above can bechosen so that for all n ge q
Aminus1n le c (280)
Proof The proof uses the uniform boundedness principle (Theorem A25in the Appendix) Specifically since the projection method converges weconclude for each u isin X that the sequence Aminus1
n PnAu n isin N converges inX as nrarrinfin and hence is norm bounded for each u isin X Thus an applicationof the uniform boundedness principle confirms (279) We note in passing thatAminus1
n PnA isin B(XXn) and by equation (278) this operator is a projection of Xonto Xn
As for (280) we argue in a similar fashion using in this case the sequenceAminus1
n Pnf n ge q where f can be chosen arbitrarily in Y because the operatorA is assumed to be onto to conclude again by the uniform boundednessprinciple that Aminus1
n Pn is bounded uniformly in n isin N Here we have thatAminus1
n Pn isin B(YXn) However since
Aminus1n = supAminus1
n y y isin Yn y = 1= supAminus1
n Pny y isin Yn y = 1le supAminus1
n Pny y isin Y y = 1 = Aminus1n Pn (281)
the claim is confirmed
Next we explore to what extent (279) and (280) are also sufficient forconvergence of a projection method To this end we introduce the notion ofdensity of the sequences of subspaces Xn n isin N in X
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22 General theory of projection methods 65
Definition 234 We say that the sequence of subspaces Xn n isin N has thedenseness property in X if for every x isin X
limnrarrinfin dist(xXn) = 0
Theorem 235 If there exists a q isin N such that the operator An isin B(XnYn)
is invertible a positive constant c such that for all n ge q Aminus1n PnA le c and
the sequence Xn n isin N has the denseness property in X then the operatorA is one to one and for each f isin R(A) the projection method converges tou isin X the unique solution to the equation Au = f
Proof The proof uses the fact that the operator Aminus1n PnA is a projection of X
onto Xn Hence we have for any v isin X the inequality
Aminus1n PnAvminus v le (1+ c)dist(vXn) (282)
and in particular for f isin R(A) we have that
un minus u le (1+ c)dist(uXn)
Although Theorem 235 shows that condition (279) nearly guaranteesconvergence it is hard to apply in practice Nevertheless further explanationof the inequality (282) will lead to some improvements Indeed for anyprojection Pn Xrarr Xn we have for any x isin X that
Pnxminus x le (1+ Pn)dist(xXn) (283)
and
dist(xXn) le Pnxminus x (284)
These inequalities lead to a useful criterion to ensure that a sequence ofprojections has the property Pn
sminusrarr I
Lemma 236 A sequence of projections Pn n isin N sube B(XXn) has theproperty Pn
sminusrarr I if and only if the sequence Pn n isin N is bounded andthe sequence of subspaces Xn n isin N has the denseness property in X
Proof This result follows directly from the uniform boundedness principleand inequalities (283) and (284)
We next comment on the denseness property For this purpose we introducethe subspace
lim supnrarrinfin
Xn =⋃misinN
⋂ngem
Xn (285)
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66 Fredholm equations and projection theory
Proposition 237 If the subspace lim supnrarrinfinXn is dense in X then thesequence of subspaces Xn n isin N has the denseness property in X
Proof If lim supnrarrinfinXn is dense in X then for any xisinX and any ε gt 0 thereexists a positive integer qisinN and a yisinX such that xminus yltε and yisin capnge q
Xn Hence for all nge q we have that dist(xXn)lt ε that is Xn nisinN hasthe denseness property in X
The next proposition presents a necessary condition for the densenessproperty
Proposition 238 If the sequence of subspaces Xn nisinN has the densenessproperty then cupnisinNXn is dense in X
Proof This result follows from the fact that for all x isin X
dist(xcupmisinNXm) le dist(xXn) for all n isin N
and the definition of the denseness property
Definition 239 We say that a sequence of subspaces Xn n isin N is nestedif for all n isin N Xn sube Xn+1
Note that when the sequence of subspaces Xn n isin N is nested it followsthat ⋃
nisinNXn = lim sup
nrarrinfinXn
Consequently Lemma 236 gives us the following fact
Proposition 240 If the sequence of subspaces Xn n isin N is nested in X
then Pnsminusrarr I if and only if the sequence Pn n isin N is bounded and the
subspace cupnisinNXn is dense in X
We may express the condition that the collection of subspaces Xn nisinNis nested in terms of the corresponding collection of projections Pn nisinNsuch that for each n isin N we have that R(Pn) = Xn Indeed Xn n isin N is anested collection of subspaces if
PnPm = Pn for m ge n (286)
We turn our attention to the adjoint projections Plowastn Xlowast rarr Xlowast To thisend we choose a basis xi i isin Nm for Xn and observe that there exists aunique collection of bounded linear functionals i i isin Nm sube Xlowast such that
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22 General theory of projection methods 67
j(xi) = δij i j isin Nm and for each x isin X we have that
Pnx =sumiisinNm
〈x i〉 xi (287)
In fact for any x isin Xn in the form x =sumiisinNmcixi where c = [ci i isin Nm] isin
Rm define bounded linear functionals j j isin Nm on Xn by
j(x) = cj
which leads to j(xi) = δij i j isin Nm and (287) We then extend the functionalsto the entire space X by the equationlang
x jrang = langPnx j
rang for all x isin X
It can easily be verified that (287) is valid for the collection of extendedfunctionals and this is unique for the requirements
From equation (287) comes the formula for the adjoint projection Plowastn namely for each isin Xlowast
Plowastn =sumjisinNm
langxj
rangj (288)
So we see that Yn =R(Plowastn )= span j j isin Nm and dimYn = dimXn Bydefinition for each x isin X and isin Xlowast we have that 〈Pnx 〉 = langxPlowastn
rang It
follows that limnrarrinfin Plowastn = for all isin Xlowast in the weak topology on Xlowastif and only if for every x isin X limnrarrinfin Pnx = x in the weak topology on X
([276] p 111)The next lemma prepares us for a result from [47] (p 14) which provides
a sufficient condition to ensure that Plowastnsminusrarr I in the norm topology on Xlowast
Recall that a normed linear space X is said to be reflexive if X may be identifiedwith its second dual
Lemma 241 If X is reflexive and for every xisinX limnrarrinfin Pnx= x in theweak topology on X then spancupnisinNR(Plowastn ) is dense in Xlowast in the normtopology
Proof We consider the subspace of Xlowast given by
W = span
⋃nisinN
R(Plowastn )
and choose any F isinWperp Since X is reflexive there is an xisinX such thatfor all isinXlowast we have that F()= 〈x 〉 Choose any misinN and observethat 0=F(Plowastm)=
langxPlowastm
rang = 〈Pmx 〉 We now let m rarr infin and useour hypothesis to conclude that 〈x 〉 = 0 Since isin Xlowast is arbitrary we
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68 Fredholm equations and projection theory
conclude that x= 0 thereby establishing that F= 0 In other words we haveconfirmed that Xperp = 0 and so W is dense in Xlowast (see Corollary A35 in theAppendix)
From this fact follows the next result
Proposition 242 If X is reflexive and for every x isin X limnrarrinfin Pnx = x inthe weak topology on X and the projection Pn satisfies (286) then Plowastn
sminusrarr Iin the norm topology on Xlowast
Proof According to equation (286) we conclude that PlowastmPlowastn =Plowastn whichimplies that the spaces Yn =R(Plowastn ) are nested for nisinN Moreover ourhypothesis ensures that for each x isin X the set Pnx nisinN is weakly boundedTherefore Corollary A39 in the Appendix implies that it is norm boundedand so by the uniform boundedness principle (see Theorem A25) the setPn n isin N is bounded But we know that Pn = Plowastn and therefore theclaim made in this proposition follows from Proposition 240 and Lemma 241above
Let us return to condition (280) which is usually more readily verifiablein applications First we comment on the relationship of (280) to (279) Ourcomment here is based on the following norm inequalities the first being
Aminus1n PnA le Aminus1
n middot Pn middot Awhich implies that (280) ensures (279) when Pn n isin N is bounded(where of course the constants in (279) and (280) will be different)Moreover when A Xrarr Y is one to one and onto then
Aminus1n Pn le Aminus1
n PnA middot Aminus1and so recalling inequality (281) we obtain the inequality
Aminus1n le Aminus1
n PnAAminus1which demonstrates that inequality (279) implies (280) at least when A is oneto one and onto We formalize this in the next proposition
Proposition 243 Let A Xrarr Y be a bounded linear operator Xn n isin Nand Yn n isin N finite-dimensional subspaces of X and Y respectively andPn Y rarr Yn a projection If Pn n isin N and Aminus1
n n isin N arebounded then so is Aminus1
n PnA n isin N If A is one to one and onto andAminus1
n PnA n isin N is bounded then Aminus1n n isin N is bounded too
Our final comments concerning projection methods demonstrate that undercertain circumstances the existence of a unique solution of the projection
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22 General theory of projection methods 69
equation (277) implies the same for the operator equation (263) In the nextlemma we provide conditions on the projection method which imply that A isone to one
Lemma 244 Let Xn n isin N and Yn n isin N be sequences of finite-dimensional subspaces of X and Y respectively and Pn Y rarr Yn Qn X rarr Xn projections If there is a q isin N and a positive constant c gt 0 suchthat for any n ge q both Pn le c and Aminus1
n le c hold and also Qnsminusrarr I
then A is one to one
Proof If u isin X satisfies Au = 0 then for n ge q we have since An =PnA|Xn that
QnuX le cAnQnuY= cPnAQnuminus PnAuYle c2AQnuminus uX
Letting nrarrinfin on both sides of this inequality we conclude that u = 0
We now present a similar result that implies the operator A is onto
Lemma 245 Let X and Y be Banach spaces with X reflexive and A isinB(XY) Let Pn Y rarr Yn be a projection such that Pn
sminusrarr I on Y
and the sequence Aminus1n n isin N is bounded If the sequence of subsets
R(Plowastn ) n isin N is nested then A is onto
Proof Choose any f isin Y and recall that un = Aminus1n Pnf Since Pn
sminusrarr I weconclude by the uniform boundedness principle (see Theorem A25) that thesequence Pn n isin N is bounded and so we obtain that un n isin Nis also bounded Moreover our hypothesis that R(Plowastn ) n isin N is nested
guarantees by the proof of Proposition 242 that Plowastnsminusrarr I on Ylowast Now since
X is reflexive we can extract a subsequence unk k isin N which convergesweakly to an element u isin X (see for example [276] p 126) We show thatAu = f To this end we first observe for any isin Ylowast that
limkrarrinfin
langAunk
rang = limkrarrinfin
langunk Alowast
rang = languAlowastrang = 〈Au 〉
that is limkrarrinfinAunk = Au weakly in Y Therefore the right-hand side of theinequality∣∣langAunk Plowastnk
rangminus 〈Au 〉∣∣ le AunkPlowastnk
minus + | langAunk minusAu rang |
goes to zero as nrarrinfin and so with the formula Anun = PnAun we obtain thatlimkrarrinfinAnk unk = Au weakly in Y Moreover by definition for all nisinN there
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70 Fredholm equations and projection theory
holds the equation Anun = Pnf and also by hypothesis limnrarrinfin Pnf = f (innorm) from which we conclude that limkrarrinfinAnk unk = f (in norm) Henceindeed we obtain the desired conclusion that Au = f
Now we turn our attention to a discussion of the numerical stability of theprojection methods The numerical stability of the approximate solution prob-lem concerns how close the approximate solution of projection equation (277)is to that of a perturbed equation of the form(
An + En)un = Pnf + gn (289)
where En isin B(XnYn) is a linear operator which affects a perturbation of An
and gn isin Yn affects a perturbation of Pnf n isin NWe begin with a formal definition of stability
Definition 246 The projection method is said to be stable if there are non-negative constants μ and ν a positive constant δ and a positive integer q suchthat for any n ge q the operator An is invertible and for any vector gn isinYn and any linear operator En isin B(XnYn) with En le δ the perturbedequation (289) always has a unique solution un isin Xn satisfying the inequality
un minus un le μEnun + νgn (290)
We next characterize the stability of the projection method
Theorem 247 If A isin B(XY) then the projection method is stable if andonly if inequality (280) holds
Proof Suppose that condition (280) is satisfied for n ge q Then for any f isinA(X) and n ge q the projection equation (277) has the unique solution un isinXn If n ge q and the perturbation satisfies the norm inequality En le c
2 thenfor any x isin Xn
(An + En)x ge c
2x (291)
Hence for any gn isin Yn and n ge q the perturbed equation (289) has a uniquesolution un isin Xn and it gives us the formula
un minus un = (An + En)minus1(gn minus Enun)
Hence with inequality (291) we get the stability estimate
un minus un le 2
c(Enun + gn)
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22 General theory of projection methods 71
Conversely suppose that the projection method is stable In this casewe choose the perturbation operator to be En= 0 Then for any f isinA(X)and gn isinYn when ngeN the projection equation (277) and its perturbedequation (289) have unique solutions un un isinXn respectively We now letvn= unminusun and observe for nge q that Anvn= gn and so the stability inequality(290) gives us the desired inequality
Aminus1n gn = vn le νgn
We now introduce an important concept in connection with the actualbehavior of approximate methods that is the condition number of a linearoperator which is used to indicate how sensitive the solution of an equationmay be to small relative changes in the input data
Definition 248 Let X and Y be Banach spaces and A Xrarr Y be a boundedlinear operator with bounded inverse Aminus1 Yrarr X The condition number ofA is defined as
cond(A) = AAminus1It is clear that the inequality cond(A) ge 1 always holds The following
proposition shows that the condition number is a suitable tool for measuringthe stability
Proposition 249 Suppose that X and Y are Banach spaces and A isin B(XY)has bounded inverse Aminus1 Let δA isin B(XY) δf isin Y be linear perturbationsof A and f and u isin X u+ δu isin X be solutions of
Au = f (292)
and
(A+ δA)(u+ δu) = f + δf (293)
respectively If δA lt 1Aminus1 and f = 0 then
δuu le
cond(A)1minus Aminus1δA
(δAA +
δff
)
Proof It follows from (292) and (293) that
(A+ δA)(u+ δu) = Au+ δf
which leads to the equation
δu = (I +Aminus1δA)minus1Aminus1(δf minus δAu) (294)
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72 Fredholm equations and projection theory
The inequality δA lt 1Aminus1 ensures the existence of the linear operator(I +Aminus1δA)minus1 and also the estimate
(I +Aminus1δA)minus1 le 1
1minus Aminus1δA
From this with (294) we conclude that
δuu le
Aminus11minus Aminus1δA
(δfu + δA
)le Aminus1A
1minus Aminus1δA(δff +
δAA
)
completing the proof
A simple fact for the condition number cond(An) = AnAminus1n of the
projection equation (277) is that if Anuminusrarr A and Aminus1
numinusrarr Aminus1
n then
limnrarrinfin cond(An) = cond(A)
We remark that if the projection method is convergent then for somepositive constant c the inequality
Aminus1n le cAminus1
holds In fact using Theorem 233 we have that
Aminus1n = supAminus1
n Pny y isin Yn y = 1le supAminus1
n Pny y isin Y y = 1le sup[Aminus1
n PnA]Aminus1y y isin Y y = 1le cAminus1
224 An abstract framework for second-kind operator equations
In this subsection we change our perspectives somewhat to a context closer tothe specific applications that we have in mind in later chapters Specificallyin this subsection our operator A will have the form I minusK where K iscompact Projection methods for this case are well developed in the literatureSpecifically it is well known that the theory of collectively compact operatorsdue to P Anselone (as presented in [6 15]) provides us with a convenientabstract setting for the analysis of many numerical schemes associated withthe Fredholm operator I minusK This theory generally requires that the sequenceof nested spaces Xn n isin N is dense in X For our purposes later it isadvantageous to improve upon this hypothesis
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22 General theory of projection methods 73
We have in mind several concrete projection methods which will beintroduced later These include discrete Galerkin PetrovndashGalerkin collocationand quadrature methods We shall describe them collectively in the followingcontext which differs somewhat from the point of view of previous sections
We begin with a Banach space X and a subspace V of it Let K isin B(XV)be a compact operator and consider the Fredholm equation of the second kind
uminusKu = f (295)
By the Fredholm alternative (see Theorem A48) this equation has a uniquesolution for all f isin V if and only if the null space of I minus K is 0 that is aslong as one is not an eigenvalue of K We always assume that this conditionholds
To set up our approximation methods for (295) unlike in the discussionsearlier we need two sequences of operators Kn n isin N sube B(XV) andPn n isin N sube B(XU) where we require that V sube U sube X As before Pn willapproximate the identity and in the present context Kn will approximate theoperator K The exact sense for which this is required shall be explained belowPostponing this issue for the moment we associate with these two sequencesof operators the approximation scheme
(I minus PnKn)un = Pnf (296)
for solving (295)For the analysis of the convergence properties of (296) we are led to
consider the existence and uniform boundedness for n isin N of the inverse ofthe operator
An = I minus PnKn
Moreover in this section we also prepare the tools to study the phenomenonof superconvergence This means that we shall approximate the solution ofequation (296) by the function
un = f +Knun (297)
We refer to un as the iterated approximation to (296) which is also called theSloan iterate and it follows directly that un satisfies the equation
(I minusKnPn)un = f (298)
Therefore we also consider in this section the existence and uniform bound-edness for n isin N of the inverse of the operators
An = I minusKnPn
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74 Fredholm equations and projection theory
Our analysis of the linear operators An and An requires several assumptionson Kn and Pn To prepare for these conditions we introduce the followingterminology
Definition 250 We say that a sequence of operators Tn n isin N sube B(XY)converges pointwise to an operator T isin B(XY) on the set S sube X providedthat for each x isin S we have limnrarrinfin Tnx minus T x = 0 Notationally weindicate this by Tn
sminusrarr T on S Similarly we say that the sequence ofoperators Tn n isin N converges to the operator T uniformly on S providedthat limnrarrinfin supTnx minus T x x isin S = 0 and indicate this by Tn
uminusrarr Ton S
Clearly when S is the unit ball in X and Tnuminusrarr T on S this means that
Tnuminusrarr T on X
Lemma 251 Let X be a Banach space and S a relatively compact subset ofX If the sequence of operators Tn n isin N sube B(X) has uniformly boundedoperator norms and Tn
sminusrarr T then Tnuminusrarr T on S
Proof Since the set S is compact it is totally bounded (Theorem A7) and sofor a given ε gt 0 there is a finite set W sube S such that for each x isin S there isa w isin W with x minus w lt ε Since W is finite by hypothesis there is a q isin N
such that for any v isin W and n ge q we have Tnv minus T v le ε In particularthis inequality holds for the choice v = w By hypothesis there is a constantc gt 0 such that for any n isin N we have Tn le c We now estimate the erroruniformly for all x isin S when n ge q
Tnxminus T x le (Tn minus T )(xminus w) + Tnwminus T wle (c+ T + 1)ε
Let us now return to our setup for approximate schemes for solvingFredholm equations We list below several conditions that we shall assumeand investigate their consequences
(H-1) The set of operators Kn n isin N sube B(XV) is collectively compactthat is for any bounded set B sube X the set cupnisinNKn(B) is relativelycompact in V
(H-2) Knsminusrarr K on U
(H-3) The set of operators Pn n isin N sube B(XU) is compact with normswhich are uniformly bounded for n isin N
(H-4) Pnsminusrarr I on V
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22 General theory of projection methods 75
As a first step we modify Proposition 17 in [6] to fit our circumstances andobtain the following fact
Lemma 252 If conditions (H-1)ndash(H-4) hold then
(i) (Pn minus I)Knuminusrarr 0 on X
(ii) (Kn minusK)PnKnuminusrarr 0 on X
(iii) (KnPn minusK)KnPnuminusrarr 0 on X
Proof (i) Let B denote the closed unit ball in X that is B = x x isin X x le1 and also set G = Knx x isin B n isin N Condition (H-1) implies that G isa relatively compact set in V while hypotheses (H-3) and (H-4) coupled withLemma 251 establish that Pn
uminusrarr I on G Consequently the inequality
(Pn minus I)Kn = sup(Pn minus I)Knx x isin B le sup(Pn minus I)x x isin G(299)
establishes (i)(ii) For any x isin V it follows from (H-4) that Pnx n isin N is a relatively
compact subset of X Therefore by Lemma 251 and the hypotheses (H-1)and (H-2) we conclude that (Kn minus K)Pn
sminusrarr 0 on V Moreover using theinequality
(Kn minusK)PnKn = sup(Kn minusK)PnKny y isin Ble sup(Kn minusK)Pnx x isin G (2100)
and specializing Lemma 251 to the choice T = 0 Tn = (Kn minusK)Pn and thechoice S = G we conclude the validity of (ii)
(iii) Hypotheses (H-1) and (H-3) guarantee that
Gprime = KnPnx x isin B n isin Nis a relatively compact subset of V Moreover from the equation
KnPn minusK = (KnPn minusKPn)+ (KPn minusK) (2101)
statement (ii) and (H-4) we obtain that KnPnminusK sminusrarr 0 on V Thus statement(iii) follows directly from equation (2101) and the relative compactness of theset Gprime
We next study the existence of the inverse operator of An and An Forthis purpose we recall a useful result about the existence and boundednessof inverse operators
Lemma 253 If X is a normed linear space with S and E in B(X) such that
Sminus1 exists as a bounded linear operator on S(X) and E lt Sminus1minus1 then
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76 Fredholm equations and projection theory
the linear operator T = S minus E has an inverse T minus1 as a bounded linearoperator on T (X) and has the property that
T minus1 le 1
Sminus1minus1 minus E
Proof For any u isin X we have that
Su = T u+ Eu
and so
Su le T u + EuThus
(Sminus1minus1 minus E)u le Su minus Eu le T ufrom which the desired result follows
We are now ready to prove the main result of this section
Theorem 254 If K isin B(XX) is a compact operator not having one asan eigenvalue and conditions (H-1)ndash(H-4) hold then there exists a positiveinteger q such that for all n ge q both (I minus PnKn)
minus1 and (I minusKnPn)minus1 are
in B(X) and have norms which are uniformly bounded Moreover if u un andun are the solutions of equations (295) (296) and (297) respectively and aconstant p gt 0 is chosen so that for all n isin N Pn le p then for all n isin N
uminus un le c(uminus Pnu + pKuminusKnu) (2102)
and
uminus un le c(K(I minus Pn)u + (K minusKn)Pnu) (2103)
Proof We first note that a straightforward computation leads to the formulas
[I + (I minusK)minus1Kn](I minus PnKn) =I minus (I minusK)minus1[(Pn minus I)Kn + (Kn minusK)PnKn]
and
[I + (I minusK)minus1KnPn](I minusKnPn) = I minus (I minusK)minus1(KnPn minusK)KnPn
By Lemma 252 there exists a q gt 0 such that for all n ge q we have
1 = (I minusK)minus1[(Pn minus I)Kn + (Kn minusK)PnKn] le 1
2
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22 General theory of projection methods 77
and
2 = (I minusK)minus1(KnPn minusK)KnPn le 1
2
It follows from Lemma 253 for all nge q that the inverse operators(I minusPnKn)
minus1 and (I minusKnPn)minus1 exist and there the inequalities
(I minus PnKn)minus1 le 1
1minus1(1+ (I minusK)minus1Kn)
and
(I minusKnPn)minus1 le 1
1minus2(1+ p(I minusK)minus1Kn)
holdSince the set of operators Kn n isin N is collectively compact it follows
that the norms Kn are uniformly bounded for n isin N and so by the aboveinequalities the norms of both (I minus PnKn)
minus1 and (I minusKnPn)minus1 are also
uniformly bounded for n isin N Therefore equations (296) and (298) haveunique solutions for every f isin X
It remains to prove the estimates (2102) and (2103) To this end we notefrom equations (295) (296) and (298) that
(I minus PnKn)un = Pn(I minusK)u
and
(I minusKnPn)un = (I minusK)u
Using these equations we obtain that
(I minus PnKn)(uminus un) = (uminus Pnu)+ Pn(KuminusKnu)
and
(I minusKnPn)(uminus un) = KuminusKnPnu = K(uminus Pnu)+ (K minusKn)Pnu
Therefore from what we have already proved we obtain the desired estimates
We remark from the estimate (2102) that the convergence rate of un to udepends only on the rate of approximations of Pn to the identity operator andKn to K Moreover it is seen from (2103) that if Kn approximates K fasterthan the convergence of Pn to the identity superconvergence of the iteratedsolution will result since the first term on the right-hand side of (2103) ismore significant than the other In fact since for each u isin X we have that
K(I minus Pn)u = K(I minus Pn)(I minus Pn)u le K(I minus Pn)(I minus Pn)u
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78 Fredholm equations and projection theory
circumstances for which limnrarrinfin K(I minus Pn) = 0 will lead to superconver-gence Examples of this phenomenon will be described in Section 41
We also remark that when X = V and Kn = K Theorem 254 leads to thefollowing well-known theorem
Theorem 255 If X is a Banach space Xn nisinN is a sequence offinite-dimensional subspaces of X K XrarrX is a compact linear operatornot having one as an eigenvalue and Pn XrarrXn is a sequence of linearprojections that converges pointwise to the identity operator I in X then thereexist an integer q and a positive constant c such that for all n ge q the equation
un minus PnKun = Pnf
has a unique solution un isin Xn and
uminus un le cuminus Pnuwhere u is the solution of equation (295) Moreover the iterated solution un
defined by (297) with Kn = K satisfies the estimate
uminus un le cK(I minus Pn)uOur final remark is that when X = V and Pn = I Theorem 254 leads to
the following theorem
Theorem 256 If X is a Banach space K isin B(XX) is a compact operatornot having one as an eigenvalue and conditions (H-1) and (H-2) hold thenthere exist an integer q and a positive constant c such that for all n ge q theequation
un minusKnun = f
has a unique solution un isin X and
uminus un le cKuminusKnuwhere u is the solution of equation (295)
23 Bibliographical remarks
The basic concepts and results of Fredholm integral equations and the projec-tion theory may be found in many well-written books on integral equationssuch as [15 110 121 177 203 253] In particular for the subjects of weaklysingular integral operators and boundary integral equations we recommendthe books [15 177 203] The functional spaces used in this book are usually
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23 Bibliographical remarks 79
covered in standard texts (for example [1 183 236 276]) Readers are referredto [15 47 177 183 203 236] for additional information on the notion of com-pact operators and weakly singular integral operators and to the Appendix ofthis book for basic elements of functional analysis Moreover readers may findadditional details on boundary integral equations in [12 15 22 121 144 150ndash153 177 203 267]
Regarding the projection methods readers can see [15 47 175ndash177 276]Especially the notion of generalized best approximation projections was origi-nally introduced in [77] For the approximate solvability of projection methodsfor operator equations Theorems 233 and 235 provide respectively necessaryand sufficient conditions which can be compared with those in [47] and [177]For the abstract framework for second-kind operator equations the theory ofcollectively compact operators [6 15] presents a convenient abstract settingfor the analysis of many numerical schemes In Section 224 we improvethe framework to fit more general circumstances For this point readers arereferred to the paper [80] More information about superconvergence of theiterated scheme may be seen in [60 246 247]
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3
Conventional numerical methods
This chapter is designed to provide readers with a background on conventionalmethods for the numerical solution of the Fredholm integral equation of thesecond kind defined on a compact domain of a Euclidean space Specificallywe discuss the degenerate kernel method the quadrature method the Galerkinmethod the collocation method and the PetrovndashGalerkin method
Let be a compact measurable domain in Rd having a piecewise smoothboundary We present in this chapter several conventional numerical methodsfor solving the Fredholm integral equation of the second kind in the form
uminusKu = f (31)
where
(Ku)(s) =int
K(s t)u(t)dt s isin
We describe the principles used in the development of the numerical methodsand their convergence analysis
31 Degenerate kernel methods
In this section we describe the degenerate kernel method for solving theFredholm integral equation of the second kind For this purpose we assumethat X is either C() or L2() with the appropriate norm middot The integraloperator K is assumed to be a compact operator from X to X
311 A general form of the degenerate kernel method
The degenerate kernel method approximates the original integral equation byreplacing its kernel with a sequence of kernels having the form
80
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31 Degenerate kernel methods 81
Kn(s t) =sumjisinNn
K1j (s)K
2j (t) s t isin (32)
where K1j K2
j isin X and may depend on n A kernel of this type is calleda degenerate kernel We require that integral operators Kn with kernels Kn
uniformly converge to the integral operator K that is Knuminusrarr K The
degenerate kernel method for solving (31) finds un isin X such that
un minusKnun = f (33)
For the unique existence and convergence of the approximate solution of thedegenerate kernel method we have the following theorem
Theorem 31 Let X be a Banach space and K isin B(X) be a compact operatornot having one as its eigenvalue If the operators Kn isin B(X) uniformlyconverge to K then there exists a positive integer q such that for all n ge q theinverse operators (I minusKn)
minus1 exist from X to X and
(I minusKn)minus1 le (I minusK)minus1
1minus (I minusK)minus1K minusKn
Moreover the error estimate
un minus u le (I minusKn)minus1(K minusKn)u
holds
Proof Note that (I minus K)minus1 exists as a bounded linear operator on X (seeTheorem A48 in the Appendix) The first result of this theorem follows fromLemma 253 with S = I minus K and E = Kn minus K For the second result wehave that
un minus u = (I minusKn)minus1f minus (I minusK)minus1f
= (I minusKn)minus1(Kn minusK)(I minusK)minus1f
= (I minusKn)minus1(Kn minusK)u
which yields the second estimate
According to the second estimate of Theorem 31 we obtain that
un minus u le (I minusKn)minus1K minusKnu
This means that the speed of convergence un minus u to zero depends on thespeed of convergence K minus Kn to zero This is determined by the choice of
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82 Conventional numerical methods
the kernels Kn and is independent of the differentiability of u It is clear thatwhen X = C() we have that
K minusKn = maxsisin
int
|K(s t)minus Kn(s t)|dt (34)
and when X = L2() we have that
K minusKn le(int
int
|K(s t)minus Kn(s t)|2dsdt
)12
(35)
We now discuss the algebraic aspects of the degenerate kernel method
Proposition 32 If un is the solution of the degenerate kernel method (33)then it can be given by
un = f +sumjisinNn
vjK1j (36)
in which [vj j isin Nn] is a solution of the linear system
vi minussumjisinNn
(K1j K2
i )vj = (f K2i ) i isin Nn (37)
where (middot middot) denotes the L2() inner product
Proof If un is the solution of equation (33) then by using (32) we have that
un(s)minussumjisinNn
K1j (s)
int
K2j (t)un(t)dt = f (s) s isin (38)
This means that the solution un can be written as (36) with
vj =int
K2j (t)un(t)dt
Multiplying (36) by K2i (s) and integrating over we find that [vj j isin Nn]
must satisfy the linear system (37)
Linear system (37) may be written in matrix form To this end we define
Kn = [(K1j K2
i ) i j isin Nn] vn = [vj j isin Nn] and fn = [(f K2i ) i isin Nn]
and let In denote the identity matrix of order n Then (37) can be rewritten as
(In minusKn)vn = fn
We next consider the invertibility of matrix In minusKn
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31 Degenerate kernel methods 83
Proposition 33 If X is a Banach space K isin B(X) is a compact operator nothaving one as its eigenvalue and the operators Kn isin B(X) uniformly convergeto K then there exists a positive integer q such that for all n ge q the coefficientmatrix In minus Kn of the linear system (37) is nonsingular
Proof It follows from Theorem 31 that there exists a positive integer q suchthat for all n ge q (I minus Kn)
minus1 exists This with Proposition 32 leads to theconclusion that (37) is solvable for any right-hand side of the form b = [bi i isin Nn] = [(f K2
i ) i isin Nn] with f isin X We proceed with this proof in twocases
Case 1 K2i i isin Nn is a linearly independent set of functions To prove that
the coefficient matrix of (37) is nonsingular it is sufficient to prove that (37)with any right-hand side b isin Rn is solvable or equivalently for any b isin Rn
there exists a function f isin X such that [(f K2i ) i isin Nn] = b To do this we
let f =sumjisinNncjK2
j and consider the equationsumjisinNn
(K2j K2
i )cj = bi i isin Nn (39)
The coefficient matrix [(K2j K2
i ) i j isin Nn] is a Gram matrix so it is positive
semi-definite Since K2i i isin Nn is linear independent this matrix is positive
definite This means that (39) is solvable and the function f exists indeedThus the coefficient matrix In minusKn of (37) is nonsingular
Case 2 K2i i isin Nn is a dependent set of functions In this case there is a
nonsingular matrix Qn such that
[K21 middot middot middot K2
n ]QTn = [K2
1 middot middot middot K2r 0 middot middot middot 0]
where K2i i isin Nr 0 lt r lt n is a linearly independent set of functions Let
[K11 middot middot middot K1
n ] = [K11 middot middot middot K1
n ]Qminus1n
and
Kr = [(K1j K2
i ) i j isin Nr]We then have that
Qn(In minusKn)Qminus1n =
[Ir minus Kr lowast
0 Inminusr
]
Noting that Ir minus Kr is the coefficient matrix associated with the degeneratekernel Kr(s t) = sum
jisinNrK1j(s)K2j(t) and K2i i isin Nr is a linearly
independent set of functions we conclude from case 1 that Ir minus Kr isnonsingular and thus In minusKn is nonsingular
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84 Conventional numerical methods
When the hypothesis of the above proposition is satisfied we solve (37) for[vj j isin Nn] and obtain from (36) the solution un of the degenerate kernelmethod (33)
312 Degenerate kernel approximations via interpolation
A natural way to construct degenerate kernel approximations is by interpo-lation We may employ polynomials piecewise polynomials trigonometricpolynomials and others as a basis to construct an interpolation for the kernelfunction
We now consider the Lagrange interpolation Let tj j isin Nn be a finitesubset of the domain and Lj j isin Nn be the Lagrange basis functionssatisfying
Lj(ti) = δij i j isin Nn
The kernel K can be approximated by the kernel Kn interpolating K withrespect to s or t That is we have
Kn(s t) =sumjisinNn
Lj(s)K(tj t)
or
Kn(s t) =sumjisinNn
K(s tj)Lj(t)
Using the former the linear system (37) becomes
vi minussumjisinNn
vj
int
Lj(t)K(ti t)dt =int
f (t)K(ti t)dt i isin Nn (310)
and the solution is given by
un = f +sumjisinNn
vjLj (311)
while using the latter the linear system (37) becomes
vi minussumjisinNn
vj
int
K(s tj)Li(s)ds =int
f (s)Li(s)ds i isin Nn (312)
and the solution is given by
un = f +sumjisinNn
vjK(middot tj) (313)
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31 Degenerate kernel methods 85
As an example of the Lagrange interpolation we consider continuouspiecewise linear polynomials that is linear splines on = [a b] Lettj = a + jh with h = (b minus a)n j isin Nn n isin N The basis functions arechosen as
Lj(t) =
1minus |t minus tj|h t isin [tjminus1 tj+1]0 otherwise
We obtain the degenerate kernel approximation by interpolating K with respectto s Specifically for t isin we have that
Kn(s t) = [(tj minus s)K(tjminus1 t)+ (sminus tjminus1)K(tj t)]h s isin [tjminus1 tj] j isin Nn
For this example we have the following error estimate
Proposition 34 If K(middot t) isin C2() for any t isin and part2Kparts2 isin C(times) then
K minusKn le 1
8h2(bminus a)
∥∥∥∥part2K
parts2
∥∥∥∥infin
Proof It can easily be derived by using the Taylor formula that
|K(s t)minus Kn(s t)| le 1
8h2∥∥∥∥part2K(middot t)
parts2
∥∥∥∥infin
for any s isin [tjminus1 tj] t isin This with (34) leads to the desired result of thisproposition
313 Degenerate kernel approximations via expansion
An alternative way to construct degenerate kernel approximations is byexpansion such as Taylor expansions and Fourier expansions of the kernelK We now introduce the latter
Let X = L2() with inner product (middot middot) which can be defined with respectto a weight function Let Fj j isin N be a complete orthonormal sequence inX Then for any x isin X we have the Fourier expansions of x with respect toFj j isin N
x =sumjisinN
(x Fj)Fj
This can be used for construction of approximate degenerate kernels of K withrespect to either variable For example we may define
Kn(s t) =sumjisinNn
Fj(s)(K(middot t) Fj(middot))
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86 Conventional numerical methods
Let Gj(t) = (K(middot t) Fj(middot)) Then the linear system (37) becomes
vi minussumjisinNn
vj(Fj Gi) = (f (t) Gi) i isin Nn (314)
and the solution is given by
un = f +sumjisinNn
vjFj (315)
Proposition 35 If K isin L2(times) and Fj j isin N is a complete orthonormalset in L2() then
K minusKn le⎛⎝ sum
jisinNNn
∥∥(K(middot bull) Fj(middot))∥∥2
⎞⎠12
Proof Note that
K(s t)minus Kn(s t) =sum
jisinNNn
Fj(s)(K(middot t) Fj(middot))
By employing the orthonormal property of the sequence Fj j isin N it followsfrom (35) that
K minusKn le(int
int
|K(s t)minus Kn(s t)|2dsdt
)12
=⎛⎝ sum
jisinNNn
∥∥(K(middot bull) Fj(middot))∥∥2
⎞⎠12
32 Quadrature methods
In this section we introduce the quadrature or Nystrom method for solving theFredholm integral equations of the second kind This method discretizes theintegral equation by directly replacing the integral appearing in the integralequation by numerical quadratures
321 Numerical quadratures
We begin by introducing numerical integrations Let isin Rd be a compact setand assume that g isin C() To approximate the integral
Q(g) =int
g(t)dt
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32 Quadrature methods 87
we consider numerical quadrature rules of the form
Qn(g) =sumjisinNn
wnjg(tnj)
where tnj isin j isin Nn are quadrature notes and wnj j isin Nn are real quadratureweights
The following are some examples of quadrature rules
Example 36 Consider the trapezoidal quadrature rule
Qn(g) = h
[1
2g(t0)+ g(t1)+ middot middot middot + g(tnminus1)+ 1
2g(tn)
]
on = [a b] where h = (bminus a)n tj = a+ jh j isin Nn When g isin C2()the error of the trapezoidal rule has the estimate ([172] p 481)
|Q(g)minus Qn(g)| le bminus a
12h2gprimeprimeinfin
Example 37 Consider the Simpson quadrature rule
Qn(g) = h
3[g(t0)+ 4g(t1)+ 2g(t2)+ middot middot middot + 2g(tnminus2)+ 4g(tnminus1)+ g(tn)]
on = [a b] where h = (bminus a)n tj = a+ jh j isin Nn and n is even Wheng isin C4() the error of the Simpson rule has the estimate ([172] p 483)
|Q(g)minus Qn(g)| le bminus a
180h4g(4)infin
Example 38 Let ψj j isin Zn+1 be a family of orthogonal polynomials ofdegree le n on = [a b] with respect to a non-negative weight function ρand tj j isin Nn be zeros of the function ψn Consider the Gaussian quadraturerule
Qn(g) =sumjisinNn
wjg(tj) (316)
for the integral Q(g) where
wj =int
ρ(t)Lnj(t)dt
and Lnj j isin Nn are the Lagrange interpolation polynomials of degree n minus 1which have the form
Lnj(t) = iisinNni =j(t minus ti)iisinNni =j(tj minus ti) j isin Nn
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88 Conventional numerical methods
When g isin C2n() the error of the Gaussian quadrature rule has the estimate([172] p 497)
Q(g)minus Qn(g) = g2n(η)
(2n)int
ρ(t)iisinNn(t minus ti)2dt
for some η isin When = [minus1 1] ρ = 1 and ψj j isin Zn+1 is chosen to be the set of
Legendre polynomials
ψj(t) = 1
2jjdj
dtj[(t2 minus 1)j] j isin Zn+1
formula (316) is called the GaussndashLegendre quadrature formula in which
wj = 2
n
1
ψnminus1(tj)ψ primen(tj)
This example shows that the Nystrom method using the Gaussian quadratureformula has rapid convergence We now turn to considering convergence of asequence of general quadrature rules
Definition 39 A sequence Qn n isin N of quadrature rules is calledconvergent if the sequence Qn n isin N converges pointwise to the functionalQ on C()
The next result characterizes convergent quadrature rules
Proposition 310 A sequence Qn n isin N of quadrature rules with theweights wnj j isin Nn converges if and only if
supnisinN
sumjisinNn
|wnj| ltinfin
and
Qn(g)rarr Q(g) nrarrinfin
for all g in a dense subset U sub C()
Proof It can easily be verified that
Qninfin =sumjisinNn
|wnj|
Thus the result of this proposition follows directly from the BanachndashSteinhaustheorem (Corollary A27 in the Appendix)
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32 Quadrature methods 89
322 The Nystrom method for continuous kernels
Using a sequence Qn n isin N of numerical quadrature rules the integraloperator
(Ku)(s) =int
K(s t)u(t)dt s isin
with a continuous kernel K isin C( times ) is approximated by a sequence ofsummation operators
(Knu)(s) =sumjisinNn
wjK(s tj)u(tj) s isin
Accordingly the integral equation (31) is approximated by a sequence ofdiscrete equations
un minusKnun = f (317)
or
un(s)minussumjisinNn
wjK(s tj)un(tj) = f (s) s isin (318)
We specify equation (318) at the quadrature points ti i isin Nn and obtain thelinear system
un(ti)minussumjisinNn
wjK(ti tj)un(tj) = f (ti) i isin Nn (319)
where the unknown is the vector [un(tj) j isin Nn] We summarize the abovediscussion in the following proposition
Proposition 311 For the solution un of (318) let unj = un(tj) j isin NnThen [unj j isin Nn] satisfies the linear system
uni minussumjisinNn
wjK(ti tj)unj = f (ti) i isin Nn (320)
Conversely if [unj j isin Nn] is a solution of (320) then the function un definedby
un(s) = f (s)+sumjisinNn
wjK(s tj)unj s isin (321)
solves equation (318)
Proof The first statement is trivial Next if [unj j isin Nn] is a solution of(320) then we have from (321) and (320) that
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90 Conventional numerical methods
un(ti) = f (ti)+sumjisinNn
wjK(ti tj)unj = uni
From (321) and the above equation we find that un satisfies (318)
Formula (321) can be viewed as an interpolation formula which extendsthe numerical solution of linear system (319) to all points s isin and is calledthe Nystrom interpolation formula
We now consider error analysis of the Nystrom method Unlike the degener-ate kernel method we do not expect uniform convergence of the sequenceKn n isin N of approximate operators to the integral operator K in theNystrom method In fact KnminusK ge K To see this for any small positiveconstant ε we can choose a function φε isin C() such that φεinfin = 1φε(tj) = 0 for all j isin Nn and φε(s) = 1 for all s isin with minjisinNn |sminus tj| ge εFor this choice of φε we have that
Kn minusK = sup(Kn minusK)vinfin v isin C() vinfin le 1ge sup(Kn minusK)(vφε)infin v isin C() vinfin le 1 ε gt 0= supK(vφε)infin v isin C() vinfin le 1 ε gt 0= supKvinfin v isin C() vinfin le 1 = K
Although the sequence Kn n isin N is not uniformly convergent it ispointwise convergent Therefore by using the theory of collectively compactoperator approximation we can obtain the error estimate for the Nystrommethod
Theorem 312 If the sequence of quadrature rules is convergent then thesequence Kn n isin N of quadrature operators is collectively compactand pointwise convergent on C() Moreover if u and un are the solutionsof equations (31) and the Nystrom method respectively then there exist apositive constant c and a positive integer q such that for all n ge q
un minus uinfin le c(Kn minusK)uinfin
Proof It follows from Proposition 310 that
C = supnisinN
sumjisinNn
|wnj| ltinfin
which leads to
Knvinfin le C maxstisin |K(s t)| vinfin (322)
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32 Quadrature methods 91
and
|(Knv)(s1)minus (Knv)(s2)| le C maxtisin |K(s1 t)minus K(s2 t)| vinfin s1 s2 isin
(323)
Noting that K is uniformly continuous on times we conclude that for anybounded set B sub C() Knv v isin B n isin N is bounded and equicontinuousThus by the ArzelandashAscoli theorem the sequence Kn n isin N is collectivelycompact
Since the sequence of quadrature rules is convergent for any v isin C()
(Knv)(s)rarr (Kv)(s) as nrarrinfin for all s isin (324)
From (323) we see that Knv n isin N is equicontinuous which with (324)leads to the conclusion that Knv n isin N is uniformly convergent that isKn n isin N is pointwise convergent on C()
The last statement of the theorem follows from Theorem 256
It follows from equations (31) and (317) that
(I minusKn)(un minus u) = (Kn minusK)u
which yields
(Kn minusK)uinfin le I minusKn un minus uinfin
This with the estimate of Theorem 312 means that the error un minus uinfinconverges to zero in the same order as the numerical integration error
(Kn minusK)uinfin = maxsisin
∣∣∣∣∣∣sumjisinNn
wjK(s tj)u(tj)minusint
K(s t)u(t)dt
∣∣∣∣∣∣
323 The Nystrom method for weakly singular kernels
In this subsection we describe the Nystrom method for the numerical solutionof integral equation (31) with weakly singular operators defined by
(Ku)(s) =int
K1(s t)K2(s t)u(t)dt
where K1 is a weakly singular kernel and K2 is a smooth kernel We considerthe important case
K1(s t) = log |sminus t|or
K1(s t) = 1
|sminus t|σ
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92 Conventional numerical methods
for some σ isin (0 d) The former is often regarded as a special case of the latterwith σ = 0
We consider a sequence Qn n isin N of numerical quadrature rules
(Qng)(s) =sumjisinNn
wnj(s)g(tnj) s isin (325)
for the integral
(Qg)(s) =int
EK1(s t)g(t)dt s isin
where the quadrature weights depend on the function K1 and the variable sThen the integral operator K is approximated by a sequence of approximateoperators defined by
(Knu)(s) = Qn(K2(s middot)u(middot))(s) s isin in terms of the quadrature rules Qn Specifically we have that
(Knu)(s) =sumjisinNn
wj(s)K2(s tj)u(tj)
where we use simplified notations wj = wnj and tj = tnj The integralequation (31) is then approximated by a sequence of linear equations
uni minussumjisinNn
wj(ti)K2(ti tj)unj = f (ti) i isin Nn (326)
and the approximate solution un is defined by
un(s) = f (s)+sumjisinNn
wj(s)K2(s tj)unj s isin (327)
Example 313 Suppose that
(Ku)(s) =int
log |sminus t|K2(s t)u(t)dt s isin = [a b]where K2 is a smooth function Let h = (bminus a)n tj = a+ jh j isin Zn+1 Fora fixed s isin we choose a piecewise linear interpolation for K2(s middot)u(middot) thatis K2(s middot)u(middot) is approximated by
[(tj minus t)K2(s tjminus1)u(tjminus1)+ (t minus tjminus1)K2(s tj)u(tj)]h
for t isin [tjminus1 tj] j isin Nn By defining the weight functions
w0(s) = 1
h
int[t0t1]
(t1 minus t) log |sminus t|dt
wj(s) = 1
h
int[tjminus1tj]
(t minus tjminus1) log |sminus t|dt + 1
h
int[tjtj+1]
(tj minus t)) log |sminus t|dt j isin Nnminus1
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32 Quadrature methods 93
and
wn(s) = 1
h
int[tnminus1tn]
(t minus tnminus1) log |sminus t|dt
we obtain the approximate operators
(Knu)(s) =sum
jisinZn+1
wj(s)K2(s tj)u(tj) s isin
The error analysis for the Nystrom method for weakly singular kernelscan be obtained in a way similar to Theorem 312 for continuous kernelsNoting that in this case the quadrature weights depend on s we need to makeappropriate modifications in Theorem 312 to fit the current case We firstmodify Proposition 310 to the following result
Proposition 314 The sequence Qn n isin N of quadrature rules defined asin (325) converges uniformly on if and only if
supsisin
supnisinN
sumjisinNn
|wnj(s)| ltinfin
and
Qn(g)rarr Q(g) nrarrinfin
uniformly on for all g in some dense subset U sub C()
With this result we describe the main result for the Nystrom method in thecase of weakly singular kernels
Theorem 315 If the sequence of quadrature rules is uniformly convergentand
limtrarrs
supnisinN
sumjisinNn
|wnj(t)minus wnj(s)| = 0 (328)
then the sequence Kn n isin N of corresponding approximate operators iscollectively compact and pointwise convergent on C() Moreover if u andun are the solutions of (equations (31)) and the Nystrom method respectivelythen there exist a positive constant c and a positive integer q such that for alln ge q
un minus uinfin le c(Kn minusK)uinfin
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94 Conventional numerical methods
Proof The proof is similar to that of Theorem 312 We only need to replaceinequality (323) by the following inequality
|(Knu)(s1)minus (Knu)(s2)| le∣∣∣sum
jisinNn
wn j(s1)[K2(s1 tj)minus K2(s2 tj)]u(tj)∣∣∣
+∣∣∣sum
jisinNn
[wnj(s1)minus wn j(s2)]K2(s2 tj)]u(tj)∣∣∣
le C maxtisin |K2(s1 t)minus K2(s2 t)|uinfin+ sup
nisinN
sumjisinNn
|wnj(s1)minus wnj(s2)|maxstisin |K2(s t)|uinfin
Then by a similar proof of Theorem 312 one can conclude the desired results
33 Galerkin methods
We have discussed projection methods for solving operator equations inSection 13 In this section and what follows we specialize the operatorequations to Fredholm integral equations of the second kind and consider threemajor projection methods namely the Galerkin method PetrovndashGalerkinmethod and collocation method for solving the equations Specifically wepresent in this section the Galerkin method the iterated Galerkin method andthe discrete Galerkin method for solving Fredholm integral equations of thesecond kind
As described in Section 132 for the operator equation in a Hilbert space theprojection method via orthogonal projections mapped from the Hilbert spaceonto finite-dimensional subspaces leads to the Galerkin method
Let X = L2() Xn n isin N be a sequence of subspaces of X satisfyingcupnisinNXn = X and Pn n isin N be a sequence of orthogonal projections fromX onto Xn The Galerkin method for solving (31) is to find un isin Xn such that
(I minus PnK)un = Pnf (329)
or equivalently
(un v)minus (Kun v) = (f v) for all v isin Xn
Under the hypotheses that s(n) = dimXn and φj j isin Ns(n) is a basis forXn the solution un of equation (329) can be written in the form
un =sum
jisinNs(n)
ujφj
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33 Galerkin methods 95
where the vector un = [uj j isin Ns(n)] satisfies the linear systemsumjisinNs(n)
uj[(φjφi)minus (Kφjφi)
] = (f φi) i isin Ns(n)
Setting
En = [(φjφi) i j isin Ns(n)]
Kn = [(Kφjφi) i j isin Ns(n)]
and
fn = [(f φj) j isin Ns(n)]
equation (329) can be written in the matrix form
(En minusKn)un = fn (330)
We call Kn the Galerkin matrixNote that Pn Xrarr Xn n isin N are orthogonal projections and cupnisinNXn = XPn = 1 and Pn converges pointwise to the identity operator I in X Accord-ing to Theorem 255 if K X rarr X is a compact linear operator not havingone as an eigenvalue then there exist an integer q and a positive constant csuch that for all n ge q equation (329) has a unique solution un isin X and
uminus un le cuminus Pnuwhere u is the solution of equation (31)
331 The Galerkin method with piecewise polynomials
Piecewise polynomial bases are often used in the Galerkin method for solvingequation (31) due to its simplicity flexibility and excellent approximationproperty In this subsection we present the standard piecewise polynomialGalerkin method for solving the Fredholm integral equation (31) of the secondkind
We begin with a description of piecewise polynomial subspaces of L2()We assume that there is a partition i i isin Zn of for n isin N which satisfiesthe following conditions
bull =⋃iisinZni and meas(j capjprime) = 0 j = jprime
bull For each i isin Zn there exists an invertible affine map φi rarr such thatφi(
0) = i where 0 is a reference element
For i isin i i isin Zn we define the parameters
hi = diam(i) ρi = the diameter of the largest circle inscribed in i
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96 Conventional numerical methods
We also assume that the partition is regular in the sense that there exists apositive constant c such that for all i isin Zn and for all n isin N
hi
ρi
le c
Let hn = maxdiam(i) i isin Zn For a positive integer k we denote byXn the space of the piecewise polynomials of total degree le k minus 1 withrespect to the partition i i isin Zn In other words every element in Xn isa polynomial of total degree le k minus 1 on each i Since the dimension of the
space of polynomials of total degree k minus 1 is given by m =(
k + d minus 1d
)
we conclude that the dimension of Xkn is mn
We next construct a basis for the space Xn We choose a collection τj j isinNm sub 0 in a general position that is the Lagrange interpolation polynomialof the total degree k minus 1 at these points is uniquely defined For j isin Nm weassume that a collection pj j isin Nm of polynomials of total degree k minus 1 ischosen to satisfy the equation
pj(τi) = δij i j isin Nm
For each i isin Zn the functions defined by
ρij(t) =(pj φminus1
i )(t) t isin i
0 t isin i(331)
form a basis for the space Xn where ldquordquo denotes the functional compositionWe let Pn L2() rarr Xn be the orthogonal projection onto Xn Then Pn isself-adjoint and Pn = 1 for all n Moreover for each x isin Hk() (cf SectionA1 in the Appendix) there exist a positive constant c and a positive integer qsuch that for all n ge q
xminus Pnx le chknxHk (332)
Associated with the projection Pn the piecewise polynomial Galerkinmethod for solving equation (31) is described as finding un isin Xn such that
(I minus PnK)un = Pnf (333)
In terms of the basis functions described in (331) the Galerkin equation (333)is equivalent to the linear system
(En minusKn)un = fn (334)
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33 Galerkin methods 97
where un isin Rmn
En = [Eijiprimejprime i iprime isin Zn j jprime isin Nm] with Eijiprimejprime = (ρiprimejprime ρij)
Kn = [Kijiprimejprime i iprime isin Zn j jprime isin Nm] with Kijiprimejprime = (Kρiprimejprime ρij)
and
fn = [fij i isin Zn j isin Nm] with fij = (f ρij)
The following result is concerned with the convergence order of the Galerkinmethod (333)
Theorem 316 Suppose that K L2()rarr L2() is a compact operator nothaving one as its eigenvalue If u isin L2() is the solution of equation (31)then there exist a positive constant c and a positive integer q such that for eachn ge q equation (333) has a unique solution un isin Xn Moreover if u isin Hk()
then un satisfies the error bound
uminus un le chknuHk
Proof By the hypothesis that one is not an eigenvalue of the compact operatorK the operator I minusK is one to one and onto Since the spaces Xn are dense inL2() we have that for x isin L2()
limhrarr0Pnxminus Ix = 0
This ensures that there exists a positive integer q such that for each n ge qequation (333) has a unique solution un isin Xn and the inverse operators (I minusPnK)minus1 are uniformly bounded
It follows from equation (333) that
uminus un = uminus Pnf minus PnKun (335)
Moreover applying the projection Pn to both sides of equation (31) yields
Pnuminus PnKu = Pnf
Substituting this equation into (335) leads to the equation
uminus un = uminus Pnu+ PnK(uminus un)
Solving for uminus un from the above equation gives
uminus un = (I minus PnK)minus1(uminus Pnu)
Therefore the desired estimate follows from the above equation the uniformboundedness of (I minus PnK)minus1 and estimate (332)
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98 Conventional numerical methods
The last theorem shows that the Galerkin method has convergence ofoptimal order That is the order of convergence is equal to the order ofapproximation from the piecewise polynomial space
We finally remark that if is the boundary of a domain the piecewisepolynomial Galerkin method is called a boundary element method
332 The Galerkin method with trigonometric polynomials
We now consider equation (31) with = [0 2π ] and the functions K andf being 2π -periodic that is K(s + 2π t) = K(s t + 2π) = K(s t) andf (s+2π) = f (s) for s t isin In this case trigonometric polynomials are oftenused as approximations for solving the equations and projection methods ofthis type are referred to as spectral methods
Let X = L2(0 2π) be the space of all complex-valued 2π -periodic andsquare integral Lebesgue measurable functions on with inner product
(x y) =int
x(t)y(t)dt
For each n isin N let Xn be the subspace of X of all trigonometric polynomialsof degree le n That is we set φj(s) = eijs s isin R
Xn = spanφj(s) j isin Zminusnnwhere i = radicminus1 and Zminusnn = minusn 0 n The orthogonal projectionof X onto Xn is given by
Pnx = 1
2π
sumjisinZminusnn
(xφj)φj
For x isin X it is well known that
xminus Pnx =⎛⎝ 1
2π
sumjisinZZminusnn
|(xφj)|2⎞⎠12
rarr 0 as nrarrinfin
To present the error analysis we introduce Sobolev spaces of 2π -periodicfunctions which are subspaces of L2(0 2π) and require for their elements acertain decay of their Fourier coefficients That is for r isin [0infin)
Hr(0 2π) =⎧⎨⎩x isin L2(0 2π)
sumjisinZ
(1+ j2)r|(xφj)|2 ltinfin⎫⎬⎭
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33 Galerkin methods 99
Note that H0(0 2π) coincides with L2(0 2π) and Hr(0 2π) is a Hilbert spacewith inner product given by
(x y)r = 1
2π
sumjisinZ
(1+ j2)r(xφj)(yφj)
and norm given by
xHr =⎛⎝ 1
2π
sumjisinZ
(1+ j2)r|(xφj)|2⎞⎠12
We remark that when r is an integer this norm is equivalent to the norm
xr =⎛⎝ sum
jisinZr+1
x(j)2⎞⎠12
It can easily be seen that for x isin Hr(0 2π)
xminus Pnx le 1
nr
⎛⎝ 1
2π
sumjisinZZminusnn
(1+ j2)r|(xφj)|2⎞⎠12
le 1
nrxHr (336)
With the help of the above estimate we have the following error analysis
Theorem 317 Suppose that K L2(0 2π) rarr L2(0 2π) is a compactoperator not having one as its eigenvalue If u isin L2(0 2π) is the solutionof equation (31) then there exist an integer N0 and a positive constant c suchthat for each n ge N0 the Galerkin approximate equation (333) has a uniquesolution un isin Xn Moreover if u isin Hr(0 2π) then un satisfies the error bound
uminus un le cnminusruHr
Proof The proof of this theorem is similar to that of Theorem 316 with (332)being replaced by (336)
333 The condition number for the Galerkin method
We discuss in this subsection the condition number of the linear systemassociated with the Galerkin equation which depends on the choice of basesof the approximate subspace To present the results we need the notion of thematrix norm induced by a vector norm For an ntimesn matrix A the matrix normis defined by
Ap = supAxp x isin Rn xp = 1 1 le p le infin
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100 Conventional numerical methods
It is well known that for A = [aij i j isin Zn]Ainfin = max
iisinZn
sumjisinZn
|aij|
A1 = maxjisinZn
sumiisinZn
|aij|
and
A2 = [ρ(ATA)]12
where ρ(A) is called the spectral radius of A and is defined as the largesteigenvalue of A
To discuss the condition number of the coefficient matrix of the linearsystem (330) for the Galerkin method we first provide a lemma on a changeof bases for the subspace Xn Let φj j isin Ns(n) and ψj j isin Ns(n) be twobases for the subspace Xn with the latter being orthonormal These bases havethe relations
n = Cnn and n = Dnn
where n = [ψj j isin Ns(n)] n = [φj j isin Ns(n)] Dn = [(φiψj) i j isin Ns(n)] and Cn = [cij i j isin Ns(n)] is the matrix determined by the firstequation
Lemma 318 If ψj j isin Ns(n) is orthonormal then
CnDn = In and DnDTn = En
where In is the identity matrix and En = [(φjφi) i j isin Ns(n)] Moreover
Dn2 = DTn 2 = En12
2 and Dminus1n 2 = DminusT
n 2 = Eminus1n 12
2
Proof The first part of this lemma follows directly from computation For thesecond part we have that
Dn2 = DTn 2 = [ρ(DnDT
n )]12 = [ρ(En)]12 = En122
The second equation can be proved similarly
The next lemma can easily be verified
Lemma 319 For any w = [wj j isin Ns(n)] isin Rs(n) let w = wTn isin XnIf the operator Qn Rs(n) rarr Xn is defined by Qnw = w then Qn is invertibleand Qn = Qminus1
n = 1
In the next theorem we estimate the condition number of the coefficientmatrix of the Galerkin method
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33 Galerkin methods 101
Theorem 320 The condition number of the coefficient matrix of the linearsystem (330) for the Galerkin method has the bound
cond(En minusKn) le cond(En)cond(I minus PnK)
Proof For any g = [gj j isin Ns(n)] let
v = (En minusKn)minus1g
It can be verified that g = gTn and v = vTn satisfy the equation
g = (I minus PnK)v
Noting that v = (DTn v)Tn = Qn(DT
n v) and g = (DTn g)Tn = Qn(DT
n g) wehave that
(En minusKn)minus1g = v = DminusT
n Qminus1n v = DminusT
n Qminus1n (I minus PnK)minus1Qn(DT
n g)
Thus using Lemmas 318 and 319 we conclude that
(En minusKn)minus1 le DminusT
n Qminus1n (I minus PnK)minus1QnDT
n le cond(En)
12(I minus PnK)minus1 (337)
Likewise we conclude from
(En minusKn)v = g = DminusTn Qminus1
n g = DminusTn Qminus1
n (I minus PnK)Qn(DTn v)
that
En minusKn le cond(En)12(I minus PnK) (338)
Combining estimates (337) and (338) yields the desired result of this theorem
We remark that if K is a compact linear operator not having one as itseigenvalue and Pn
sminusrarr I then PnKuminusrarr K and (IminusPnK)minus1 uminusrarr (IminusK)minus1
which yields
cond(I minus PnK)rarr cond(I minusK) as nrarrinfin
334 The iterated Galerkin method
We presented the iterated projection scheme (297) in Section 224 Accordingto Theorem 255 if K X rarr X is a compact linear operator not having oneas an eigenvalue then there exist a positive constant c and a positive integer q
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102 Conventional numerical methods
such that for all n ge q the iterated solution defined by (297) with Kn = Ksatisfies the estimate
uminus un le cK(I minus Pn)uSince (I minus Pn)
2 = I minus Pn we have that
K(I minus Pn)u le K(I minus Pn)(I minus Pn)uSince K is compact its adjoint Klowast is also compact As a result we obtain that
K(I minus Pn) = [K(I minus Pn)]lowast = (I minus Pn)Klowast rarr 0 as nrarrinfin
Thus we conclude that
uminus un le c(I minus Pn)Klowast(I minus Pn)uand see that u minus un converges to zero more rapidly than u minus un doesMoreover we have that
K(I minus Pn)u = (K(s middot) (I minus Pn)u(middot)) = ((I minus Pn)K(s middot) (I minus Pn)u(middot))which leads to
uminus un le c ess sup(I minus Pn)K(s middot) s isin (I minus Pn)uThis shows that the additional order of convergence gained from iterationis attributed to approximation of the integral kernel from the approximatesubspace
335 Discrete Galerkin methods
The implementation of the Galerkin method (329) requires evaluating theintegrals involved in (330) There are two types of integral required forevaluation the integral that defines the operator K and the inner product (middot middot)of the space L2() These integrals usually cannot be evaluated exactly andthus they require numerical integration The Galerkin method with integralscomputed using numerical quadrature is called the discrete Galerkin method
We choose quadrature nodes τj j isin Nqn with qn ge s(n) and discreteoperators Kn defined by
(Knx)(t) =sum
jisinNqn
wj(t)x(τj) t isin
to approximate the operator K We require that the sequence Kn n isin N ofdiscrete operators is collectively compact and pointwise convergent on C()
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33 Galerkin methods 103
For sufficient conditions to ensure collective compactness see Theorem 315Suppose that we use a quadrature formula to approximate integrals that isint
x(t)dt asympsum
jisinNqn
λjx(τj) with λj gt 0 j isin Nqn
and define a discrete semi-definite inner product
(x y)n =sum
jisinNqn
λjx(τj)y(τj) x y isin C()
and the corresponding discrete semi-norm
xn = (x x)12n x isin C()
We require that the rank of the matrix n = [φi(τj) i isin Ns(n) j isin Nqn ] isequal to s(n) It follows that there is a subset of quadrature nodes say τj j isinNs(n) such that the matrix [φi(τj) i j isin Ns(n)] is nonsingular Thus for anydata bj j isin Ns(n) there exists a unique φ isin Xn satisfying
φ(τj) = bj j isin Ns(n)
We see that middot n is a semi-norm on C() and is a norm on XnThe discrete Galerkin method for solving (31) is to find un isin Xn such that
(un v)n minus (Knun v)n = (f v)n for all v isin Xn (339)
The analysis of the discrete Galerkin method requires the notation of thediscrete orthogonal projection
Definition 321 Let Xn be a subspace of C() The operator Pn C() rarrXn defined for x isin C() by
(Pnx y)n = (x y)n for all y isin Xn
is called the discrete orthogonal projection from C() onto Xn
Proposition 322 If λj gt 0 j isin Nqn and rank n = s(n) then the discreteorthogonal projection Pn C() rarr Xn is well defined and is a linearprojection from C() onto Xn If in addition qn = s(n) then Pn is aninterpolating projection satisfying for x isin C()
(Pnx)(τj) = x(τj) j isin Ns(n)
Proof For x isin C() let Pnx =sumjisinNs(n)xjφj and consider the linear systemsum
jisinNs(n)
xj(φjφi)n = (xφi)n i isin Ns(n) (340)
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104 Conventional numerical methods
The coefficient matrix Gn = [(φjφi)n i j isin Ns(n)] is a Gram matrix Thusfor any x = [xj j isin Ns(n)] isin Rs(n) we have that
xTGnx =∥∥∥ sum
jisinNs(n)
xjφj
∥∥∥2
nge 0
Since λj gt 0 for all j isin Nqn xTGnx = 0 if and only ifsum
jisinNs(n)xjφj(τi) =
0 i isin Nqn The latter is equivalent to x = 0 since rank n = s(n) Thus weconclude that the matrix Gn is positive definite and equation (340) is uniquelysolvable that is Pn is well defined From Definition 321 we can easily verifythat Pn C()rarr Xn is a linear projection
When qn = s(n) we define for x isin C() Inx = [x(τj) j isin Ns(n)] isin Rs(n)We next show that
In(Pnx) = Inx
We first have that
In(Pnx) = In
( sumjisinNs(n)
xjφj
)= T
n x
It follows from equation (340) that
Gnx = nnInx
where n is the diagonal matrix diag(λ1 λs(n)) The definition of thenotation Gn leads to Gn = nn
Tn Thus combining the above equations
we obtain that
In(Pnx) = Tn Gminus1
n nnInx = Inx
which completes the proof
The discrete orthogonal projection Pn is self-adjoint on C() with respectto the discrete inner product that is
(Pnx y)n = (xPny)n x y isin C()
and is bounded on C() with respect to the discrete semi-norm that is
Pnxn le xn x isin C()
We remark that the latter does not mean the uniform boundedness of theoperators Pn n isin N However if Xn are piecewise polynomial spaces andquadrature nodes are obtained by using affine mappings from a set of quadra-ture nodes in a reference element then we have the uniform boundednesssupPninfin n isin N ltinfin (see Section 354)
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34 Collocation methods 105
The following proposition provides convergence of the discrete orthogonalprojection
Proposition 323 If the sequence of discrete orthogonal projections Pn n isin N is uniformly bounded on C() then there is a positive constant c suchthat for all x isin C()
xminus Pnxinfin le c infxminus vinfin v isin XnProof Since Pn is a linear projection from C() onto Xn for any v isin Xn
xminus Pnx = xminus vminus Pn(xminus v)
This yields the estimate
xminus Pnxinfin le (1+ supPninfin n isin N)xminus vinfin
Thus the desired result follows from the above estimate and the hypothesisthat Pn is uniformly bounded
With the help of discrete orthogonal projections the discrete Galerkinmethod (339) for solving (31) can be rewritten in the form of (296) that is
(I minus PnKn)un = Pn f
Thus the error analysis for this method follows from the same framework asTheorem 254
34 Collocation methods
We consider in this section the collocation method for solving the Fredholmintegral equation of the second kind According to the description in Sec-tion 221 for the operator equation in the space C() the projection methodvia interpolation projections into finite-dimensional subspaces leads to thecollocation method
Let Xn n isin N be a sequence of subspaces of C() with s(n) = dimXnand let Pn n isin N be a sequence of interpolation projections from C()onto Xn defined for x isin C() by
(Pnx)(tj) = x(tj) for all j isin Ns(n)
where tj j isin Ns(n) is a set of distinct nodes in The collocation methodfor solving (31) is to find un isin Xn such that
(I minus PnK)un = Pnf (341)
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106 Conventional numerical methods
or equivalently
un(ti)minusint
K(ti t)un(t)dt = f (ti) for all i isin Ns(n) (342)
Suppose that φj j isin Ns(n) is a basis for Xn Let un =sumjisinNs(n)ujφj Equation
(342) can be written in the matrix form
(En minusKn)un = fn (343)
where
En = [φj(ti) i j isin Ns(n)]
Kn = [(Kφj)(ti) i j isin Ns(n)]
un = [uj j isin Ns(n)]
and fn = [f (tj) j isin Ns(n)]
Kn is called the collocation matrixWe remark that the set of collocation points tj j isin Ns(n) should be chosen
such that the subspace Xn is unisolvent that is the interpolating function isuniquely determined by its values at the interpolating points It is clear that thisrequirement is equivalent to the condition det(En) = 0 Since the collocationmethod is interpreted as a projection method with the interpolating operatorthe general convergence results for projection methods are applicable
When the Lagrange basis Lj j isin Ns(n) for Xn is used then
un(t) =sum
jisinNs(n)
ujLj(t) with uj = un(tj)
and the linear system (343) becomes
ui minussum
jisinNs(n)
uj
int
K(ti t)Lj(t)dt = f (ti) for all i isin Ns(n)
Note that the coefficient matrix is the same as that for the degenerate kernelmethod (310) In other words the operator PnK is a degenerate kernel integraloperator
PnKu(t) =int
Kn(s t)u(t)dt with Kn(s t) =sum
jisinNs(n)
K(tj t)Lj(s)
We then have the estimate
K minus PnK = maxsisin
int
|K(s t)minus Kn(s t)|dt (344)
We observe that the computational cost of the collocation method is muchlower than that of the Galerkin method since it reduces the calculation ofintegrations involved
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34 Collocation methods 107
341 The collocation method with piecewise polynomials
In this subsection we consider the collocation method with the subspace Xn
being a piecewise polynomial space Suppose that there is a regular partitioni i isin Nn of for n isin N which satisfies
=⋃iisinNn
i and meas(j capjprime) = 0 j = jprime
and for each i isin Nn there exists an invertible affine map φi which maps areference element 0 onto i For a positive integer k let Xn be the space ofpiecewise polynomials of total degree le k minus 1 with respect to the partitioni i isin Nn We choose m distinct points τj isin 0 j isin Nm such thatthe Lagrange interpolation polynomial of total degree k minus 1 at these points isuniquely defined Then we can find pj isin Pk polynomials of total degree k minus 1such that
pj(τi) = δij i j isin Nm
For each n isin Nn the functions defined by
ρij(t) =(pj φminus1
i )(t) t isin i
0 t isin i(345)
form a basis for the space Xn and the points tij = φi(τj) i j isin Nm form a setof collocation nodes satisfying
ρij(tiprimejprime) = δiiprimeδjjprime
We let Pn C()rarr Xn be the interpolating projection onto Xn We then have
Pnxinfin le maxPnxinfini i isin NnNoting that
Pnx(t) =sumjisinNm
x(tij)ρij(t) =sumjisinNm
x(tij)pj(τ ) t isin i τ = φminus1i (t) isin 0
we conclude that
Pn lesumjisinNm
pjinfin
which means that the sequence of projections Pn is uniformly boundedMoreover for each x isin Wkinfin() (cf Section A1 in the Appendix) thereexists a positive constant c and a positive integer q such that for all n ge q
xminus Pnxinfin le c infxminus vinfin v isin Xn le chknxWkinfin (346)
where hn = maxdiam(i) i isin Zn We have the following theorem
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108 Conventional numerical methods
Theorem 324 Suppose that K C() rarr C() is a compact operator nothaving one as its eigenvalue If u isin C() is the solution of equation (31)then there exist a positive constant c and a positive integer q such that foreach n ge q equation (341) has a unique solution un isin Xn Moreover ifu isin Wkinfin() then there exists a positive constant c such that for all n
uminus uninfin le chknuWkinfin()
342 The collocation method with trigonometric polynomials
We consider in this subsection equation (31) in which = [0 2π ] and K andf are 2π -periodic and describe the collocation method for solving the equationusing trigonometric polynomials
Let X = Cp(0 2π) be the space of all 2π -periodic continuous functions onR with uniform norm middot infin and choose the approximate subspace as
Xn = span1 cos t sin t cos nt sin nt n isin N
To define an interpolating projection from X onto Xn we recall the Dirichletkernel
Dn(t) = sin(n+ 12 )t
2 sin t2
= 1
2+sumjisinNn
cos jt
and observe that for tj = jπn+ 1
2 j isin Z2n+1
Dn(tj) = 1
2 + n j = 00 j isin Z2n+1 0
This means that functions j defined by
j(t) = 2
2n+ 1Dn(t minus tj) j isin Z2n+1
satisfy
j(ti) = δij
and form a Lagrange basis for Xn We then define the interpolating projectionPn Xrarr Xn for x isin X by
Pnx =sum
jisinZ2n+1
x(tj)j
The following estimate is known (cf [279])
Pn = O(log n) (347)
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34 Collocation methods 109
Hence from the principle of uniform boundedness there exists x isin X forwhich Pnx does not converge to x The bound (347) leads to the fact that forx isin X
Pnxminus xinfin le (1+ Pn) infxminus vinfin v isin Xnle O(log n) infxminus vinfin v isin Xn
We next consider the estimate of KminusPnK Assume that the kernel satisfiesthe α-Holder continuity condition
|K(s1 t)minus K(s2 t)| le c|s1 minus s2|α for all s1 s2 t isin
for some positive constant c Then using (344) we conclude that
K minus PnK le cnminusα log n
343 The condition number for the collocation method
We now turn our attention to consideration of the condition number cond(En minus Kn) of the coefficient matrix of the linear system (343) obtained fromthe collocation method In this case the condition number is defined in termsof the infinity norm of a matrix A specifically
cond(A) = AinfinAminus1infin
Theorem 325 If det(En) = 0 then the condition number of the linear system(343) of the collocation method satisfies
cond(En minusKn) le Pn2infincond(En) cond(I minus PnK) (348)
Proof For g = [gj j isin Ns(n)] let
v = [vj j isin Ns(n)] = (En minusKn)minus1g
Choose g isin C() such that
ginfin = ginfin and g(tj) = gj j isin Ns(n)
Set
v = (I minus PnK)minus1Png
Then we have that
v(ti) =sum
jisinNs(n)
vjφj(ti) i isin Ns(n)
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110 Conventional numerical methods
Letting v = [v(ti) i isin Ns(n)] in matrix notation the above equation can berewritten as
Env = v (349)
We then conclude from
(En minusKn)minus1g = v = Eminus1
n v
that
(En minusKn)minus1ginfin le Eminus1
n infinvinfin le Eminus1n infinvinfin
Since
vinfin = (I minus PnK)minus1Pnginfin le (I minus PnK)minus1Pnginfin
we conclude that
(En minusKn)minus1infin le PnEminus1
n infin(I minus PnK)minus1 (350)
Moreover for v = [vj j isin Ns(n)] let
g = [gj j isin Ns(n)] = (En minusKn)v
We choose g isin C() as before and also set v = (I minus PnK)minus1Png Notingthat
Png(tj) = g(tj) = gj j isin Ns(n)
we have ginfin le Pnginfin Thus
(En minusKn)vinfin = ginfin le Pnginfin = (I minus PnK)vinfin (351)
Choose v isin C() such that
vinfin = vinfin and v(tj) = v(tj) j isin Ns(n)
Then we have that v = Pnv and thus
vinfin le Pnvinfin = Pninfinvinfin le PnEninfinvinfin (352)
Combining estimates (351) and (352) yields
(En minusKn)infin le PnEninfin(I minus PnK)infin
which with (350) leads to the desired result of this theorem
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34 Collocation methods 111
344 Discrete collocation methods
Before investigating discrete collocation methods we remark on the iteratedcollocation method It was known that the iterated collocation method may notlead to superconvergence In contrast with the iterated Galerkin method
K(I minus Pn) ge Kholds This means that the iterated collocation method converges more rapidlyonly in the case that for the solution u K(I minusPn)u has superconvergence (seeSection 224) This is the case when the approximate subspaces Xn are chosenas piecewise polynomials of even degree and the kernel K and solution u aresufficiently smooth (cf [15] for details)
We now begin to discuss discrete collocation methods This approachreplaces the integrals appearing in the collocation equation (342) by finitesums to be chosen depending on the specific numerical methods to be used Tothis end we define
(Knu)(s) =sum
jisinNqn
wjK(s τj)u(τj) s isin
Then the discrete collocation method for solving (31) is to find un isin Xn suchthat
(I minus PnKn)un = Pnf
or equivalently
un(ti)minussum
jisinNqn
wjK(ti τj)un(τj) = f (ti) for all i isin Ns(n) (353)
Some assumptions should be imposed to guarantee the unique solvability ofthe resulting system The iterated discrete collocation solution is defined by
un = f +Knun
which is the solution of the equation
(I minusKnPn)un = f (354)
The analysis of the discrete collocation method can be done by using theframework given in Section 224 with X = V = C()
We close this subsection by giving a relationship between the iterateddiscrete collocation solution and the Nystrom solution That is if τj j isinNqn sube tj j isin Ns(n) then the iterated discrete collocation solution un is theNystrom solution satisfying
(I minusKn)un = f (355)
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112 Conventional numerical methods
In fact by the definition of the interpolating projection for x isin C()
Pnx(τj) = x(τj) j isin Nqn
This leads to
KnPnx(s) =sum
jisinNqn
wjK(s τj)Pnx(τj) =sum
jisinNqn
wjK(s τj)x(τj) = Knx(s)
which with (354) yields (355)
35 PetrovndashGalerkin methods
In this section we establish a theoretical framework for the analysis of conver-gence for the PetrovndashGalerkin method and superconvergence for the iteratedPetrovndashGalerkin method for Fredholm integral equations of the second kind
Unlike the standard Galerkin method the PetrovndashGalerkin method employsa sequence of finite-dimensional subspaces to approximate the solution space(the trial space) of the equation and a different sequence to approximate theimage space of the integral operator (the test space) This feature provides uswith great freedom in choosing a pair of space sequences in order to improvethe computational efficiency of the standard Galerkin method while preservingits convergence order However the space of sequences cannot be chosenarbitrarily They must be coupled properly This motivates us to develop atheoretical framework for convergence analysis of the PetrovndashGalerkin methodand the iterated PetrovndashGalerkin method
It is revealed in [77] that for the PetrovndashGalerkin method the roles of thetrial space and test space are to approximate the solution space of the equationand the range of the integral operator (or in other words the image space)respectively Therefore the convergence order of the PetrovndashGalerkin methodis the same as the approximation order of the trial space and it is independentof the approximation order of the test space This leads to the followingstrategy of choosing the trial and test spaces We may choose the trial spaceas piecewise polynomials of a higher degree and the test space as piecewisepolynomials of a lower degree but keep them with the same dimension Thischoice of the trial and test spaces results in a significantly less expensivenumerical algorithm in comparison with the standard Galerkin method withthe same convergence order which uses the same piecewise polynomials asthose for the trial space The saving comes from computing the entries of thematrix and the right-hand-side vector of the linear system that results fromthe corresponding discretization Note that an entry of the Galerkin matrix is
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35 PetrovndashGalerkin methods 113
the inner product of the integral operator applied to a basis function for thetrial space against a basis function for the same space which is a piecewisepolynomial of a higher degree while an entry of the PetrovndashGalerkin matrix isthat against a basis function for the test space which is a piecewise polynomialof a lower degree Computing the latter is less expensive than computing theformer due to the use of lower-degree polynomials for the test space In factthe PetrovndashGalerkin method interpolates between the Galerkin method and thecollocation method
351 Analysis of PetrovndashGalerkin and iteratedPetrovndashGalerkin methods
1 The PetrovndashGalerkin methodLet X be a Banach space with the norm middot and let Xlowast denote its dualspace Assume that K X rarr X is a compact linear operator We considerthe Fredholm equation of the second kind
uminusKu = f f isin X (356)
where u isin X is the unknown to be determinedWe choose two sequences of finite-dimensional subspaces Xn sub X n isin
N and Yn sub Xlowast n isin N and suppose that they satisfy condition (H) Foreach x isin X and y isin Xlowast there exist xn isin Xn and yn isin Yn such that xnminusx rarr 0and yn minus y rarr 0 as nrarrinfin and
s(n) = dim Xn = dim Yn n isin N (357)
The PetrovndashGalerkin method for equation (356) is a numerical method forfinding un isin Xn such that
(un minusKun y) = (f y) for all y isin Yn (358)
Let
Xn = spanφ1φ2 φs(n)Yn = spanψ1ψ2 ψs(n)
and
un =sum
jisinNs(n)
αjφj
Equation (358) can be written assumjisinNs(n)
αj[(φjψi)minus (Kφjψi)] = (f ψi) i isin Ns(n)
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114 Conventional numerical methods
If XnYn is a regular pair (see Definition 230) in the sense that there is alinear operator n Xn rarr Yn with nXn = Yn and satisfying the conditions
x le c1(xnx)12 and nx le c2x for all x isin Xn
where c1 and c2 are positive constants independent of n then equation (358)may be rewritten as
(un minusKunnx) = (f nx) for all x isin Xn
Furthermore using the generalized best approximation projection Pn X rarrXn (see Definition 225) which is defined by
(xminus Pnx y) = 0 for all y isin Yn
equation (358) is equivalent to the operator equation
un minus PnKun = Pnf (359)
This equation indicates that the PetrovndashGalerkin method is a projectionmethod Using Theorem 255 we obtain the following result
Theorem 326 Let X be a Banach space and K Xrarr X be a compact linearoperator Assume that one is not an eigenvalue of the operator K Suppose thatXn and Yn satisfy condition (H) and XnYn is a regular pair Then thereexists an N0 gt 0 such that for n ge N0 equation (359) has a unique solutionun isin Xn for any given f isin X that satisfies
un minus u le cuminus Pnu n ge N0
where u isin X is the unique solution of equation (356) and c gt 0 is a constantindependent of n
2 The iterated PetrovndashGalerkin methodWe now turn our attention to studying superconvergence of the iterated PetrovndashGalerkin method for integral equations of the second kind
Let X be a Banach space and let Xn sub X n isin N and Yn sub Xlowast n isin N be two sequences of finite-dimensional subspaces satisfying condition(H) Assume that Pn X rarr Xn are the linear projections of the generalizedbest approximation from Xn to X with respect to Yn Consider the projectionmethod (359) Suppose that un isin Xn is the unique solution of equation (359)which approximates the solution of equation (356) The iterated projectionmethod is defined by
uprimen = f +Kun (360)
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35 PetrovndashGalerkin methods 115
It can easily be verified that the iterated projection approximation uprimen satisfiesthe integral equation
uprimen minusKPnuprimen = f (361)
In order to analyze uprimen as the solution of equation (361) we need tounderstand the convergence of the approximate operator KPn The next lemmais helpful in this regard
Lemma 327 Suppose that X is a Banach space and Xn sub X and Yn sub Xlowastsatisfy condition (H) Let Pn X rarr X be the sequence of projections of thegeneralized best approximation from X to Xn with respect to Yn that convergespointwise to the identity operator I in X Then the sequence of dual operatorsPlowastn converges pointwise to the identity operator Ilowast in Xlowast
Proof It follows from condition (H) that for any v isin Xlowast there exists asequence vn isin Yn such that vn minus v rarr 0 as nrarrinfin Consequently
Plowastn vminus v le Plowastn vminus vn + vn minus v le (Pn + 1)vn minus v rarr 0
where the first inequality holds because Plowastn Xlowast rarr Yn are also projectionsThat is Plowastn rarr Ilowast pointwise The second inequality uses the general resultPlowastn = PnTheorem 328 Suppose that X is a Banach space and Xn sub X and Yn sub Xlowastsatisfy condition (H) Assume that K is a compact operator in X Let Pn XrarrX be the projections of the generalized best approximation from X to Xn withrespect to Yn that converges pointwise to the identity operator I in X Then
KPn minusK rarr 0 as nrarrinfin
Proof Note that
KPn minusK = [KPn minusK]lowast = PlowastnKlowast minusKlowastSince K is compact we also have that Klowast is compact Using Lemmas 327 and252 we conclude the result of this theorem
352 Equivalent conditions in Hilbert spaces for regular pairs
From the last section we know that the notion of regular pairs plays anessential role in the analysis of the PetrovndashGalerkin method Therefore it isnecessary to re-examine this concept from different points of view In thissubsection we first study regular pairs from a geometric point of view and
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116 Conventional numerical methods
second characterize them in terms of the uniform boundedness of the sequenceof projections defined by the generalized best approximation
In what follows we confine the space X to be a Hilbert space with innerproduct (middot middot) from which a norm middot is induced In this case Xlowast is identifiedto be X via the inner product We assume XnYn sub X satisfy condition (H)The structure of Hilbert spaces allows us to define the angle between spacesXn and Yn which is done by the orthogonal projection from X onto Yn Foreach x isin X we define the best approximation ylowastn from Yn by
xminus ylowastn = infxminus y y isin YnSince Yn is a finite-dimensional Hilbert subspace in X there exists a bestapproximation from Yn to x isin X We furthermore define the best approxi-mation operator Yn by
Ynx = ylowastn for each x isin X
It is well known that for any x isin X Ynx satisfies the equation
(xminus Ynx y) = 0 for all y isin Yn (362)
In other words the operator Yn is the orthogonal projection from X onto YnTo define the angle between two spaces Xn and Yn we denote
γn = inf
Ynxx x isin Xn
We call
θn = arccos γn
the angle between spaces Xn and Yn The next theorem characterizes a regularpair XnYn in a Hilbert space X in terms of the angles between Xn and Yn
Theorem 329 Let X be a Hilbert space and let Xn and Yn be two subspacesof X satisfying condition (H) and dimXn = dimYn lt infin for n isin N ThenXnYn is a regular pair if and only if there exists a positive number θ0 lt π2such that
θn le θ0 n isin N
Proof We first prove the sufficiency Assume that there exists a positivenumber θ0 lt π2 such that θn le θ0 for all n isin N Thus
γn = inf
Ynxx x isin Xn
ge cos θ0 gt 0
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35 PetrovndashGalerkin methods 117
Using the characterization of the best approximation we have that
(xYnx) = Ynx2 ge cos2 θ0x2 for all x isin Xn
This implies that YnXn = Yn and condition (H-1) holds with c1 = 1 cos θ0Moreover since the operator Yn is the orthogonal projection we conclude that
Ynx le x for all x isin Xn
Hence condition (H-2) holds with c2 = 1We now show the necessity It follows from the definition of a regular pair
that
x2 le c21(xnx) le c2
1xnx le c21c2x2 for all x isin Xn
Thus we obtain
0 lt1
c21c2le 1
It can be seen that there exists an xprime isin Xn with xprime = 0 such that
Ynxprimexprime = inf
Ynxx x isin Xn
= cos θn
By the characterization of the best approximation we obtain that
xprime2 le c21(xprimenxprime) = c2
1(Ynxprimenxprime) le c21Ynxprimenxprime le c2
1c2 cos θnxprime2
Therefore
cos θn ge 1
c21c2
gt 0
and
θn le arccos1
c21c2
ltπ
2
The proof is complete
We now turn to establishing the equivalence of the regular pair and theuniform boundedness of the projections Pn when they are well defined Weneed two preliminary results to prove this equivalence
Lemma 330 Let X be a Hilbert space Assume that XnYn sub X withdim Xn = dim Yn ltinfin satisfy condition (273) Then
Yn = YnPn
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118 Conventional numerical methods
Proof Let x isin X Then Pnx and Ynx satisfy equations (272) and (362)respectively It follows that
(Pnxminus Ynx y) = 0 for all y isin Yn
By the definition of best approximation from Yn to Pnx we conclude that
Ynx = YnPnx for all x isin X
The proof of the lemma is complete
In Hilbert spaces we can interpret condition (273) from many differentpoints of view The next proposition lists eight equivalent statements
Proposition 331 Let X be a Hilbert space and XnYn sub X with dim Xn =dim Yn ltinfin Then the following statements are equivalent
(i) Yn cap Xperpn = 0(ii) det[(φiψj)] = 0 where φl l isin Nm and ψl l isin Nm are bases for
Xn and Yn respectively(iii) Xn cap Yperpn = 0(iv) If x isin Xn with x = 0 then Ynx = 0(v) γn gt 0
(vi) YnXn = Yn(vii) PnYn = Xn and Pn|Yn = (Yn|Xn)
minus1(viii) If y isin Yn with y = 0 then Pny = 0
Proof The implications that (i) implies (ii) and (ii) implies (iii) follow fromthe proof of Proposition 226
We prove that (iii) implies (iv) Let x isin Xn with x = 0 Using the definitionof Yn and (iii) we conclude that
(Ynx y) = (x y) = 0 for all y isin Yn with y = 0
Thus Ynx = 0To prove the implication that (iv) implies (v) we use (iv) to conclude that
on the closed unit sphere x x isin Xn x = 1 Ynx gt 0 Thus we have
γn = infYnx x isin Xn x = 1 gt 0
and statement (v) is provedTo establish (vi) it suffices to show that if φl l isin Nm is a basis for Xn
then Ynφl l isin Zm is a basis for Yn Note that Ynφi isin Yn and dimYn = NIt remains to show that Ynφ1 YnφN is linearly independent To this end
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35 PetrovndashGalerkin methods 119
assume that there are not all zero constants c1 cN such thatsumjisinNN
ciYnφi = 0
Let x =sumjisinNNciφi Then x isin Xn with x = 0 but Ynx = 0 Hence γn = 0 a
contradiction to (v)We now show that (vi) implies (vii) Since PnYn = PnYnXn it is sufficient
to prove PnYnXn = Xn For any x isin Xn applying the definition of Yn gives
(Ynxminus x y) = 0 for all y isin Yn
The definition of Pn implies that x = PnYnx for all x isin Xn Hence weconclude that PnYnXn = Xn and (vii) is established
The implication of (vii) to (viii) is obviousFinally we prove that (viii) implies (i) Let y isin Yn cap Xperpn By the definition
of Pn we find
y2 = (y y) = (Pny y) = 0
This ensures that y = 0
The next theorem shows that XnYn is a regular pair if and only if thesequence of projections Pn is uniformly bounded
Theorem 332 Let X be a Hilbert space Assume that XnYn sub X satisfycondition (H) and equation (273) Let Pn be a sequence of projectionsdefined by the generalized best approximation (272) Then XnYn is aregular pair if and only if there exists a positive constant c for which
Pn le c for all n isin N
Proof We have proved in Proposition 231 that if XnYn is a regular pairthen Pn is uniformly bounded It remains to prove the converse For thispurpose we let Yn Xrarr Yn be the orthogonal projection By our conventionspaces Xn and Yn satisfy condition (273) Thus Proposition 331 ensuresthat YnXn = Yn The validity of condition (H-2) follows from a propertyof best approximation in Hilbert spaces We now prove condition (H-1) bycontradiction Assume to the contrary that condition (H-1) is not valid Thento each ε with 0 lt ε lt 1c where c is the constant that gives the bound forPn there exist n isin N and x isin Xn such that
(xYnx) lt ε2x2
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120 Conventional numerical methods
It follows from the characterization of best approximation in the Hilbert spaceX that
(xYnx) = Ynx2
We then have
Ynx lt εxLet x0 = Ynx Clearly x0 isin Yn Since x isin Xn satisfies the equation
(x0 minus x y) = (Ynxminus x y) = 0 for all y isin Yn
we conclude that x = Pnx0 Consequently
x0 lt εPnx0In other words
Pnx0 gt cx0which contradicts the assumption that Pn le c This contradiction shows thatcondition (H-1) must hold
In the remaining part of this subsection we discuss regular pairs from analgebraic point of view
Definition 333 Let X = φi i isin Zm Y = ψi i isin Zm be two finite(ordered) subsets of the Hilbert space X The correlation matrix between X andY is defined to be the mtimes m matrix
G(X Y) = [(φiψj) i j isin Zm]Note that GT(X Y) the transpose of G(X Y) is G(Y X) For the special
case X = Y we use G(X) for G(X X) and recall that G(X) is the Gram matrixfor the set X The matrix G(X) is positive semi-definite Generally the matrixG(X Y) is not symmetric We use G+(X Y) to denote the symmetric part ofG(X Y) Specifically we set
G+(X Y) = 1
2[G(X Y)+G(Y X)]
We use the standard ordering on m times m symmetric matrices A = [aij i j isinZm] B = [bij i j isin Zm] and write A le B provided thatsum
iisinZm
sumjisinZm
xiaijxj = xTAx le xTBx
for all x = [xi i isin Zm] isin Rm When the strict inequality holds above exceptfor x = 0 we write A lt B
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35 PetrovndashGalerkin methods 121
Definition 334 Let X be a Hilbert space Suppose for any n isin N Xn = φi i isin Zs(n) and Yn = ψi i isin Zs(n) are finite subsets of X where s(n) denotesthe cardinality of Xn We say that Xn Yn forms a regular pair provided thatthere are constants σ gt 0 and σ prime gt 0 such that for all n isin N we have
0 lt G(Xn) le σG+(Xn Yn) (363)
and
0 lt G(Yn) le σ primeG(Xn) (364)
Thus given any finite sets X and Y of linearly independent elements inX of the same cardinality the constant pair X Y is regular if and only ifG+(X Y) gt 0 Moreover when we only have that det G(X Y) = 0 wecan form from X and Y a constant regular pair by modifying either one ofthe sets X and Y To explain this we suppose that X = φi i isin Zn andY = ψi i isin Zn Let W = ωi i isin Zn where the elements of this set aredefined by the formula
ωi =sumjisinZn
(φjψi)φj i isin Zn
Then
G(W Y) = G(X Y)TG(X Y)
and so W Y is a constant regular pair when det G(X Y) = 0 and the elementsof X and Y are linearly independent In the special case that the elements of Xare orthonormal then ωi = Xψi i isin Zn where X is the orthogonal projectionof X onto spanX
Let Xn = span Xn and Yn = span Yn When Xn Yn form a regularpair of finite sets and for every xisinX limnrarrinfin dist(xXn)= 0 the subspacesXnYn form a regular pair of subspaces in the terminology of Definition 230Conversely whenever two subspaces XnYn form a regular pair thesesubspaces have bases which as sets form a regular pair The notion of regularpairs of subspaces from Definition 230 is independent of the bases of thesubspaces However Definition 334 is dependent upon the specific sets usedand may fail to hold if these sets are transformed into others by lineartransformations
Let us observe that (363) and (364) imply
G(Yn) le σσ primeG+(Xn Yn) (365)
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122 Conventional numerical methods
Moreover for any a = [ai i isin Zs(n)] isin Rs(n) by the CauchyndashSchwarzinequality and (363) we have that
aTG+(Xn Yn)a =⎛⎝ sum
jisinZs(n)
ajφjsum
jisinZs(n)
ajψj
⎞⎠le [aTG(Xn)a]12[aTG(Yn)a]12
le σ 12[aTG+(Xn Yn)a]12[aTG(Yn)a]12
This inequality implies that
G+(Xn Yn) le σG(Yn)
Using this inequality and (363) we conclude that
G(Xn) le σG+(Xn Yn) le σ 2G(Yn) (366)
Therefore it follows that whenever Xn Yn is a regular pair then so is Yn XnWhen the sets Xn Yn form a regular pair with constants σ σ prime the
generalized best approximation projection Pn X rarr Xn with respect to Yn
enjoys the bound
Pn le p = σσ prime12 (367)
To confirm this inequality for each x isin X we write Pnx in the form
Pnx =sum
jisinZs(n)
ajφj
where the vector a = [aj j isin Zs(n)] is the solution of the linear equations
(xψi) =( sum
jisinZs(n)
ajφjψi
) i isin Zs(n)
Hence multiplying both sides of these equations by ai summing over i isin Zs(n)
and using (363) and (364) we get that
Pnx2 = aTG(Xn)a le σaTG+(Xn Yn)a
= σ(Pnx
sumjisinZs(n)
ajψj
)= σ
(xsum
jisinZs(n)
ajψj
)le σx
∥∥∥ sumjisinZs(n)
ajψj
∥∥∥ le σσ prime12x∥∥∥ sum
jisinZs(n)
ajφj
∥∥∥ = σσ prime12xPnx
Now we divide the first and last terms in the above inequality by Pnx toyield the desired inequality Pn le p
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35 PetrovndashGalerkin methods 123
Recall that
γn = inf
Ynxx x isin Xn
and
θn = arccos γn n isin N
Note that γn isin [0 1] and θn isin [0π2] By definition θn is the angle betweenthe two subspaces Xn and Yn Let us observe for any x isin Xn that the equation
PnYnx = x
holds This leads to
cos θn = inf
YnxPnYnx x isin Xn
ge inf
yPny y isin Yn
ge 1
Pn
Therefore we conclude that when XnYn is a pair of subspaces with basesXn Yn which form a regular pair with constants σ σ prime the inequality
cos θn ge pminus1 gt 0
holds In other words in this case for all n isin N we have that θn isin [0 θlowast)where θlowast lt π2
353 The discrete PetrovndashGalerkin method and its iterated scheme
The PetrovndashGalerkin method for Fredholm integral equations of the secondkind was studied in the last sections To use the PetrovndashGalerkin method inpractical computation we have to be able to efficiently compute the integralsoccurring in the method In this subsection we take an approach to discretizinga given integral equation by a discrete projection and a discrete inner productThe iterated solution suggested in this section is also fully discrete
In this subsection we describe discrete PetrovndashGalerkin methods for Fred-holm integral equations of the second kind with weakly singular kernels Forthis purpose we consider the equation
(I minusK)u = f (368)
where K Linfin()rarr C() is a compact linear integral operator defined by
(Ku)(s) =int
K(s t)u(t)dt s isin (369)
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124 Conventional numerical methods
sub Rd is a bounded closed domain and K is a function defined on times
which is allowed to have weak singularities We assume that one is not aneigenvalue of the operator K to guarantee the existence of a unique solutionu isin Linfin() Some additional specific assumptions will be imposed later inthis subsection
We first recall the PetrovndashGalerkin method for equation (368) In thisdescription we let X = L2() with an inner product (middot middot) Let Xn and Ynbe two sequences of finite-dimensional subspaces of X such that
dimXn = dimYn = s(n)
Xn = spanφ1φ2 φs(n)and
Yn = spanψ1ψ2 ψs(n)We assume that XnYn is a regular pair It is known from the last sectionthat the necessary and sufficient condition for a generalized best approximationfrom Xn to x isin X with respect to Yn to exist uniquely is YncapXperpn = 0 If thiscondition holds then Pn is a projection and XnYn forms a regular pair ifand only if Pn is uniformly bounded The PetrovndashGalerkin method for solvingequation (368) is a numerical scheme to find a function
un(s) =sum
jisinNs(n)
αjφj(s) isin Xn
such that
((I minusK)un y) = (f y) for all y isin Yn (370)
or equivalentlysumjisinNs(n)
αj[(φjψi)minus (Kφjψi)
] = (f ψi) i isin Ns(n) (371)
Using the generalized best approximation Pn X rarr Xn we write equa-tion (370) in operator form as
(I minus PnK)un = Pnf (372)
It is also proved in the last section that if XnYn is a regular pair then forsufficiently large n equation (372) has a unique solution un isin Xn whichsatisfies the estimate
un minus u le C infuminus x x isin Xn
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35 PetrovndashGalerkin methods 125
Solving equation (372) requires solving the linear system (371) Of coursethe entries of the coefficient matrix of (371) involve the integrals (Kφjψi)which are normally evaluated by a numerical quadrature formula Roughlyspeaking the discrete PetrovndashGalerkin method is the scheme (371) withthe integrals appearing in the method computed by quadrature formulasHowever we shall develop our discrete PetrovndashGalerkin method independentof the PetrovndashGalerkin method (372) In other words we do not assumethat the PetrovndashGalerkin method (372) has been previously constructed toavoid the ldquoregular pairrdquo assumption which is crucial for the solvability andconvergence of the PetrovndashGalerkin method We take a one-step approachto fully discretize equation (368) directly We first describe the method inldquoabstractrdquo terms without specifying the bases and the concrete quadratureformulas Later we specialize them using the piecewise polynomial spacesThe only assumption that we have to impose later to guarantee the solvabilityand convergence of the resulting concrete method is a local condition on thereference element and thus it is easy to verify it
In our description we use function values f (t) at given points t isin for anLinfin function f We follow [21] to define them precisely Let C() denote thesubspace of Linfin() which consists of functions each of which is equal to anelement in C() ae The point evaluation functional δt on the space C() isdefined by
δt(f ) = f (t) t isin f isin C()
where f on the right-hand side is chosen to be the representative function f isinC() which is continuous By the HahnndashBanach theorem the point evaluationfunctional δt can be extended from C() to the whole Linfin() in such a waythat the norm is preserved We use dt to denote such an extension and define
f (t) = dt(f ) for f isin Linfin()
We remark that the extension is not unique but that is usually immaterial Whatis important is that it exists and preserves many of the properties naturallyassociated with the point evaluation functional For example at a point ofcontinuity of f the extended point evaluation is uniquely defined and has thenatural value and moreover dt is continuous at such points The reader isreferred to [21] for more details on this extension
We now return to our description of the discrete PetrovndashGalerkin methodAs in the description of the (continuous) PetrovndashGalerkin method we choosetwo subspaces Xn = span φj j isin Ns(n) and Yn = span ψj j isin Ns(n) ofthe space Linfin() such that dim Xn = dim Yn = s(n) We choose mn pointsti isin and two sets of weight functions w1i w2i i isin Nmn We define the
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126 Conventional numerical methods
discrete inner product
(x y)n =sum
iisinNmn
w1ix(ti)y(ti) x y isin Linfin() (373)
which will be used to approximate the inner product (x y) = intD x(t)y(t)dtand define discrete operators by
(Knu)(s) =sum
iisinNmn
w2i(s)u(ti) u isin Linfin() (374)
which will be used to approximate the operator K With these notationsthe discrete PetrovndashGalerkin method for equation (368) is a numericalscheme to find
un(s) =sum
jisinNs(n)
αnjφj(s) (375)
such that
((I minusKn)un y)n = (f y)n for all y isin Yn (376)
In terms of basis functions equation (376) is written as
sumjisinNs(n)
αnj
⎡⎣ sumisinNmn
w1φj(t)ψi(t)minussumisinNmn
w1
summisinNmn
w2m(t)φj(tm)ψi(t)
⎤⎦=sumisinNmn
w1f (t)ψi(t) i isin Ns(n) (377)
Upon solving the linear system (377) we obtain s(n) values αnj Substitutingthem into (375) yields an approximation to the solution u of equation (368)Equation (376) can also be written in the operator form by a discretegeneralized best approximation Qn which we define next Let Qn Xrarr Xn
be defined by
(Qnx y)n = (x y)n for all y isin Yn (378)
If Qnx is uniquely defined for every x isin X equation (376) can be written inthe form
(I minusQnKn)un = Qnf (379)
We postpone a discussion of the unique existence of Qnx until laterThe iterated PetrovndashGalerkin method has been shown to have a supercon-
vergence property where the additional order of convergence gained froman iteration is attributed by approximation of the kernel from the test space
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35 PetrovndashGalerkin methods 127
The convergence order of the iterated PetrovndashGalerkin method is equal to theapproximation order of space Xn plus the approximation order of space YnIt is of interest to study the superconvergence of the iterated discrete PetrovndashGalerkin method which we define by
uprimen = f +Knun (380)
Equation (380) is a fully discrete algorithm which can be implemented easilyinvolving only multiplications and additions It can be shown that uprimen satisfiesthe operator equation
(I minusKnQn)uprimen = f (381)
This form of equation allows us to treat the iterated discrete PetrovndashGalerkinmethod as an operator equation whose analysis is covered by the theorydeveloped in Section 224
Up to now the discrete PetrovndashGalerkin method has been described inabstract terms without specifying the spaces Xn and Yn In the remainder ofthis section we specialize the discrete PetrovndashGalerkin method by specifyingthe spaces Xn and Yn and defining operators Qn and Kn in terms of piecewisepolynomials We assume that is a polyhedral region and construct a partitionTn for by dividing it into Nn simplices ni i isin NNn such that
h = maxdiam ni i isin NNn rarr 0 as nrarrinfin (382)
=⋃
iisinNNn
ni
and
meas(ni capnj) = 0 i = j
When the dependence of the simplex ni on n is well understood we dropthe first index n in the notation and simply write it as i For each positiveinteger n the set Tn forms a partition for the domain We also require thatthe partition is regular in the sense that any vertex of a simplex in Tn is not inthe interior of an edge of a face of another simplex in the set It is well knownthat for each simplex there exists a unique one to one and onto affine mappingwhich maps the simplex onto a unit simplex 0 called a reference element
Let Fi i isin NNn denote the invertible affine mappings that map the referenceelement 0 one to one and onto the simplices i Then the affine mappingsFi have the form
Fi(t) = Bit + bi t isin 0 (383)
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128 Conventional numerical methods
where Bi is a d times d invertible matrix and bi a vector in Rd and they satisfy
i = Fi(0)
On the reference element 0 we choose two piecewise polynomial spacesS1k1(
0) and S2k2(0) of total degree k1minus1 and k2minus1 respectively such that
dim S1k1(0) = dim S2k2(
0) = μ
The partitions 1 and 2 of 0 associated respectively with S1k1(0) and
S2k2(0) may be different they are arranged according to the integers k1 k2
and d Assume that the numbers of sub-simplices contained in the partitions1 and 2 are denoted by ν1 and ν2 We have to choose these pairs of integersk1 ν1 and k2 ν2 such that(
k1 minus 1+ d
d
)ν1 =
(k2 minus 1+ d
d
)ν2 = μ
because the dimension of the space of polynomials of total degree k is(k+d
d
)
We shall not provide a detailed discussion on how the partitions 1 and 2
are constructed Instead we assume that we have chosen bases for these twospaces so that
S1k1(0) = spanφj j isin Nμ
and
S2k2(0) = spanψj j isin Nμ
We next map these piecewise polynomial spaces on 0 to each simplex i byletting
φij(t) =φj Fminus1
i (t) t isin i
0 t isin i
and
ψij(t) =ψj Fminus1
i (t) t isin i
0 t isin i
for i isin NNn and j isin Nμ Using these functions as bases we define the trialspace and the test space respectively by
Xn = spanφij i isin NNn j isin Nμand
Yn = spanψij i isin NNn j isin Nμ
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35 PetrovndashGalerkin methods 129
It follows from (382) that
C() sube⋃
Xn
and
C() sube⋃
Yn
Moreover we have that if x isin Wk1infin() then there exists a constant c gt 0 suchthat for all n
infxminus φ φ isin Xn le chk1
and if x isin Wk2infin() then likewise there exists a constant c gt 0 such that forall n
infxminus φ φ isin Yn le chk2
However the space X =⋃Xn does not equal Linfin() it is a proper subspaceof Linfin() because the space Linfin() is not separable Due to this fact theexisting theory of collectively compact operators (cf [6]) does not applydirectly to this setting Some modifications of the theory are required
We next specialize the definition of the discrete inner product (373) anddescribe a concrete construction of the approximate operators Kn To this endwe introduce a third piecewise polynomial space S3k3(
0) of total degreek3 minus 1 on 0 We divide the reference element 0 into ν3 sub-simplices
3 = ei i isin Nν3and also assume that the partition 3 is regular On each of the simplicesei we choose m = (k3minus1+d
d
)points τij j isin Nm such that they admit a
unique Lagrange interpolating polynomial of total degree k3 minus 1 on ei Formultivariate Lagrange interpolation by polynomials of total degree see [83]and the references cited therein Let pij be the polynomial of total degree k3minus1on ei satisfying the interpolation conditions
pij(τiprimejprime) = δiiprimeδjjprime i iprime isin Nν3 j jprime isin Nm
We assemble these polynomials to form a basis for the space S3k3(0) by
letting
ζ(iminus1)m+j(t) =
pij(t) t isin ei0 t isin ei
i isin Nν3 j isin Nm
Set γ = mν3 which is equal to the dimension of S3k3(0) and
t(iminus1)m+j = τij i isin Nν3 j isin Nm
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130 Conventional numerical methods
Then ζi isin S3k3(0) and satisfy the interpolation conditions
ζi(tj) = δij i j isin Nγ
This set of functions forms a basis for the space S3k3(0) It can be used to
introduce a piecewise polynomial space on by mapping the basis ζj j isin Nγ
for S3k3(0) from 0 into each i Specifically we define
ζij(t) =ζj Fminus1
i (t) t isin i0 t isin i
where Fi is the affine map defined by (383) Let
Zn = spanζij i isin NNn j isin Nγ Hence Zn is a piecewise polynomial space of dimension γNn For each i wedefine
tij = Fi(tj) = Bitj + bi
where Bi and bi are respectively the matrix and vector appearing in thedefinition of the affine map Fi Furthermore we define the linear projectionZn Xrarr Zn by
Zng =sum
iisinNNn
sumjisinNγ
dtij(g)ζij
where dt is the extension of the point evaluation functional δt satisfyingdt = 1 which was discussed earlier and satisfies the condition
dtij(ζiprimejprime) = δiiprimeδjjprime i iprime isin NNn j jprime isin Nγ
Moreover we have that
Zn = ess suptisin
sumiisinNNn
sumjisinNγ
|ζij(t)| = ess suptisin
sumjisinNγ
|ζj(t)|
That is Zn is uniformly bounded for all n It follows from the uniformboundedness of Zn that for any y isin Wk3infin() there holds an estimate
yminus Zny le C infyminus φφ isin Zn le Chk3 (384)
Using the projection Zn defined above we have a quadrature formulaint
g(t)dt =sum
iisinNNn
sumjisinNγ
wijdtij(g)+O(hk3)
where
wij =int
ζij(t)dt
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35 PetrovndashGalerkin methods 131
If we set
wi =int0
ζi(t)dt i isin Nγ
then we have
wij =inti
ζj(Fminus1i (t))dt = det(Bi)
int0
ζj(t)dt = det(Bi)wj
Without loss of generality we assume that
det(Bi) gt 0 i isin Nγ
Employing this formula we introduce the following discrete inner product
(x y)n =sum
iisinNNn
sumisinNγ
wix(ti)y(ti) (385)
Formula (385) is a concrete form for (373) When x y isin Wk3infin() we havethe error estimate
|(x y)minus (x y)n| le Chk3
With this specific definition of the spaces Xn Yn and the discrete inner productwe obtain a construction of the operators Qn using equation (378)
Finally to describe a concrete construction of the approximate operatorsKn we impose a few additional assumptions on the kernel K of the integraloperator K Roughly speaking we assume that K is a product of two kernelsone of them is continuous but perhaps involves a complicated function and theother has a simple form but has a singularity In particular we let
K(s t) = K1(s t)K2(s t) s t isin
where K1 is continuous on times and K2 has a singularity and satisfies theconditions
K2(s middot) isin L1() s isin supsisin
int
|K2(s t)|dt lt +infin (386)
K2(s middot)minus K2(sprime middot)1 rarr 0 as sprime rarr s (387)
Moreover we assume that the integration of the product of K2(s t) and apolynomial p(t) with respect to the variable t can be evaluated exactly Manyintegral operators K that appear in practical applications are of this type
Using the linear projection Zn we define Kn Xrarr X by
(Knx)(s) =int
Zn(K1(s t)x(t))K2(s t)dt
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132 Conventional numerical methods
which approximates the operator K For un isin Xn we have that
(Knun)(s) =sum
iisinNNn
sumjisinNγ
wij(s)K1(s tij)un(tij)
where
wij(s) =inti
ζij(t)K2(s t)dt
This concrete construction of the trial space Xn the test space Yn and operatorsQn Kn yields a specific discrete PetrovndashGalerkin method which is describedby equation (379) This is the method that we shall analyze in the nextsubsection
354 The convergence of the discrete PetrovndashGalerkin method
In this subsection we follow the general theory developed in Section 224 toprove the convergence results of the discrete PetrovndashGalerkin method when apiecewise polynomial approximation is used Throughout the remaining part ofthis subsection we let X = Linfin() V = C() Xn and Yn be the piecewisepolynomial spaces defined in the last subsection and X = cupnXn Our maintask is to verify that the operators Qn and Kn with the spaces Xn Yn defined inthe last subsection by the piecewise polynomials satisfy the hypotheses (H-1)ndash(H-4) of Section 224 so that Theorem 254 can be applied For this purposewe define the necessary notation Let
= [φi(tj) i isin Nμ j isin Nγ ] and = [ψi(tj) i isin Nμ j isin Nγ ]where φi and ψi are the bases we have chosen for the piecewise polynomialspaces S1k1(
0) and S2k2(0) and tj are the interpolation points in the
reference element 0 chosen in the last subsection Noting that wi are theweights of the quadrature formula on the reference element developed inSection 421 we set
W = diag(w1 wγ ) and M = WT
The next proposition presents a necessary and sufficient condition for thediscrete generalized best approximation to exist uniquely
Proposition 335 For each x isin Linfin() the discrete generalized bestapproximation Qnx from Xn to x with respect to Yn defined by (378) existsuniquely if and only if
det(M) = 0 (388)
Under this condition Qn is a projection that is Q2n = Qn
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35 PetrovndashGalerkin methods 133
Proof Let x isin Linfin() be given Showing that there is a unique Qnx isin Xn
satisfying equation (378) is equivalent to proving that the linear systemsumiisinNNn
sumjisinNμ
cij(φijψiprimejprime)n = (xψiprimejprime)n iprime isin NNn jprime isin Nμ (389)
has a unique solution [c11 c1μ cNn1 cNnμ] This in turn is equiv-alent to the fact that the coefficient matrix M of this system is nonsingularIt is easily seen that
M = diag(det(B1)M det(BNn)M)
Thus the first result of this proposition follows from hypothesis (388)It remains to show that Qn is a projection By definition we have for every
x isin Linfin() that
(Qnx y)n = (x y)n for all y isin Yn
In particular this equation holds when x is replaced by Qnx That is
(Q2hx y)n = (Qnx y)n for all y isin Yn
It follows for each x isin X that
Q2nx = Qnx
That is Qn is a projection
Condition (388) is on the choice of points tj on the reference elementThey have to be selected in a careful manner so that they match with the choiceof the bases φi and ψi This condition has to be verified before a concreteconstruction of the projection Qn is given This is not a difficult task since thecondition is on the reference element it is independent of n and in practicalapplications the numbers μ and γ are not too large
The next proposition gives two useful properties of the projection Qn
Proposition 336 Assume that condition (388) is satisfied Let Qn be definedby (378) with the spaces Xn Yn and the discrete inner product constructed interms of the piecewise polynomials described in the last subsection Then thefollowing statements hold
(i) Qn is uniformly bounded that is there exists a constant c gt 0 such thatQn le c for all n
(ii) There exists a constant c gt 0 such that for all n
Qnxminus xinfin le c infxminus φinfinφ isin Xnholds for all x isin Linfin() Thus for each x isin C() Qnx minus xinfin rarr 0holds as nrarrinfin
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134 Conventional numerical methods
Proof (i) For any x isin Linfin() we have the expression
Qnx =sum
iisinNNn
sumjisinNμ
cijφij (390)
where the coefficients cij satisfy equation (389) It follows that
Qnxinfin le cinfiness supsisin
sumiisinNNn
sumjisinNμ
|φij(s)| = cinfinmaxsisin0
sumjisinNμ
|φj(s)| (391)
where
c = [c11 c1μ cNn1 cNnμ]T
and the discrete norm of c is defined by cinfin = max|cij| i isin NNn j isin NμBy definition the vector c is dependent on n although we do not specify it inthe notation However we prove that cinfin is in fact independent of n To thisend we use system (389) and hypothesis (388) to conclude that
cinfin = Mminus1dinfin (392)
where
d = [(xψ11)n (xψ1μ)n (xψNn1)n (xψNnμ)n]T
and
Mminus1 = diag(
det(B1)minus1Mminus1 det(BNn)
minus1Mminus1)
Let
di = [(xψi1)n (xψiμ)n]T isin Rμ
Then it follows from (392) that the following estimate of cinfin holds in termsof blocks di and Mminus1
cinfin le maxiisinNNn
det(Bi)minus1Mminus1diinfin (393)
This inequality reduces estimating cinfin to bounding each block di By thedefinition of the discrete inner product we have the estimate for the normof di
diinfin le xinfinmaxjisinNμ
sumisinNγ
wi|ψij(ti)| = det(Bi)xinfinmaxjisinNμ
sumisinNγ
w|ψj(t)|
(394)
From (391)ndash(394) we conclude that
Qnxinfin le cxinfin for all n
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35 PetrovndashGalerkin methods 135
where c is a constant independent of n with the value
c = Mminus1infinmaxsisinE
sumjisinNμ
|φj(s)|maxjisinNμ
sumisinNγ
w|ψj(t)|
(ii) Let φ isin Xn Since Qn is a projection we have for each x isin Linfin that
Qnxminus xinfin le xminus φinfin + Qnφ minusQnxinfin le (1+ c)xminus φinfin
Thus we obtain the estimate
Qnxminus xinfin le c infxminus φinfinφ isin XnThis estimate with the relation C() sube cupnXn implies that Qnxminus xinfin rarr 0as nrarrinfin for each x isin C()
In the next proposition we verify that the operators Kn defined in thelast subsection by the piecewise polynomial approximation satisfy hypotheses(H-1) and (H-2)
Proposition 337 Suppose that Kn is defined as in the last subsection by thepiecewise polynomial approximation Then the following statements hold
(i) The set of operators Kn is collectively compact(ii) For each x isin X KnxminusKxinfin rarr 0 as nrarrinfin
(iii) If x isin Wk3infin() and K1 isin C()timesWk3infin() then
KxminusKnxinfin le chk3
Proof (i) By the continuity of the kernel K1 and condition (387) there existconstants c1 and c2 such that
K1(s middot)infin le c1 and K2(s middot)1 le c2
Thus we have that
|(Knx)(s)| =∣∣∣∣int
Zn(K1(s t)x(t))K2(s t)dt
∣∣∣∣ le c0c1c2xinfin (395)
Moreover
|(Knx)(s)minus (Knx)(sprime)|=∣∣∣∣int
Zn(K1(s t)x(t))K2(s t)dt minusint
Zn(K1(sprime t)x(t))K2(s
prime t)dt
∣∣∣∣le∣∣∣∣int
Zn(K1(s t)x(t))[K2(s t)minus K2(sprime t)]dt
∣∣∣∣
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136 Conventional numerical methods
+∣∣∣∣int
[Zn(K1(s t)x(t))minus Zn(K1(sprime t)x(t))]K2(s
prime t)dt
∣∣∣∣le c0xinfin
(c1K2(s middot)minus K2(s
prime middot)1 + c2K1(s middot)minus K1(sprime middot)infin
)
Since K2(s middot)minusK2(sprime middot)1 and K1(s middot)minusK1(sprime middot)infin are uniformly continu-ous on we observe that Knx is equicontinuous on By the ArzelandashAscolitheorem we conclude that Kn is collectively compact
(ii) For any x isin X
|(Knx)(s)minus (Kx)(s)| =∣∣∣∣int
[Zn(K1(s t)x(t))minus K1(s t)x(t)]K2(s t)dt
∣∣∣∣le c2Zn(K1(s middot)x)minus K1(s middot)xinfin
Note that K1x is piecewise continuous as is x By the definition of Zn wehave that the right-hand side of the above inequality converges to zero as nrarrinfin We conclude that the left-hand side converges uniformly to zero on thecompact set That is KnxminusKx rarr 0 as nrarrinfin
(iii) If x isin Wk3infin() by the approximate order of the interpolation projectionZn we have
KnxminusKxinfin le c supsisin(Zn(K1(s middot)x(middot)))(middot)minus K1(s middot)x(middot)infin le chk3
The estimate above follows immediately from the fact that K1 isin C() timesWk3infin() and inequality (384)
Using Propositions 336 and 337 and Theorem 254 we obtain the follow-ing theorem
Theorem 338 The following statements are valid
(i) There exists N0 gt 0 such that for all n gt N0 the discrete PetrovndashGalerkin method using the piecewise polynomial approximation describedin Section 421 has a unique solution un isin Xn
(ii) If u isin Wαinfin() with α = mink1 k3 then
uminus uninfin le chα
Proof By Propositions 336 and 337 we conclude that conditions(H-1)ndash(H-4) are satisfied Hence from Theorem 254 statement (i) followsimmediately and the estimate
uminus uninfin le c (uminusQnuinfin + KuminusKnuinfin) (396)
holds Now let u isin Wαinfin() Again Proposition 336 ensures that
uminusQnuinfin le c infuminus φinfinφ isin Xn le chα (397)
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35 PetrovndashGalerkin methods 137
By (iii) of Proposition 337 we have that
KuminusKnuinfin le chα (398)
Substituting estimates (397) and (398) into inequality (396) yields theestimate in (ii)
355 Superconvergence of the iterated approximation
We present in this subsection a superconvergence property of the iterateddiscrete PetrovndashGalerkin method when the kernel is smooth
To obtain superconvergence we require furthermore that the partitions 1
and 3 of 0 associated with the spaces S1k(0) and S3k3(
0) respectivelyare exactly the same In the main theorem of this section we prove that thecorresponding iterated discrete PetrovndashGalerkin approximation has a super-convergence property when the kernels are smooth In particular we assumethat the kernel K = K1 and K2 = 1 in the notation of the last subsection Wefirst establish a technical lemma
Lemma 339 Let x isin Linfin() and k1 isin C() times Wk3infin() If 1 = 3 thenthere exists a positive constant c such that for all n
(K minusKn)Qnxinfin le chk3
Proof Since Qnx is not a continuous function Proposition 337 (iii) doesnot apply to this case However it follows from the proof of Proposition 337(ii) that
|(KnQnx)(s)minus (KQnx)(s)| le crsinfin
where
rs(t) = (Zn(K1(s middot)(Qnx)(middot))(t)minus K1(s t)(Qnx)(t)
Hence it suffices to estimate rs(t)Using the definition of the projection Qn we write
(Qnx)(t) =sum
iisinNNn
sumjisinNμ
cijφij(t) t isin (399)
where φij are the basis functions for Xn given in Subsection 354 and thecoefficients cij satisfy the linear system (389) Consequently we have that
(Zn(K1(s middot)(Qnx)(middot))(t) =sum
iisinNNn
sumjisinNμ
cij(Zn(K1(s middot)φij(middot))(t) (3100)
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138 Conventional numerical methods
By the construction of the functions φij we have that φij(tiprimejprime) = 0 if i = iprimeThus it follows that
(Zn(K1(s middot)φij(middot))(t) =sum
iprimeisinNNn
sumjprimeisinNγ
K1(s tiprimejprime)φij(tiprimejprime)ζiprimejprime(t)
=sum
jprimeisinNγ
K1(s tijprime)φij(tijprime)ζijprime(t)
Substituting this equation into (3100) yields
(Zn(K1(s middot)(Qnx)(middot))(t) =sum
iisinNNn
sumjisinNμ
cij
sumjprimeisinNγ
K1(s tijprime)φij(tijprime)ζijprime(t) t isin
(3101)
We now assume that for some point t isin iprime rsinfin = |rs(t)| For this point tthere exists a point τ in the reference element 0 such that t = Fiprime(τ ) Hence
rsinfin =∣∣∣∣∣∣sumjisinNμ
ciprimej
⎡⎣sumjprimeisinNγ
K1(sFiprime(tjprime))φj(tjprime)ζjprime(τ )minus K1(sFiprime(τ ))φj(τ )
⎤⎦∣∣∣∣∣∣ Because 0 = cupiisinNν3
ei the point τ must be in some ei For each integerjprime isin Nγ assume that positive integers i0 and j0 with i isin Nν3 j0 isin Nm are suchthat (i0 minus 1)m+ j0 = jprime Therefore we have
ζjprime(t) =
pi0j0(t) t isin ei00 t isin ei0
and tjprime = τi0j0
so that
rsinfin =∣∣∣∣∣∣sumjisinNμ
cij
⎡⎣ sumi0isinNν3
sumj0isinNm
K1(sFiprime (τi0j0 ))φj(τi0j0 )pi0j0 (τ )minus K1(sFiprime (τ ))φj(τ )
⎤⎦∣∣∣∣∣∣=∣∣∣∣∣∣sumjisinNμ
cij
⎡⎣ sumj0isinNm
K1(sFiprime (τij0 ))φj(τij0 )pij0 (τ )minus K1(sFiprime (τ ))φj(τ )
⎤⎦∣∣∣∣∣∣ We identify that the function in the blanket of the last term is the error ofpolynomial interpolation of the function K1(sFiprime(τ ))φj(τ ) on ei which wecall the error term on ei Since 1 = 3 K1(sFiprime(τ ))φj(τ ) as a function of
τ is in the space Wk3infin(ei) We conclude that the error term on ei is bounded bya constant time k3(K1(sFiprime(middot))φj(middot)infin The latter is bounded by a constanttime |det(Biprime)|k3 le chk3 Hence we obtain
rsinfin le ccinfinhk3
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35 PetrovndashGalerkin methods 139
By the proof of Proposition 336 we know that cinfin le c Therefore we haversinfin le chk3
We are now ready to establish the main result of this subsection concerningthe superconvergence of the iterated solution
Theorem 340 If β = mink1 + k2 k3 u isin Wβinfin() and K isin C() timesWk3infin() then there exists a constant c gt 0 such that for all n
uminus uprimeninfin le chβ
Proof It follows from Theorem 254 that
uminus uprimeninfin le c ((K minusKn)Qnuinfin + K(I minusQn)uinfin) (3102)
Because 1 = 3 by applying Lemma 339 we have that
(K minusKn)Qnuinfin le chk3 (3103)
Moreover since K(s middot) isin Wk3infin() and 1 = 3 we conclude that
K(uminusQnu)infin le (K(s t) u(t)minus (Qnu)(t))ninfin + chk3 (3104)
It remains to establish an upper bound for (K(s t) u(t)minus (Qnu)(t))ninfin Forthis purpose we note that for any y isin Yn
(y uminusQnu)n = 0
holds It follows that
|(K(s t) u(t)minus (Qnu)(t))n| = |(K(s t)minus y(t) u(t)minusQnu)(t))n|le infK(s t)minus y(t)infin y isin YnuminusQnuinfin
This implies that
(K(s t) u(t)minus (Qnu)(t))ninfin le chk2 hk1 = chk1+k2 (3105)
Combining inequalities (3102)ndash(3105) we establish the estimate of thistheorem
We remark that when k1 lt k3 lt k1+k2 the optimal order of convergence ofun is O(hk1) while the iterated solution uprimen has an order of convergence O(hk3)This phenomenon is called superconvergence
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140 Conventional numerical methods
356 Numerical examples
In this subsection we present two numerical examples to illustrate the theo-retical estimates obtained in the previous subsections The kernel in the firstexample is weakly singular while the kernel in the second example is smoothThe second example is presented to show the superconvergence property of theiterated solution We restrict ourselves to simple one-dimensional equationswhose exact solutions are known
In both examples we use piecewise linear functions and piecewise constantfunctions for the spaces Xn and Yn respectively Specifically we define thetrial space by
Xn = spanφj j isin N2nwhere
φ2j+1(t) =
nt minus j jn le t le j+1
n 0 otherwise
j isin Zn
and
φ2j+2(t) =
j+ 1minus nt jn le t le j+1
n 0 otherwise
j isin Zn
The test space is then defined by
Yn = spanψj j isin N2nwhere
ψj(t) =
1 jminus12n le t le j
2n 0 otherwise
j isin N2n
Example 1 Consider the integral equation with a weakly singular kernel
u(s)minusint π
0log | cos sminus cos t|u(t)dt = 1 0 le s le π
This equation is a reformulation of a third boundary value problem of the two-dimensional Laplace equation and it has the exact solution given by
u(s) = 1
1+ π log 2
See [12] for more details on this example By changes of variable t = π tprimes = πsprime we have an equivalent equation
u(πs)minus π
int 1
0log | cos(πs)minus cos(π t)|u(π t)dt = 1 0 le s le 1
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35 PetrovndashGalerkin methods 141
We write the kernel
log | cos(πs)minus cos(π t)| =4sum
i=1
Ki1(s t)Ki2(s t)
where
K11(s t) = log
⎛⎝ sin(π(tminuss)
2
)sin(π(t+s)
2
)π3 (tminuss)
2 (t + s)(2minus t minus s)
⎞⎠
K12(s t) = K21(s t) = K31(s t) = K41(s t) = 1
K22(s t) = log |π(sminus t)| K32(s t) = log(π(2minus sminus t))
and
K42(s t) = log(π(s+ t))
In Table 31 we present the error en of the approximate solution and the erroreprimen of the iterated approximate solution where we use q and qprime to represent thecorresponding orders of approximation respectively In our computation wechoose k3 = 2
The order of approximation agrees with our theoretical estimate Theiteration does not improve the accuracy of the approximate solution for thisexample due to the nonsmoothness of the kernel
Example 2 We consider the integral equation with a smooth kernel
u(s)minusint 1
0sin s cos tu(t)dt = sin s(1minus esin 1)+ esin s 0 le s le 1
It is not difficult to verify that u(s) = esin s is the unique solution of thisequation In the notation of Section 353 we have K1(s t) = sin s cos t andK2(s t) = 1 In this case we choose k3 = 3 for the quadrature formula Thenotation in Table 32 is the same as that in Table 31
In this example the iteration improves the accuracy of approximation by theorder as estimated in Theorem 340
Table 31
n 4 8 16 32
en 1504077e-6 3879971e-7 9877713e-8 2957718e-8q 1954761 1973797 1739639eprimen 3186220e-6 8005914e-7 1973337e-7 5153006e-8qprime 1992708 2020429 193715
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142 Conventional numerical methods
Table 32
n 4 8 16 32
en 168156e-2 410275e-3 101615e-3 300353e-4q 2035137 2013478 175839eprimen 678911e-5 416056e-6 258946e-7 161679e-8qprime 4028373 4006054 4001447
36 Bibliographical remarks
The material presented in this chapter on conventional numerical methods forsolving Fredholm integral equations of the second kind is mainly taken fromthe books [15 177 178] Analysis of the quadrature method may be found in[6] As related issues of the collocation method valuation of an Linfin functionf at a given point the reader is referred to [21] and multivariate Lagrangeinterpolation by polynomials may be found in [83] and the references citedtherein For the theoretical framework for analysis of the PetrovndashGalerkinmethod readers are referred to [64 77] Superconvergence of the iteratedPetrovndashGalerkin method was originally analyzed in [77] The discrete PetrovndashGalerkin method and its iterated scheme presented in Section 353 are takenfrom [68 80] The iterated Galerkin method a special case of the iteratedPetrovndashGalerkin method for Fredholm integral equations of the second kindwas studied by many authors (see [23 165 246] and the references citedtherein) Reference [241] gives a nice review of the iterated Galerkin methodand iterated collocation method
We would like to mention other developments on this topic not includedin this book Boundary integral equations of the second kind with periodiclogarithmic kernels were solved by a Nystrom scheme-based extrapolationmethod in [271] where asymptotic expansions for the approximate solutionsobtained by the Nystrom scheme were developed to analyze the extrapolationmethod The generalized airfoil equation for an airfoil with a flap was solvednumerically in [204] In [49] it was shown that the dense coefficient matrixobtained from a quadrature rule for boundary integral equations with logarithmkernels can be replaced by a sparse one if appropriate graded meshes areused in the quadrature rules A fast numerical method was developed in [266]for solving the two-dimensional Fredholm integral equation of the secondkind More information about the Galerkin method using the Fourier basis forsolving boundary integral equations may be found in [25 81] Fast numericalalgorithms for this method were developed recently in [37 154 155 263]A singularity-preserving Galerkin method was developed in [41] for the
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36 Bibliographical remarks 143
Fredholm integral equation of the second kind with weakly singular kernelswhose solutions have singularity The method was extended in [38 229] tosolve the Volterra integral equation of the second kind with weakly singularkernels which was also used in [111 112] to solve fractional differentialequations
A singularity-preserving collocation method for solving the Fredholm inte-gral equation of the second kind with weakly singular kernels was developedin [39] In [16] a discretized Galerkin method was obtained using numericalintegration for evaluation of the integrals occurring in the Galerkin methodand in [23] by considering discrete inner product and discrete projectionsthe authors treated more appropriately kernels with discontinuous derivativesA discrete convergence theory and its applications to the numerical solutionof weakly singular integral equations were presented in [256] Finally weremark that it may be obtained by a similar analysis provided by [165] forthe superconvergence of the iterated Galerkin method when the kernels areweakly singular
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4
Multiscale basis functions
Since a large class of physical problems is defined on bounded domains wefocus on integral equations on bounded domains As we know a boundeddomain in Rd may be well approximated by a polygonal domain which isa union of simplexes cubes and perhaps L-shaped domains To develop fastGalerkin PetrovndashGalerkin and collocation methods for solving the integralequations we need multiscale bases and collocation functionals on polygonaldomains Simplexes cubes or L-shaped domains are typical examples ofinvariant sets This chapter is devoted to a description of constructions ofmultiscale basis functions including multiscale orthogonal bases interpolatingbases and multiscale collocation functionals The multiscale basis functionsthat we construct here are discontinuous piecewise polynomials For thisreason we describe their construction on invariant sets which can turn to baseson a polygon
To illustrate the idea of the construction we start with examples on[0 1] which is the simplest example of invariant sets This will be donein Section 41 Constructions of multiscale basis functions and collocationfunctionals on invariant sets are based on self-similar partitions of thesets Hence we discuss such partitions in Section 42 Based on such self-similar partitions we describe constructions of multiscale orthogonal basesin Section 43 For the construction of the multiscale interpolating basis werequire the availability of the multiscale interpolation points Section 44 isdevoted to the notion of refinable sets which are a base for the constructionof the multiscale interpolation points Finally in Section 45 we present theconstruction of multiscale interpolating bases
144
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41 Multiscale functions on the unit interval 145
41 Multiscale functions on the unit interval
This section serves as an illustration of the idea for the construction oforthogonal multiscale piecewise polynomial bases on an invariant set Weconsider the simplest invariant set = [0 1] in this section The essentialaspect of this construction is the recursive generation of partitions of andthe multiscale bases based on the partitions
Let L2() denote the Hilbert space equipped with inner product
(u v) =int
u(t)v(t)dt u v isin L2()
and the induced norm middot = radic(middot middot) We now describe a sequence of finite-dimensional subspaces of L2() For two positive integers k and m we let Sk
mdenote the linear space of all functions which are polynomials of degree atmost k minus 1 on
Im =[
m+ 1
m
] isin Zm
The functions in Skm are allowed to be discontinuous at the knots jm for j isin
Nmminus1 Hence the dimension of the space Skm is km When mprime divides m that
is m = mprime for some positive integer then
Skmprime sube Sk
m
since the knot sequence mprime isin Zmprime for the space Skmprime is contained in the
sequence m isin Zm for the space Skm In particular for m = 2k we have
that
Sk1 sube Sk
2 sube middot middot middot sube Sk2n (41)
In this context we reinterpret the unit interval and its partition Recall thatthe unit interval is the invariant set with respect to the maps
φε(t) = ε + t
2 t isin ε isin Z2
in the sense that
= φ0() cup φ1() and meas(φ0() cap φ1()) = 0
where meas(A) denotes the Lebesgue measure of the set A Note that themaps φ0 and φ1 are contractive and they map onto [0 12] and [12 1]respectively The partition I2k isin Z2k of can be re-expressed in termsof the contractive maps φε ε isin Z2 as
φε1middotmiddotmiddotεk() εj isin Z2
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146 Multiscale basis functions
0 1
f
0 1φ0()
T0 f
0 1φ1()
T1 f
Figure 41 φe and Te
Associated with the contractive maps φε ε isin Z2 we introduce two mutuallyorthogonal isometries on L2() that will be used to recursively generate basesfor the spaces in the chain (41) For each ε isin Z2 we set ε = [ε2 (ε+1)2]and define the isometry Tε by setting for f isin L2()
Tε f = radic2(
f φminus1ε
)χε =
radic2f (2t minus ε) t isin ε
0 t isin ε (42)
where χA denotes the characteristic function of the set A Figure 41 illustratesthe results of applications of operators Tε to a function
For each ε isin Z2 we use T lowastε for the adjoint operator of Tε We have thefollowing result concerning the adjoint operator T lowastε
Proposition 41 (1) If f isin L2() then
T lowastε f =radic
2
2f φε
(2) For any ε εprime isin Z2
T lowastε Tεprime = δεεprimeI (43)
(3) For any ε εprime isin Z2 and f g isin L2()
(Tε f Tεprimeg) = δεεprime( f g)
Proof (1) For f g isin L2() by the definition of Tε we have thatint
g(x)(Tε f )(x)dx = radic2intε
g(x)( f φminus1ε )(x)dx
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41 Multiscale functions on the unit interval 147
We make a change of variables t = φminus1ε (x) to conclude thatint
g(x)(Tε f )(x)dx =radic
2
2
int
(g φε)(x)f (x)dx
The formula for T lowastε g follows(2) For f isin L2() by (1) of this proposition we observe that
(T lowastεprime Tε f )(x) =radic
2
2(Tε f )(φεprime(x))
By the definition of the operator Tε if εprime = ε then T lowastεprime Tε = 0 and if εprime = ε
then T lowastεprime Tε = I
(3) The formula in this part follows directly from (43)
It is clear from their definitions that the operators Tε ε isin Z2 preserve thelinear independence of a set of functions in L2() Moreover it follows fromProposition 41 (2) that functions resulting from applications of the operatorsTε with different ε are orthogonal We next show how to use the operators Tε ε isin Z2 to generate recursively the bases for spaces
Xn = Sk2n n isin N0
To this end when S1 and S2 are subsets of L2() such that (u v) = 0 for allu isin S1 v isin S2 we introduce the notation S1 cupperp S2 which denotes the union ofS1 and S2
Proposition 42 If X0 is an orthonormal basis for Sk1 then
Xn =⋃εisinZ2
perpTεXnminus1 n isin N (44)
is an orthonormal basis for Sk2n
Proof We prove by induction on n Suppose that fj j isin Nk2nminus1 form anorthonormal basis for Xnminus1 By Proposition 41 Tε fj jisinNk2nminus1 ε isinZ2 arealso orthonormal It can be shown that these k2n functions are elements inXn Moreover dimXn = k2n which equals the number of these elementsTherefore they form an orthonormal basis for the space Xn
We now turn to the construction of our multiscale basis for space XnRecalling Xnminus1 sube Xn we have that
Xn = Xnminus1 oplusperpWn
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148 Multiscale basis functions
where Wn is the orthogonal complement of Xnminus1 in Xn Since the dimensionof Xn is k2n the dimension of Wn is given by
dimWn = k2nminus1
Repeating this process leads to the decomposition
Xn = X0 oplusperpW1 oplusperp middot middot middot oplusperpWn (45)
for space Xn In order to construct a multiscale orthonormal basis it sufficesto construct an orthonormal basis Wj for space Wj for each j isin Nn We firstchoose the Legendre polynomials of degree le k minus 1 on as an orthonormalbasis for X0 = Sk
1 and denote by X0 the basis We then use Proposition 42 toconstruct an orthonormal basis X1 for space X1 that is
X1 =⋃εisinZ2
perpTεX0
Since both X0 and X1 are finite-dimensional we can use the GramndashSchmidtprocess to find an orthonormal basis W1 for W1 Specifically we form a linearcombination of the basis functions in X1 and require it to be orthogonal toall elements of X0 This gives us k linearly independent elements which areorthogonal to X0 We then orthonormalize these k functions and they serve asan orthonormal basis for W1 For construction of basis Wj when j ge 2 weappeal to the following proposition
Proposition 43 If W1 is given as an orthonormal basis for W1 then
Wn+1 =⋃εisinZ2
perpTεWn n isin N (46)
is an orthonormal basis for Wn+1
Proof We prove that Wn is an orthonormal basis for Wn by induction on nWhen n = 1 W1 is an orthonormal basis for W1 by hypothesis Assume thatWj is an orthonormal basis for Wj for some j ge 1 we show that Wj+1 is anorthonormal basis for Wj+1
Let W = T0Wj cup T1Wj Since Wj sube Xj by Proposition 42 we concludethat W sube Xj+1 Proposition 41 with the induction hypothesis that WjperpXjminus1ensures Wperp(T0Xjminus1 cup T1Xjminus1) = Xj which implies that W subeWj+1 Becausethe elements in Wj are orthonormal by Proposition 41 the elements in W arealso orthonormal Moreover
card W = dimWj+1
holds Therefore W is a basis for Wj+1
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41 Multiscale functions on the unit interval 149
The proposition above gives a recursive generation of the multiscale basisfunctions for spaces Wn once orthonormal basis functions for W1 areavailable It is useful for us in what follows to index the functions in the waveletbases for Xn and to clearly have in mind the interval of their ldquosupportrdquo To thisend we set W0 = X0 and define
w(n) = dimWn and s(n) = dimXn n isin N0
Thus we have that
w(0) = k w(n) = k2nminus1 and s(n) = k2n n isin N0
For i isin N0 we write Wi = wij j isin Zw(i) where we use double subscriptsfor the basis functions with the first representing the level of the scale of thesubspaces and the second indicating the location of its support There are twoproperties of the functions in the set wij (i j) isin U where U = (i j) i isinN0 j isin Zw(i) which are important to us The first is that they form a completeorthonormal system for the space L2() In particular we have that
(wij wiprimejprime) = δiiprimeδjjprime (i j) (iprime jprime) isin U
Embodied in this fact is the useful property that the wavelet basis wij (i j) isinU has vanishing moments of order k that is
((middot)r wij) = 0 for r isin Zk j isin Zw(i) i isin N
The second property is the ldquoshrinking supportrdquo (as the level i increases)of the multiscale basis functions To pin down this fact we take the point ofview that the k functions in W1 have ldquosupportrdquo on Thereafter the waveletbasis at level i will be grouped into 2iminus1 sets of k functions each having thesame ldquosupport intervalrdquo For future reference we identify a set off which wij
vanishes For this purpose we write j isin Zw(i) i isin N uniquely in the formj = vk + l where l isin Zk and v isin Z2iminus1 i isin N Then
wij(t) = 0 t isin Iv2iminus1 j = vk + l (47)
and therefore we have for j isin Zw(i) that
meas(supp wij) le 1
2iminus1
To see this fact clearly we express v in its dyadic expansion
v = 2iminus2ε1 + middot middot middot + 2εiminus2 + εiminus1
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150 Multiscale basis functions
where ε1 ε2 εiminus1 isin Z2 The recursion (46) then gives the formula
wij = Tε1 middot middot middot Tεiminus1w1l
which confirms (47)We end this section by presenting bases for spaces X0 and W1 for four con-
crete examples piecewise constant linear quadratic and cubic polynomials
Piecewise constant functions This case leads to the Haar wavelet We have abasis for X0 given by
w00(t) = 1 t isin [0 1]and a basis for W1 given by (see also Figure 42)
w10(t) =
1 t isin [0 12]minus1 t isin (12 1]
We illustrate in Figure 42 the graph of the functions w00 and w10
Piecewise linear polynomials In this case k = 2 and dimX0 = dimW1 = 2We have an orthonormal basis for X0 given by
w00(t) = 1 w01(t) = radic3(2t minus 1)
and an orthonormal basis for W1 given by
w10(t) =
1minus 6t t isin [0 12 ]
5minus 6t t isin ( 12 1] w11(t) =
radic3(1minus 4t) t isin [0 1
2 ]radic3(4t minus 3) t isin ( 1
2 1]We illustrate in Figure 43 the graph of the functions w00 w01 w10 and w11
0 02 04 06 08 10
02
04
06
08
1
w00
0 02 04 06 08 1
minus1
minus05
0
05
1
w10
Figure 42 Basis functions for piecewise constant functions
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41 Multiscale functions on the unit interval 151
0 02 04 06 08 10
02
04
06
08
1
w00
0 02 04 06 08 1minus2
minus1
0
1
2
w01
0 02 04 06 08 1
minus2
minus1
0
1
2
w10
0 02 04 06 08 1minus2
minus1
0
1
2
w11
Figure 43 Basis functions for piecewise linear polynomials
Piecewise quadratic polynomials In this case k= 3 and dimX0= dimW1= 3 An orthonormal basis for X0 is given by
w00(t) = 1 w01(t) = radic3(2t minus 1) w02(t) = radic5(6t2 minus 6t + 1)
and an orthonormal basis for W1 is given by
w10(t) =
1minus 6t t isin [0 12 ]
5minus 6t t isin ( 12 1]
w11(t) =⎧⎨⎩radic
9331 (240t2 minus 116t + 9) t isin [0 1
2 ]radic93
31 (3minus 4t) t isin ( 12 1]
w12(t) =⎧⎨⎩radic
9331 (4t minus 1) t isin [0 1
2 ]radic93
31 (240t2 minus 364t + 133) t isin ( 12 1]
In Figure 44 we illustrate the graph of the bases for X0 and W0 in this case
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152 Multiscale basis functions
0 05 10
02
04
06
08
1
w00
0 05 1minus2
minus1
0
1
2
w01
0 05 1minus15
minus1
minus05
0
05
1
15
2
25
w02
0 05 1minus2
minus1
0
1
2
w10
0 05 1minus2
minus1
0
1
2
3
4
w11
0 05 1minus2
minus1
0
1
2
3
4
w12
Figure 44 Basis functions for piecewise quadratic polynomials
Piecewise cubic polynomials In this case we have that k = 4 and dimX0 =dimW1 = 4 An orthonormal basis for X0 is given by
w00(t) = 1 w01(t) = radic3(2t minus 1)
w02(t) = radic5(6t2 minus 6t + 1) w03(t) = radic7(20t3 minus 30t2 + 12t minus 1)
and a basis for W1 is given by
w10(t) = radic
515 (240t2 minus 90t + 5) t isin [0 1
2 ]minusradic
515 (240t2 minus 390t + 155) t isin ( 1
2 1]
w11(t) = radic
3(30t2 minus 14t + 1) t isin [0 12 ]radic
3(30t2 minus 46t + 17) t isin ( 12 1]
w12(t) = radic
7(160t3 minus 120t2 + 24t minus 1) t isin [0 12 ]
minusradic7(160t3 minus 360t2 + 264t minus 63) t isin ( 12 1]
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42 Multiscale partitions 153
0 02 04 06 08 10
02
04
06
08
1
w00
0 02 04 06 08 1minus2
minus1
0
1
2
w01
0 02 04 06 08 1minus15
minus1
minus05
0
05
1
15
2
25
w02
0 02 04 06 08 1minus3
minus2
minus1
0
1
2
3
w03
Figure 45 Basis functions for piecewise cubic polynomials
w13(t) =⎧⎨⎩
14radic
2929 (160t3 minus 120t2 + 165
7 t minus 1314 ) t isin [0 1
2 ]14radic
2929 (160t3 minus 360t2 + 1845
7 t minus 87714 ) t isin ( 1
2 1]The bases for X0 and W1 are shown respectively in Figures 45 and 46
42 Multiscale partitions
Because a polygonal domain in Rd is a union of a finite number of invariantsets in this section we focus on multiscale partitioning of an invariant set in Rd
421 Invariant sets
We introduce the notion of invariant sets following [148] Let M be a completemetric space For any subset A of M and x isinM we define the distance of x to
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154 Multiscale basis functions
0 02 04 06 08 1
minus3
minus2
minus1
0
1
2
3
w10
0 02 04 06 08 1minus2
minus1
0
1
2
3
w11
0 02 04 06 08 1minus3
minus2
minus1
0
1
2
3
w12
0 02 04 06 08 1minus3
minus2
minus1
0
1
2
3
w13
Figure 46 Basis functions for piecewise cubic polynomials
A and the diameter of A respectively by
dist (x A) = infd(x y) y isin Aand
diam (A) = supd(x y) x y isin AA mapping from M to M is called contractive if there exists a γ isin (0 1) suchthat for all subsets A of M
diam (φε(A)) le γ diam (A) ε isin Zμ (48)
For a positive integer μ gt 1 we suppose that = φε ε isin Zμ is a familyof contractive mappings on M We define the subset of M by
(A) =⋃εisinZμ
φε(A)
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42 Multiscale partitions 155
According to [148] there exists a unique compact subset of M such that
() = (49)
We call the set the invariant set relative to the family of contractivemappings
Generally an invariant set has a complex fractal structure For examplethere are choices of for which is the Cantor subset of the interval [0 1]the Sierpinski gasket contained in an equilateral triangle or the twin dragonsfrom wavelet analysis In Figures 47 and 48 we illustrate the generation ofthe Cantor set of [0 1] and the Sierpinski gasket respectively
In the context of numerical solutions of integral equations we are interestedin the cases when has a simple structure including for example the cubeand simplex in Rd With these cases in mind we make the following additionalrestrictions on the family of mappings
Figure 47 Generation of the Cantor set
Figure 48 Generation of the Sierpinski gasket
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156 Multiscale basis functions
(a) For every ε isin Zμ the mapping φε has a continuous inverse on (b) The set has nonempty interior and
meas (φε() cap φεprime()) = 0 ε εprime isin Zμ ε = εprime
We present several simple examples of invariant sets
Example 44 For the metric space R and an integer μ gt 1 consider thefamily of contractive mappings = φε ε isin Zμ where
φε(t) = t + ε
μ t isin R ε isin Zμ
The unit interval = [0 1] is the invariant set relative to which satisfies
=⋃εisinZμ
φε()
When μ= 2 this example is discussed in Section 41 Figure 49 illustrates thecase when μ = 3 Note that in this case
φ0() =[
11
3
] φ1() =
[1
3
2
3
] φ2() =
[2
3 1
]and clearly
[0 1] = φ0() cup φ1() cup φ2()
Example 45 In the metric space R2 we consider four contractive affinemappings
φ0(s t) = 1
2(s t) φ1(s t) = 1
2(s+ 1 t)
φ2(s t) = 1
2(s t + 1) φ3(s t) = 1
2(1minus s 1minus t) (s t) isin R2
The invariant set relative to these mappings is the unit triangle with verticesat (0 0) (1 0) and (0 1) since
= φ0() cup φ1() cup φ2() cup φ3()
This is illustrated in Figure 410
0 1φ0() φ1() φ2()
Figure 49 The invariant set in Example 1 with μ = 3
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42 Multiscale partitions 157
(0 0) (1 0)
(0 1)
φ0() φ1()
φ2()
φ3()
Figure 410 The unit triangle as an invariant set
Figure 411 The unit L-shaped domain as an invariant set
Example 46 In the metric space R2 we consider four contractive affinemappings
φ0(s t) = 1
2(s t) φ1(s t) = 1
2(2minus s t)
φ2(s t) = 1
2(s 2minus t) φ3(s t) = 1
2(s+ 12 t + 12) (s t) isin R2
The invariant set relative to these mappings is the L-shaped domain illustratedin Figure 411
Example 47 As the last example in the metric space R3 we consider eightcontractive affine mappings
φ0(x y z) = 1
2(x y z) φ1(x y z) = 1
2(y z x+ 1)
φ2(x y z) = 1
2(x z y+ 1) φ3(x y z) = 1
2(x y z+ 1)
φ4(x y z) = 1
2(x y+ 1 z+ 1) φ5(x y z) = 1
2(y x+ 1 z+ 1)
φ6(x y z) = 1
2(z x+ 1 y+ 1) φ7(x y z) = 1
2(x+ 1 y+ 1 z+ 1)
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158 Multiscale basis functions
0 02 04 06 08 1
002
0406
081
0
01
02
03
04
05
06
07
08
09
1
S
0 02 04 06 08 1
0
05
10
01
02
03
04
05
06
07
08
09
1
S[000]
0 02 04 06 081
0
05
10
01
02
03
04
05
06
07
08
09
1
S[100]
S S(000) S(100)
0 02 04 06 08 1
002
0406
0810
01
02
03
04
05
06
07
08
09
1
S[010]
0 02 04 0608 1
002
0406
0810
01
02
03
04
05
06
07
08
09
1
S[001]
0 02 04 06 08 1
002
0406
0810
01
02
03
04
05
06
07
08
09
1
S[011]
S(010) S(001) S(011)
0 02 04 06 08 1
002
0406
0810
01
02
03
04
05
06
07
08
09
1
S[101]
0 02 04 06 08 1
0
05
10
01
02
03
04
05
06
07
08
09
1
S[110]
0 02 04 06 08 1
0
05
10
01
02
03
04
05
06
07
08
09
1
S[111]
S(101) S(110) S(111)
Figure 412 A three-dimensional unit simplex as an invariant set
The invariant set relative to these eight mappings is the simplex in R3
defined by
S = (x y z) 0 le x le y le z le 1This is illustrated in Figure 412
422 Multiscale partitions by contractive mappings
The contractive mappings that define the invariant set naturally forma partition for the invariant set Repeatedly applying the mappings to theinvariant set generates a sequence of multiscale partitions for the invariant set
We next show how the contractive mappings are used to generate asequence of multiscale partitions n n isin N0 of the invariant set which
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42 Multiscale partitions 159
is defined by For notational convenience we introduce the notation
Znμ = Zμ times middot middot middot times Zμ n times
For each e = [ej j isin Zn] isin Znμ we define the composition mapping
φe = φe0 φe1 middot middot middot φenminus1
and the number
μ(e) = μnminus1e0 + middot middot middot + μenminus2 + enminus1
Note that every i isin Zμn can be written uniquely as i = μ(e) for some e isin Znμ
From equation (49) and conditions (a) and (b) it follows that the collection ofsets
n = ne ne = φe() e isin Znμ (410)
forms a partition of We require that this partition has the following property
(c) There exist positive constants cminus c+ such that for all n isin N0
cminusμminusnd le maxd(ne) e isin Znμ le c+μminusnd (411)
where d(A) represents the diameter of set A that is d(A) = sup|xminusy| x y isinA with | middot | being the Euclidean norm in the space Rd
If a sequence of partitions n n isin N0 has property (c) we say that it formsa sequence of multiscale partitions for
Proposition 48 If the Jacobian of the contractive affine mappings φe e isinZμ satisfies
|Jφe | = O(μminus1)
then the sequence of partitions n n isin N0 is of multiscale
Proof For any s t isin φe() there exist s t isin such that s = φe(s) andt = φe(t) and thus we have that
|sminus t| = |Jφe |1d|sminus t|This with the hypothesis on the Jacobian of the mappings ensures that for anye isin Zμ
d(1e) = O(μminus1d)
By induction we may find that for any e isin Znμ
d(ne) = O(μminusnd) (412)
proving the result
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160 Multiscale basis functions
423 Multiscale partitions of a multidimensional simplex
For the purpose of solving integral equations on a polygonal domain in Rd wedescribe in this subsection multiscale partitions of a simplex in Rd for d ge 1For a vector x isin Rd we write x = [xj xj isin R j isin Zd] The unit simplex S inRd is the subset
S = x x isin Rd 0 le x0 le x1 le middot middot middot le xdminus1 le 1This set is the invariant set relative to a family of μd contractive mappingsIn order to describe these contractive mappings for a positive integer μ wedefine a family of counting functions χj Zd
μ rarr Zd+1 j isin Zμ for e = [ej j isin Zd] isin Zd
μ by
χj(e) =sumiisinZd
δj(ei) (413)
where δj(k) = 1 when j = k and otherwise δj(k) = 0 Note that the value ofχj(e) is exactly the number of components of e that equals j Given e isin Zd
μ
we identify a vector c(e) = [cj j isin Zμ+1] isin Zμ+1d+1 by
c0 = 0 cj =sumiisinZj
χi(e) j isin Nμ (414)
We remark that c(e) is always nondecreasing since each χj takes a non-negativevalue and cμ is always equal to d For e isin Zd
μ and j lt k we define the index
set kj = el j le l lt k el = ek Then we define the permutation vector
Ie = [ik k isin Zd] isin Zdd of e by
ik = cek + card(k0) (415)
where we assume card(empty) = 0 We have the following lemma about Ie
Lemma 49 For any e isin Zdμ the permutation vector Ie has the following
properties
(1) For k isin Zd cm le ik lt cm+1 if and only if m = ek(2) For any j k isin Zd ij lt ik if and only if ej lt ek or ej = ek with j lt k(3) The equality ij = ik holds if and only if j = k(4) The vector Ie is a permutation of vd = [j j isin Zd]Proof According to the definition of Ie we have for any k isin Zk that
cek le ik lt cek + card(ej j isin Zd ej = ek) = cek+1 (416)
This implies that if m = ek cm le ik lt cm+1 On the contrary if there is anm such that cm le ik lt cm+1 it is unique because the components of c(e) are
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42 Multiscale partitions 161
nondecreasing It follows from the uniqueness of m and (416) that m = ekThus property (1) is proved
We now turn to proving property (2) If ej lt ek then ej + 1 le ek and hencecej+1 le cek since the components of c(e) form a nondecreasing sequence By(416) we conclude that ij lt cej+1 le cek le ik If ej= ek with jlt k then ik minusij= card(k
j )ge 1 hence ij lt ik It remains to prove that if ij lt ik then ej lt ek
or ej= ek jlt k Since in general for j kisinZd one of the following cases holdsej lt ek ej= ek with jlt k ej= ek with jge k or ej gt ek it suffices to show thatif ej gt ek or ej= ek with jge k then ijge ik If ej gt ek by the proof we showedearlier in this paragraph we conclude that ij gt ik If ej= ek with j ge k we
have that ij minus ik= card( jk)ge 0 that is ijge ik Thus we complete a proof for
property (2)The above analysis also implies that the only possibility to have ij = ik is
j = k This proves property (3)Noticing that ek isin Zμ for k isin Zd and 0 le cek le ik lt cek+1 le d we
conclude that Ie is a permutation of vd
We also need conjugate permutations in order to define the contractivemappings A permutation matrix has exactly one entry in each row and columnequal to one and all other entries zero Hence a permutation matrix is anorthogonal matrix For any permutation Ie of vd there is a unique permutationmatrix Pe such that Ie = Pevd We call the vector
Ilowaste = [ilowastj j isin Zd] = PTe vd
the conjugate permutation of Ie Thus Ilowaste itself is also a permutation of vdIt follows from the definition above that for l isin Zd ilowastl = k if and only ifik = l We define the conjugate vector elowast = [elowastj j isin Zd] of e by settingelowastl = eilowastl l isin Zd Utilizing the above notations we define the mapping Ge by
Ge(x) = x =[
xl =xilowastl + elowastl
μ l isin Zd
] x isin S (417)
It is clear that mappings Ge e isin Zdμ are affine and contractive
We next identify the set Ge(S) To this end associated with each e isin Zdμ we
define a set in Rd by
Se =
x isin Rd 0 le xi0 minuse0
μle xi1 minus
e1
μle middot middot middot le xidminus1 minus
edminus1
μle 1
μ
(418)
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162 Multiscale basis functions
where ik k isin Zd are the components of the permutation vector Ie of e SinceIe is a permutation of vd Se is a simplex in Rd In the next lemma we identifyGe(S) with the simplex Se
Lemma 410 For all e isin Zdμ there holds Ge(S) = Se
Proof For k isin Zd we let l = ik and observe by definition that ilowastl = k elowastl = ekThus xl = xk+ek
μ or
xk = μxl minus ek = μxik minus ek (419)
If x isin S then 0 le x0 le x1 le middot middot middot le xdminus1 le 1 which implies that
0 le μxi0 minus e0 le μxi1 minus e1 le middot middot middot le μxidminus1 minus edminus1 le 1
or
0 le xi0 minuse0
μle xi1 minus
e1
μle middot middot middot le xidminus1 minus
edminus1
μle 1
μ
so that x isin Se Moreover given x isin Se we define x = [xk k isin Zd] byequation (419) Thus x isin S and x = Ge(x) Therefore Ge(S) = Se
In the following lemma we present properties of the simplices Se e isin Zdμ
Lemma 411 The simplices Se e isin Zdμ have the following properties
(1) For any x isin Se there holds
k
μle xck le xck+1 le middot middot middot le xck+1minus1 le k + 1
μ k isin Zμ (420)
(2) For any e isin Zdμ Se sub S
(3) If e1 e2 isin Zdμ with e1 = e2 then int(Se1) cap int(Se2) = empty
(4) For any e isin Zdμ meas(Se) = 1(μdd) where meas() denotes the
Lebesgue measure of set
Proof In order to prove (420) it suffices to show
0 le xck minusk
μle xck+1 minus k
μle middot middot middot le xck+1minus1 minus k
μle 1
μ k isin Zμ (421)
or equivalently
0 le xp minus k
μle xq minus k
μle 1
μ
for any ck le p lt q lt ck+1 In fact since Ie is a permutation of vd for anyintegers ck le p lt q lt ck+1 there exists a unique pair pprime qprime isin Zd such that
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42 Multiscale partitions 163
ipprime = p iqprime = q It follows from Lemma 49 that epprime = eqprime = k and pprime lt qprimeThus (418) states that
0 le xp minus k
μ= xipprime minus
epprime
μle xiqprime minus
eqprime
μ= xq minus k
μle 1
μ
which concludes property (1)Property (2) is a direct consequence of (1) and the definition of SFor the proof of (3) we first notice that
int(Se) =
x isin Rd 0 lt xi0 minuse0
μlt xi1 minus
e1
μlt middot middot middot lt xidminus1 minus
edminus1
μlt
1
μ
(422)
Moreover by a proof similar to that for (420) we utilize (422) to concludefor any x isin int(Se) that
k
μlt xck lt xck+1 lt middot middot middot lt xck+1minus1 lt
k + 1
μ k isin Zμ (423)
For j = 1 2 we let ej = [ejk k isin Zd] Iej = [ijk k isin Zd] c(ej) = [cj
k k isinZμ+1]
Assume to the contrary that int(Se1)cap int(Se2) is not empty We consider twocases In case 1 that c(e1) = c(e2) we let k be the smallest integer such thatc1
k = c2k and assume c1
k lt c2k without loss of generality For any x isin int(Se1)cap
int(Se2) by (423) we have xc1kgt k
μand xc2
kminus1 ltkμ
Moreover because x isin Swe have that xc1
kle xc2
kminus1 a contradiction In case 2 that c(e1) = c(e2) since
e1 = e2 we let k be the smallest integer such that e1k = e2
k Hence e1j = e2
j for
j lt k and we assume that e1k lt e2
k without loss of generality Thus we havethat i1k lt c1
e1k+1le c2
e2kle i2k There exists a unique p isin Zd such that i1p = i2k since
Ie1 is a permutation and p ge k because i1j = i2j = i2k for all j lt k Furthermore
it follows from Lemma 49 c(e1) = c(e2) and i1p = i2k that e1p = e2
k = e1k
which implies p = k Therefore for any x isin int(Se1) there holds
xi1kminus e1
k
μlt xi1p
minus e1p
μ= xi2k
minus e2k
μ
However there is a unique q isin Zd such that q gt k i2q = i1k and for anyx isin int(Se2)
xi2kminus e2
k
μlt xi2q
minus e2q
μ= xi1k
minus e1k
μ
again a contradiction This completes the proof of property (3)
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164 Multiscale basis functions
For property (4) we find by direct computation that meas(Sprimee) = 1(μdd)where
Sprimee =
x isin Rd 0 le xi0 le xi1 le middot middot middot le xidminus1 le1
μ
Notice that Se is the translation of simplex Sprimee via the vector eμ
Sincethe Lebesgue measure of a set is invariant under translation we concludeproperty (4)
Theorem 412 The family S(Zdμ) = Se e isin Zd
μ is an equivolume partitionof the unit simplex S
Proof By Lemma 411 we see that for any e isin Zdμ Se sub S and for e1 e2 isin Zd
μ
with e1 = e2 int(Se1) cap int(Se2) = empty and meas(Se1) = meas(Se2) It remainsto prove that S sube⋃eisinZd
μSe
To this end for each x isin S we find e isin Zdμ such that x isin Se Note that for
each x isin S we have that 0 le x0 le x1 le middot middot middot le xdminus1 le 1 For each k isin Zμ wedenote by ck the subscript of the smallest component xj greater than or equalto k
μ We order the elements in set xj j isin Zd cup k
μ k isin Zμ+1 in increasing
order We then obtain that
0 le x0 le middot middot middot le xc1minus1 lt1
μle xc1 le middot middot middot le xcμminus1minus1
ltμminus 1
μle xcμminus1 le middot middot middot le xcμminus1 = xdminus1 le 1
In other words we have that
0 le xck minusk
μle xck+1 minus k
μle middot middot middot le xck+1minus1 minus k
μle 1
μ k isin Zμ (424)
Let pj = maxk ck le j It follows from (424) that the set xj minus pjμ
j isinZd sub [0 1
μ] We sort the elements of this set into
0 le xi0 minuspi0
μle xi1 minus
pi1
μle middot middot middot le xidminus1 minus
pidminus1
μle 1
μ (425)
Notice that the vector I = [ik k isin Zd] is a permutation of vd Let e = [ek k isin Zd] be a vector such that ej = pij It is easy to verify ij = cej + | j
0|Hence I = Ie which together with (425) shows x isin Se
The expression of the inverse mapping Gminus1e has been given by equa-
tion (419) which is written formally as
x = Gminus1e (x) = [xk = μxik minus ek k isin Zd] x isin Se (426)
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42 Multiscale partitions 165
For any e isin Zdμ and xprime xprimeprime isin Rd
Ge(xprime)minus Ge(xprimeprime)p = 1
μxprime minus xprimeprimep (427)
and
Gminus1e (xprime)minus Gminus1
e (xprimeprime)p = μxprime minus xprimeprimep (428)
hold where middot p is the standard p-norm on Rd for 1 le p le infin
Proposition 413 The family S(Zdμ) is a uniform partition of the unit simplex
S in the sense that all elements of S(Zdμ) have an identical diameter
Proof We let = maxxprimexprimeprimeisinS xprime minus xprimeprimep It suffices to prove for any e isin Zdμ
that
maxxprimexprimeprimeisinSe
xprimee minus xprimeprimeep =
μ
It follows from formula (428) that for any xprimee xprimeprimee isin Se
μxprimee minus xprimeprimeep = Gminus1e (xprimee)minus Gminus1
e (xprimeprimee )p le
Moreover suppose that xprime xprimeprime isin S such that xprime minus xprimeprimep = and let xprimee =Ge(xprime) and xprimeprimee = Ge(xprimeprime) By (427) we have that
xprimee minus xprimeprimeep =1
μxprime minus xprimeprimep =
μ
which completes the proof
When a partition of the unit simplex has been established it is notdifficult to obtain a corresponding partition of a general simplex in Rd Fora nondegenerate simplex Sprime in Rd in the sense Vol(Sprime) = 0 there exists anaffine mapping F Rd rarr Rd such that F(Sprime) = S It can be shown that for1 le p le infin there are two positive constants c1 and c2 such that
c1xprime minus xprimeprimep le F(xprime)minus F(xprimeprime)p le c2xprime minus xprimeprimep (429)
for any xprime xprimeprime isin Sprime For any e isin Zdμ we define Gprimee = Fminus1 Ge F Thus the
family of simplices Gprimee(Sprime) e isin Zdμ is a partition of Sprime Furthermore for any
xprime xprimeprime isin Rd and e isin Zdμ
c1
c2μxprime minus xprimeprimep le Gprimee(xprime)minus Gprimee(xprimeprime)p le
c2
c1μxprime minus xprimeprimep
holds For E = [ej j isin Zm] isin (Zdμ)
m we define the composite mappings
GE = Ge0 middot middot middot Gemminus1 and GprimeE = Gprimee0 middot middot middot Gprimeemminus1
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166 Multiscale basis functions
and observe that GprimeE = Fminus1 GE F In the next theorem we show that thepartition Gprimee(Sprime) e isin Zd
μ of Sprime is uniform To this end we let SE = GE(S)and SprimeE = GprimeE(Sprime) Also we use diamp to denote the diameter of a domain inRd with respect to the p-norm
Theorem 414 For any xprime xprimeprime isin Rd and E isin (Zdμ)
m there hold
c1
c2
(1
μ
)m
xprime minus xprimeprimep le GprimeE(xprime)minus GprimeE(xprimeprime)p lec2
c1
(1
μ
)m
xprime minus xprimeprimep
diamp(SE) =(
1
μ
)m
diamp(S)
and
c1
c2
(1
μ
)m
diamp(Sprime) le diamp(S
primeE) le
c2
c1
(1
μ
)m
diamp(Sprime)
43 Multiscale orthogonal bases
In this section we describe the recursive construction of multiscale orthogonalbases for spaces L2() on the invariant set
431 Piecewise polynomial spaces
On the partition n we consider piecewise polynomials in a Banach space X
which has the norm middot Choose a positive integer k and let Xn be the spaces ofall functions such that their restriction to any cell ne e isin Zn
μ is a polynomialof total degree le kminus 1 Here we use the convention that for n = 0 the set isthe only cell in the partition and so
m = dim X0 =(
k + d minus 1d
)
It is easily seen that
x(n) = dimXn = mμn n isin N0
To generate the spaces Xn by induction from X0 we introduce linearoperators Te Xrarr X e isin Zμ defined by
(Tev)(t) = c0v(φminus1e (t))χφe()(t) (430)
where χA denotes the characteristic function of set A and c0 the positiveconstant such that Te = 1 Thus we have that
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43 Multiscale orthogonal bases 167
Xn =opluseisinZμ
TeXnminus1 n isin N (431)
where A oplus B denotes the direct sum of the spaces A and B It is easily seenthat the sequence of spaces has the property of nestedness that is
Xnminus1 sub Xn n isin N (432)
Assume that there exists a basis of elements in X0 denoted byψ0ψ1 ψmminus1 such that
X0 = spanψj j isin ZmIt is clear that
Xn = spanTeψj j isin Zm e isin Znμ (433)
432 A recursive construction
Noting that the subspace sequence Xn n isin N0 is nested we can define foreach n isin N0 subspaces Wn+1 sub Xn+1 such that
Xn+1 = Xn oplusperpWn+1 n isin N0 (434)
Thus by setting W0 = X0 we have the multiscale space decomposition thatfor any n isin N
Xn =oplus
iisinZn+1
perpWi (435)
and
L2() =oplusiisinN0
perpWi (436)
It can be computed that the dimension of Wn is given by
w(n) = dimWn = x(n)minus x(nminus 1) = m(μminus 1)μnminus1 n isin N (437)
Now the family of operators Te L2()rarr L2() e isin Zμ is defined by
Tev = |Jφe |minus12 v φminus1
e χφe()
In the next proposition we see that the operators Te e isin Zμ are isometricsfrom L2() to L2()
Proposition 415 If e eprime isin Zμ then for all u v isin L2()
(Teu Teprimev) = δeeprime(u v)
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168 Multiscale basis functions
Proof When e = eprime the intersection of the support of Teu and that of Teprimev hasmeasure zero Hence in this case we have that
(Teu Teprimev) = 0
Now we consider the case e = eprime By the definition of the operator Te we havethat
(Teu Teprimev) = |Jφe |minus1intφe()
(u φminus1e )(t)(v φminus1
e )(t)dt
Using a change of variable we conclude that
(Teu Teprimev) =int
u(t)v(t)dt = (u v)
which completes the proof
The following proposition shows that the spaces Wn can be generatedrecursively by W1
Proposition 416 It holds that
Wn+1 =opluseisinZμ
perpTeWn n isin N
Proof We remark first that by Proposition 415 the direct sum above satisfiesthe orthogonal property It follows from (434) that
Wn sub Xn and WnperpXnminus1
Thus by (431) and Proposition 415 we conclude that
TeWn sub Xn+1 and TeWnperpXn
These relations with (434) ensure that for any e isin Zμ
TeWn subWn+1
Since from (437) we have that
dimopluseisinZμ
perpTeWn = μ dim Wn = m(μminus 1)μn = dim Wn+1
the result of this proposition holds
It can easily be seen from the above proposition and its proof that thefollowing proposition holds
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44 Refinable sets and set wavelets 169
Proposition 417 If W1 is a basis of space W1 then for n isin N the recursivelygenerated set
Wn+1 =⋃
eisinZμ
perpTeWn =
⋃eisinZn
μ
perpTeW1
is the basis of space Wn+1
Now it is clear that as soon as we choose an orthogonal basis of W0 denotedby w0j j isin Zm and get an orthogonal basis of W1 denoted by w1j j isin Zrwhere r = w(1) then we can generate recursively an orthogonal basis wij j isin Zw(i) of the space Wi by
wij = Tew1l j = μ(e)r + l e isin Ziminus1μ l isin Zr iminus 1 isin N (438)
Functions wij (i j) isin U are wavelet-like functions which are also calledorthogonal wavelets
44 Refinable sets and set wavelets
In Section 43 we showed how to construct multiscale orthonormal bases oninvariant sets These wavelet-like functions are discontinuous nonetheless theyhave important applications to the numerical solution of integral equations Inthe next section we explore similar recursive structures for multiscale functionrepresentation and approximation by focusing on the analogous situation forinterpolation on an invariant set Thus in this section we seek a mechanism togenerate sequences of points which have a multiscale structure that can then beused to efficiently generate interpolating functions and multiscale functionals
We first develop a notion of refinable sets give a complete characterizationof refinable sets in a general metric space setting and illustrate the generalcharacterization with several examples of practical importance Next we showhow refinable sets lead to a multiresolution structure relative to set inclusionwhich is analogous to the multiresolution analysis associated with refinablefunctions This set-theoretic multiresolution structure leads us to what we callset wavelets which are generated by a successive application of the contractivemappings to an initial set wavelet These will lead us in particular in Section45 to the construction of interpolation that has the desired multiscale structureand in Chapter 7 to the construction of multiscale functionals for developingfast multiscale collocation methods for solving integral equations
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170 Multiscale basis functions
441 Refinable sets
This subsection is devoted to a study of the refinable set relative to a familyof contractive mappings A complete characterization of refinable sets willbe presented which will be illustrated by several examples of practicalimportance
Assume that X is a complete metric space = φε ε isin Zμ is a familyof contractive mappings on X and is the invariant set relative to the family of mappings
For a subset V sub X we let
(V) =⋃εisinZμ
φε(V)
Definition 418 A subset V of X is said to be refinable relative to themappings if V sube (V)
For every k isin N and ek = [εj j isin Zk] isin Zkμ we define the contractive
composition mapping φek = φε0φε1middot middot middotφεkminus1 and letk = φek ek isin Zkμ
in particular1 = Observe that the union of any collection of refinable setsis likewise refinable Moreover if V is a refinable subset of X then k(V) =cupekisinZk
μφek(V) is also a refinable subset of X for all k isin N One of our main
objectives is to identify refinable sets of finite cardinalityBefore we present a characterization of these sets we look at some examples
on the real line which will be helpful to illuminate the general resultFor the metric space R and integer μ gt 1 we consider the mappings
ψε(t) = t + ε
μ t isin R ε isin Zμ (439)
The invariant set for this family of mappings is the unit interval [0 1] Our firstexample of a refinable set relative to the family of mappings = ψε ε isinZμ given in (439) comes next
Proposition 419 The set U0 =
jk jisinZk+1
is refinable relative to the
mappings
Proof It is sufficient to consider j gt 0 since 0 = ψ0(0) For every j isin Zk+1
we write the integer μj uniquely in the form μj = kε + where minus 1 isin Zk
and ε isin N0 Since μj le μk we conclude that ε isin Zμ Moreover we have thatjk = ψε
(k
)and so U0 is refinable
In some applications the exclusion of the endpoints 0 and 1 from a refinableset is desirable As an example of this case we present the following fact
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44 Refinable sets and set wavelets 171
Proposition 420 The set U0 =
jk+1 jminus 1 isin Zk
is refinable relative to
the mappings if and only if μ and k + 1 are relatively prime
Proof Suppose that μ and k + 1 have a common multiple mgt 1 that isμ=m1 and k+1=m2 for some integers 1 and 2 and U0 is refinable relativeto the mappings Then we have that 2 minus 1isinZk and 1 isinZμ Moreover wehave that ψ1(0)= 1
μ= 2
k+1 This equation implies that ψ1(0)isinU0 SinceU0 is refinable there exist ε0 isinZμ and u isin U0 such that ψ1(0)=ψε0(u)It follows from the equation above that 1= u + ε0 Thus either 1= ε0 andu= 0 or ε0 + 1= 1 and u= 1 In either case we conclude that either 0 or 1 isin U0 But this is a contradiction since U0 contains neither 0 nor 1 Hence theintegers μ and k + 1 must be relatively prime
Conversely suppose μ and k + 1 are relatively prime For every jminus 1 isin Zk
there exist integers ε and such that jμ = (k + 1)ε + where minus 1 isin Zk+1Since jμ le (k + 1)μ it follows that ε isin Zμ Moreover because μ and k + 1are relatively prime it must also be the case that minus 1 isin Zk Furthermore
since jk+1 = ψε
(
k+1
)we conclude that U0 is refinable
Our third special construction of refinable sets U0 in [0 1] relative tothe mappings is formed from cyclic μ-adic expansions To describe thisconstruction we introduce two additional mappings The first mapping π Zinfinμ rarr [0 1] is defined by
π(e) =sumjisinN
εjminus1
μj e = [εj j isin N0] isin Zinfinμ
and we also write it as π(e)= ε0ε1ε2 middot middot middot This mapping takes an infinitevector eisinZinfinμ and associates it with a number in [0 1]whoseμ-adic expansionis read off from the components of e The mapping π is not invertibleReferring back to the definition (439) we conclude for any ε isinZμ and e isin Zinfinμthat ψε(π(e))= εε0ε1 middot middot middot We also make use of the ldquoshiftrdquo map σ Zinfinμ rarrZinfinμ Specifically for e=[εj j isin N0] isinZinfinμ we set σ(e) =[εj jisinN] isinZinfinμ Thus the mapping σ discards the first component of e while the mapping ψε
restores the corresponding digit that is
ψε0(π σ(e)) = π(e) (440)
For any k isin N and ek = [εj j isin Zk] isin Zkμ we let ε0ε1 middot middot middot εkminus1 denote the
number π(e) where e = [εj j isin N0] isin Zinfinμ and εi+k = εi i isin N0 Note thatfor such an infinite vector e we have that σ k(e) = e where σ k = σ middot middot middot σis the k-fold composition of σ and also the number ε0ε1 middot middot middot εkminus1 is the uniquefixed point of the mapping ψek
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172 Multiscale basis functions
Proposition 421 Choose k isin N and ek = [εj j isin Zk] isin Zkμ such that
at least two components of ek are different Let e = [εj j isin N0] isin Zinfinμwith εi+k = εi i isin N0 Then the set U0(π(e)) = π σ(e) isin Zk isrefinable relative to the mappings and has cardinality le k Moreover if kis the smallest positive integer such that εi+k = εi i isin N0 then U0(π(e)) hascardinality k
Proof If ε0ε1 middot middot middot εkminus1 = εprime0εprime1 middot middot middot εprimekminus1 for εi εprimei isin Zμ i isin Zk then εi = εprimei i isin Zk Hence it follows that all the elements of U0(π(e)) are distinct Alsoby using (440) for any isin Zk we have that π σ(e) = ψε(π σ+1(e))Note that trivially π σ+1(e) isin U0(π(e)) for isin Zkminus1 and π σ k(e) =π(e) isin U0(π(e)) Thus U0(π(e)) is indeed refinable
Various useful examples can be generated from this proposition We mentionthe following possibilities for μ = 2
U0(01) =
1
3
2
3
U0(001) =
1
7
2
7
4
7
U0(0011) =
1
5
2
5
3
5
4
5
We now present a characterization of refinable sets relative to a given family of contractive mappings on any metric space M To state this result for everyk isin N = 1 2 and ek = [εj j isin Zk] isin Zk
μ we define the contractivemapping φek = φε0 φε1 middot middot middot φεkminus1 and let k = φek ek isin Zk
μ inparticular 1 = We let xek be the unique fixed point of the mapping φek that is
φek(xek) = xek
and set
Fk = xek ek isin Zkμ
We also define Zinfinμ to be the set of infinite vectors e = [εj j isin N0] εi isin Zμi isin N0 With every such vector e isin Zinfinμ and k isin N we let ek = [εj j isin Zk] isinZkμ It was shown in [148] that the limit of xek as krarrinfin exists and we denote
this element of the metric space X by xe In other words we have that
limkrarrinfin xek = xe
Moreover we let er =[εj jisinZ Zr] for r isin Z and use xer to denote thefixed point of the composition mapping φer where φer =φεr φεr+1 middot middot middot φεminus1 when r isin Z and φer is the identity mapping when r =
Theorem 422 Let = φε ε isin Zμ be a family of contractive mappingson a complete metric space X and let V0 sube X be a nonempty set of cardinalityk isin N Then V0 is refinable relative to if and only if V0 has the following
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44 Refinable sets and set wavelets 173
property For every v isin V0 there exist integers m isin Zk+1 with lt m andεi isin Zμ i isin Zm such that v = φe0 (xem) and the points
vr = φer (xem) isin V0 r isin Z
v+r = φe+rm(xem) isin V0 r isin Zmminus (441)
Moreover in this case we have that Vi = i(V0) sube i isin N and also
V0 sube (Fmminus) (442)
Proof Assume that V0 is refinable and v isin V0 Let v0 = v By the refinabilityof V0 there exist points vj+1 isin V0 and εj isin Zμ for j isin Zk such that vj =φεj(vj+1) j isin Zk Therefore we have that vr = φers(vs) r isin Zs s isin Zk+1Since the cardinality of V0 is k there exist two integers m isin Zk+1 with lt mfor which v = vm Hence in particular we conclude that v = vm = xem Itfollows that
vr = φer (v) = φer (xem) r isin Z
and
v+r = φe+rm(vm) = φe+rm(xem) r isin Zmminus
These remarks establish the necessity and also the fact that v0 isin(Fmminus)subeConversely let V0 be a set of points with the property and let v be a typical
element of V0 Then we have that either v = φe0 (xem) with gt 0 or v = xe0m
with = 0 In the first case since v = φε0(φe1 (xem)) and φe1 (xem) isin V0we have that v isin (V0) In the second case since xe0m is the unique fixedpoint of the mapping φe0m we write v = φε0(φe1m(xe0m)) By our hypothesisφe1m(xe0m) isin V0 and thus in this case we also have that v isin (V0) Thereforein either case v isin (V0) and so V0 is refinable These comments complete theproof of the theorem
We next derive two consequences of this observation To present the firstwe go back to the definition of the point xe in the metric space X wheree = [εj j isin N0] isin Zinfinμ and observe that when the vector e is s-periodic thatis its coordinates have the property that s is the smallest positive integer suchthat εi = εi+s i isin N0 we have xe = xes where es = [εj j isin Zs] Converselygiven any es isin Zs
μ we can extend it as an s-periodic vector e isin Zinfinμ andconclude that xe = xes
Let us observe that the powers of the shift operator σ act on s-periodic vectors in Zinfinμ as a cyclic permutation of vectors in Zs
μ Also thes-periodic orbits of σ that is vectors e isin Zinfinμ such that σ s(e) = e are exactly
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174 Multiscale basis functions
the s-periodic vectors in Zinfinμ With this viewpoint in mind we can draw thefollowing conclusion from Theorem 422
Theorem 423 A finite set V0 in a metric space X is refinable relative to themappings if and only if for every v isin V0 there exists an e isin Zinfinμ such thatv = xe and xσ k(e) isin V0 for all k isin N
Proof For convenience we define the notation πlowast(ε0ε1ε2 middot middot middot ) = [εj j isinN0] isin Zinfinμ for εi isin Zμ The proof requires a formula from [148] (p 727) whichin our notation takes the form
φε(xe) = xπlowast(ψε(π(e))) ε isin Zμ e isin Zinfinμ
where ψε are the concrete mappings defined by (439) Using this formula thenumber π(e) associated with the vector e in Theorem 422 is identified as
π(e) = ε0ε1 middot middot middot εminus1ε middot middot middot εmminus1
An immediate corollary of this result characterizes refinable sets on R
relative to the mappings defined by (439)
Theorem 424 Let U0 be a subset of R having cardinality k Then U0 isrefinable relative to the mappings (439) if and only if for every point u isin U0there exist integers m isin Zk+1 with lt m and εi isin Zμ i isin Zm such thatu = ε0 middot middot middot εminus1ε middot middot middot εmminus1 and for any cyclic permutation η ηmminus1 ofε εmminus1 and r isin Z the point εr middot middot middot εminus1η middot middot middot ηmminus1 is in U0
It is the vectors e isin Zinfinμ which are pre-orbits of σ that is for some k isin N0
the vector σ k(e) is periodic which characterize refinable sets Thus there is anobvious way to build from refinable sets U0 relative to the mappings (439) onR refinable sets relative to any finite contractive mappings on a metric spaceFor example let U0 be a finite subset of cardinality k in the interval [0 1] Werequire for each number u in this set that there is an e isin Zinfinμ such that u = π(e)and for every j isin N0 π(σ j(e)) isin U0 In other words U0 is refinable relativeto the mappings (439) We define a set V0 in X associated with U0 by theformula
V0 = xe π(e) isin U0where xe isin X is the limit of xek This set is a refinable subset of X relative to thecontractive mappings We may use this association to construct examplesof practical importance in the finite element method and boundary integralequation method
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44 Refinable sets and set wavelets 175
Example Let sub R2 be the triangle with vertices at y0 = (0 0) y1 = (1 0)and y2 = (0 1) Set y3 = (1 1) and consider four contractive affine mappings
φε(x) = 1
2(yε + (minus1)τ(ε)x) ε isin Z4 x isin R2 (443)
where τ(ε) = 0 ε isin Z3 and τ(3) = 1 The invariant subset of R2 relative tothese mappings is the triangle and the following sets are refinable(
1
3
1
3
)
(1
7
4
7
)
(2
7
1
7
)
(4
7
2
7
)(
1
15
2
15
)
(2
15
4
15
)
(4
15
8
15
)
(8
15
1
15
)with respect to these mappings Also we record for any ek = [εj j isin Zk] isinZkμ and x isin R2 that
φek(x) =1
2k
⎡⎣(minus1)τk x+sumjisinZk
(minus1)τj 2kminusjyεj
⎤⎦
where τj =sumjminus1=0 τ(ε) j isin Zk+1 From this equation it follows that
xek =1
2k + (minus1)τk+1
sumjisinZk
(minus1)τj 2kminusjyεj
These formulas can be used to generate the above sets
442 Set wavelets
In this subsection we generate a sequence W = Wn n isin N0 of finite sets ofa metric space X which have a wavelet-like multiresolution structure We callan element of W a set wavelet and demonstrate in subsequent sections that setwavelets are crucial for the construction of interpolating wavelets on certaincompact subsets of Rd
The generation of set wavelets begins with an initial finite subset of distinctpoints V0 = vj j isin Zm in X We use this subset and the finite set ofcontractive mappings to define a sequence of subsets of X given by
Vi = (Viminus1) i isin N (444)
Assume that a compact set in X is the unique invariant set relative tothe mappings When V0 sube it follows for each i isin N that Vi sube Furthermore using the set k = φek ek isin Zk
μ of contractive mappings
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176 Multiscale basis functions
introduced in the last subsection and for every subset A of X we definethe set
k(A) =⋃
ekisinZkμ
φek(A)
and so in particular 1(A) = (A) Therefore equation (444) implies thatVi = i(V0) i isin N
The next lemma is useful to us
Lemma 425 Let be a finite family of contractive mappings on X Assumethat sube X is the invariant set relative to the mappings If V0 is a nonemptyfinite subset of X then
sube⋃iisinN0
Vi
where Vi is generated by the mappings by (444)
Proof Let x isin and δ gt 0 Since is a compact set in X we choosean integer n gt 0 such that γ ndiam ( cup V0) lt δ where γ is the contractionparameter appearing in equation (48) According to the defining property (49)of the set there exists an en isin Zn
μ such that x isin φen() sube φen( cup V0)Since V0 is a nonempty set of X there exists a y isin φen(V0) sube φen( cup V0)Moreover by the contractivity (48) of the family we have that
d(x y) le diamφen( cup V0) le γ ndiam ( cup V0) lt δ
This inequality proves the result
Proposition 426 Let V0 be a nonempty refinable set of X relative to a finitefamily of contractive mappings and Vi i isin N0 be the collection of setsgenerated by definition (444) Then
=⋃iisinN0
Vi
Proof This result follows directly from Lemma 425 and Theorem 443
Let us recall the construction of the invariant set given a family ofcontractive mappings (see [148]) The invariant set is given by eitherone of the formulas
= xe e isin Zinfinμ
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44 Refinable sets and set wavelets 177
or
=⋃kisinN
Fk
The above proposition provides another way to construct the unique invariantset relative to a finite family of contractive mappings In other words westart with a refinable set V0 and then form Vi i isin N recursively by (444)
We say that a sequence of sets Ai i isin N0 is nested (resp strictly nested)provided that Aiminus1 sube Ai i isin N (resp Aiminus1 sub Ai i isin N) The next lemmashows the importance of the notion of the refinable set
Lemma 427 Let be the invariant set in X relative to a finite family ofcontractive mappings Suppose that is not a finite set and V0 is a nonemptyfinite subset of X Then the collection of sets Vi i isin N0 defined by (444) isstrictly nested if and only if the set V0 is refinable relative to
Proof Suppose that V0 is refinable relative to Then it follows by inductionon i isin N that Viminus1 sube Vi
It remains to prove that this inclusion is strict for all i isin N Assume to thecontrary that for some i isin N we have that Viminus1 = Vi By the definition of Viwe conclude that Viminus1 = Vj for all j ge i and thus we have that⋃
jisinN0
Vj = Viminus1
This conclusion contradicts Proposition 426 and the fact that does not havefinite cardinality
When the sequence of sets Vi i isin N0 is strictly nested we let Wi =Vi Viminus1 i isin N that is Vi = Viminus1 cupperp Wi i isin N where we use the notationA cupperp B to denote A cup B when A cap B = φ By Lemma 427 if the set V0 isrefinable relative to we have that Wi = empty i isin N Similarly we use thenotation
perp(A) =⋃εisinZμ
perpφε(A)
when φε(A) cap φεprime(A) = empty ε εprime isin Zμ ε = εprime The sets Wi i isin N0 give us thedecomposition
Vn =⋃
iisinZn+1
perpWi (445)
where W0 = V0 The next theorem shows that when the set W1 is specifiedthe sets Wi i isin N can be constructed recursively and the set has a
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178 Multiscale basis functions
decomposition in terms of these sets This result provides a multiresolutiondecomposition for the invariant set For this reason we call the sets Wii isin N set wavelets the set W1 the initial set wavelet and the decomposition of in terms of Wi i isin N the set wavelet decomposition of
Theorem 428 Let be the invariant set in X relative to a finite family of contractive mappings Suppose that each of the contractive mappings φε ε isin Zμ in has a continuous inverse on X and they have the property that
φε(int) cap φεprime(int) = empty ε εprime isin Zμ ε = εprime (446)
Let V0 be refinable with respect to and W1 sub int Then
Wi+1 = perp(Wi) i isin N (447)
and the compact set has the set wavelet decomposition
=⋃
nisinN0
perpWn (448)
where W0 = V0
Proof Our hypotheses on the contractive mappings φε ε isin Zμ guarantee thatthey are topological mappings Hence for any subsets A and B of X we havefor any ε isin Zμ that
intφε(A) = φε(int A) (449)
and
φε(A) cap φε(B) = φε(A cap B) (450)
Let us first establish that when W1 sub int the sets Wi i isin N defined bythe recursion (447) are all in int We prove this fact by induction on i isin NTo this end we suppose that Wi sube int i isin N Then the invariance property(49) of and (449) imply that
(Wi) sube (int) = int() = int
Therefore we have advanced the induction hypothesis and proved that Wi+1 subeint for all i isin N Using the fact that Wi+1 sube int we conclude from ourhypothesis (446) for any i isin N ε εprime isin Zμ ε = εprime that φε(Wi)cap φεprime(Wi) = emptywhich justifies the ldquoperprdquo in formula (447) It follows from (444) (447) andW1 = V1 V0 that Wi+1 sube Vi+1 i isin N0
Next we wish to confirm that
Vi Viminus1 = Wi i isin N (451)
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44 Refinable sets and set wavelets 179
Again we rely upon induction on i and assume that
Vi Viminus1 = Wi (452)
Therefore we obtain that
Vi cupWi+1 = (Viminus1) cup(Wi) =⋃εisinZμ
(φε(Viminus1) cup φε(Wi))
=⋃εisinZμ
φε(Vi) = Vi+1
which implies that Vi+1 Vi sube Wi To confirm that equality holds we observethat
Vi capWi+1 = (Viminus1) cap(Wi) =⋃εisinZμ
⋃εprimeisinZμ
(φε(Viminus1)) cap (φεprime(Wi)) (453)
For ε = εprime we can use (449) and hypothesis (446) to conclude thatφε()capφεprime(int) = empty To see this we assume to the contrary that there existsx isin φε() cap φεprime(int) Then there exist y isin and yprime isin int such thatx = φε(y) = φεprime(yprime) Condition (446) insures that y isin int Hence byequation (449) it follows from the first equality that x isin int and fromthe second equality that x isin int a contradiction Consequently we have that
(φε(Viminus1)) cap (φεprime(Wi)) = empty (454)
When ε= εprime we use (450) to obtain that (454) still holds Hence equa-tion (453) implies that Vi cap Wi+1 = empty This establishes (451) advances theinduction hypothesis and proves the result
We end this section by considering the following converse question to theone we have considered so far Given a finite set in a metric space is itrefinable relative to some finite set of contractive mappings The motivationfor this question comes from practical considerations As is often the case incertain numerical problems associated with interpolation and approximationwe begin on an interval of the real line with prescribed points for exampleGaussian points or the zeros of Chebyshev polynomials We then want to findmappings to make these prescribed points refinable relative to them We shallonly address this question in the generality of the space Rd relative to theinfin-norm It is easy to see that given any subset V0 = vi i isin Zk ofRd there is a family of contractive mappings on Rd such that V0 is refinablerelative to them For example the mappings φi(x) = 1
2 (x + vi) i isin Zkx isin Rd will do since clearly the fixed point of the mapping φi is vi for i isin ZkHowever almost surely the associated invariant set will have an empty interior
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180 Multiscale basis functions
and therefore Theorem 428 will not apply For instance in the example ofa triangle mentioned above the general prescription described applied to thevertices of the triangle will yield the Serpinski gasket This invariant set isa Cantor set and is formed by successively applying the maps (443) to thetriangles throwing away the middle triangle which is the image of the fourthmap used in the example To overcome this we must add to the above familyof mappings another set of contractive mappings ldquowhich fill the holesrdquo Todescribe this process we review some facts about parallelepipeds
A finite set I = ti i isin Zn+1 with t0 lt t1 lt middot middot middot lt tnminus1 lt tn iscalled a partition of the interval I = [t0 tn] and divides it into subintervalsIi = [ti ti+1] i isin Zn where the points in Icap(t0 tn) appear as endpoints of twoadjacent subintervals For every finite set U0 of (0 1) there exists a partition Isuch that the points of U0 lie in the interior of the corresponding subintervalsThe lengths of these subintervals can be chosen as small as desired
Likewise for any two vectors x = [xi i isin Zd] y = [yi i isin Zd] in Rdwhere xi lt yi i isin Zd which we denote by x ≺ y (also x y when xi le yii isin Zd) we can partition the set
otimesiisinZd[xi yi] ndash called a parallelepiped and
denoted by 〈x y〉 ndash into (sub)parallelepipeds formed from the partition
Id =otimesiisinZd
Ii = [tj j isin Zd] ti isin Ii i isin Zd (455)
where each Ii is a partition of the interval [xi yi] i isin Zd If Ii j j isin Zni is theset of subintervals associated with the partition Ii then a typical parallelepipedassociated with the partition Id corresponds to a lattice point i = [ij j isin Zd]where ij isin Znj j isin Zd defined by
Ii =otimesjisinZd
Iij j (456)
Given any finite set V0 sub Rd contained in the interior of a parallelepipedP we can partition it as above so that the interior of the subparallelepipedscontains the vectors of V0 We can if required choose the volume of thesesubparallelepipeds to be as small as desired
The set of all parallelepipeds is closed under translation as the simple rule
〈x y〉 + z = w+ z w isin 〈x y〉 = 〈x+ z y+ z〉 valid for any x y z isin Rd with x ≺ y For any x y isin Rd we associate an affinemapping A on Rd defined by the formula
At = Xt + y t isin Rd (457)
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44 Refinable sets and set wavelets 181
where X = diag(x0 x1 xdminus1) Such an affine map takes a parallelepipedone to one and onto a parallelepiped (as long as the vector x has no zerocomponents) Conversely given any two parallelepipeds P and Pprime there existsan affine mapping of the form (457) which takes P one to one and onto PprimeMoreover if there exists a z isin Rd such that Pprime + z sub int P then A is acontraction relative to the infin-norm on Rd given by
[xi i isin Zd]infin = max|xi| i isin ZdFor any two parallelepipeds P = 〈x y〉 and Pprime = langxprime yprime
rangwith Pprime sube P we
can partition their set-theoretic difference into parallelepipeds in the followingway For each i isin Zd we partition the interval [xi yi] into three subintervals byusing the partition Ii = xi xprimei yprimei yi The associated partition Id decomposesP into subparallelepipeds such that one and only one of them corresponds toPprime itself In other words if Pi i isin ZN with N = 3d are the subparallelepipedswhich partition P and PNminus1 = Pprime then we have
P Pprime =⋃
iisinZNminus1
Pi
We can now state the theorem
Theorem 429 Let m be a positive integer and V0 a finite subset of cardinalitym in the metric space (Rd middot infin) There exists a finite set of contractivemappings of the form (457) such that V0 is refinable relative to and theinvariant set for is a parallelepiped
Proof First we put the set V0 into the interior of a parallelepiped P whichwe partition as described above into subparallelepipeds so that the vectors inV0 are in the union of the interior of these subparallelepipeds Specifically wesuppose that
V0 = vi i isin Zm P =⋃
iisinZM
Pi
with m lt M vi isin int Pi i isin Zm and V0 cap int Pi = empty i isin ZM ZmFor each i isin Zm we choose a vector zi = [zi j j isin Zd] isin (0 1)d with
sufficiently small components zi j so that the affine mapping
Ait = Qi(t minus vi)+ vi t isin Rd (458)
where Qi = diag(zi0 zi1 zidminus1) has the property that the parallelepipedQi = AiP is contained in Pi Since Aivi = vi i isin Zm the set V0 is refinablerelative to any set of mappings including those in (458) We wish to appendto these m mappings another collection of one to one and onto contractive
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182 Multiscale basis functions
affine mappings of the type (457) so that the extended family has P as theinvariant set
To this end for each i isin Zm we partition the difference set Pi Qi intoparallelepipeds in the manner described above
Pi Qi =⋃
jisinZNminus1
Pi j
where N= 3d Thus we have succeeded in decomposing P into subparal-lelepipeds so that exactly m of them are the subparallelepipeds Qi i isin Zm Inother words we have
P =⋃iisinZk
Wi
where m lt k and Wi = Qi i isin Zm Finally for every i isin Zk Zm we choosea one to one and onto contractive affine mapping Ai such that AiP = Wi Thisimplies that
P =⋃iisinZk
AiP
and therefore P is the invariant set relative to the one to one and ontocontractive mappings Ai i isin Zk
In the remainder of this section we look at the above result for the real lineand try to economize on the number of affine mappings needed to make a givenset V0 refinable
Theorem 430 Let k be a positive integer and V0 = vl l isin Zk a subset ofdistinct points in [0 1] of cardinality k Then there exists a family of one-to-oneand onto contractive affine mappings φε ε isin Zμ of the type (457) for some2 le μ le 4 when k = 1 2 and 3 le μ le 2k minus 1 when k ge 3 such that V0 isrefinable relative to these mappings
Proof Since the mappings φ0(t) = t2 and φ1(t) = t+1
2 have the fixed pointst = 0 and t = 1 respectively we conclude that when k = 1 and V0 consists ofeither 0 or 1 and when k = 2 and V0 consists of 0 and 1 these two mappingsare the desired mappings When V0 consists of one interior point v0 we needat least three mappings For example we choose
φ0(t) = v0
2t φ1(t) = 1
2(t minus v0)+ v0
and
φ2(t) = 1minus v0
2t + v0 + 1
2 for t isin [0 1]
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44 Refinable sets and set wavelets 183
When V0 consists of two interior points of (0 1) we need four mappingsconstructed by following the spirit of the construction for the case k ge 3which is given below
When k ge 3 regardless of the location of the points there exist 2k minus 1mappings that do the job We next construct these mappings specificallyWithout loss of generality we assume that v0 lt v1 lt middot middot middot lt vkminus1 We firstchoose a parameter γ1 such that
0 lt γ1 lt min
v1 minus v0
v1
v2 minus v1
1minus v1
and consider the mapping φ1(t) = γ1(t minus v1) + v1 t isin [0 1] Therefore ifwe let α1 = φ1(0) and β1 = φ1(1) then v0 lt α1 lt β1 lt v2 Next we letγ0 = (α1 minus v0)(1minus v0) and introduce the mapping φ0(t) = γ0(tminus v0)+ v0t isin [0 1] Clearly by letting α0 = φ0(0) and β0 = φ0(1) we have 0 le α0 lt
β0 = α1The remaining steps in the construction proceed inductively on k For this
purpose we assume that the affine mapping φjminus2 has been constructed We letβjminus2 = φjminus2(1) and define
φjminus1(t) = γjminus1(t minus vjminus1)+ vjminus1 t isin [0 1] j = 3 4 k minus 1
where the parameters γjminus1 are chosen to satisfy the conditions
0 lt γjminus1 lt min
vjminus1 minus βjminus2
vjminus1
vj minus vjminus1
1minus vjminus1
j = 3 4 k minus 1
It is not difficult to verify that φjminus1([0 1]) sub (βjminus2 vj) or equivalently βjminus2 lt
αjminus1 lt βjminus1 lt vj by letting αjminus1 = φjminus1(0) and βjminus1 = φjminus1(1) Next we
let φkminus1(t) = γkminus1(t minus vkminus1)+ vkminus1 t isin [0 1] where 0 lt γkminus1 = vkminus1minusβkminus2vkminus1
and let αkminus1 = φkminus2(0) and βkminus1 = φkminus1(1) Then we have that βkminus2 =αkminus1 lt βkminus1 le 1 By the construction above we find two sets αi i isin Zkand βi i isin Zk of numbers that satisfy the condition
0 le α0 lt β0 = α1 lt β1 lt middot middot middot lt βkminus2 = αkminus1 lt βkminus1 le 1
and the union of the images of the interval [0 1] under mappings φj jisinZk isU =
⋃jisinZk
[αjβj]
Notice that the set U is not the whole interval [0 1] There are at most k minus 1gaps which need to be covered It is straightforward to construct these k minus 1additional mappings
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184 Multiscale basis functions
The family of mappings of cardinality at most 2k minus 1 that we haveconstructed above has [0 1] as the invariant set and V0 is a refinable set relativeto them
When the points in a given set V0 have special structure the number ofmappings may be reduced
45 Multiscale interpolating bases
In this section we present the construction of multiscale bases for bothLagrange interpolation and Hermite interpolation based on refinable sets andset wavelets developed in the previous section
451 Multiscale Lagrange interpolation
In this subsection we describe a construction of the Lagrange interpolatingwavelet-like basis using the set wavelets constructed previously For thispurpose we let X = Rd and assume that = φε ε isin Zμ is a family ofcontractive mappings that satisfies the hypotheses of Theorem 449 We alsoassume that sub X is the invariant set relative to with meas ( int) = 0Let k be a positive integer and assume that
V0 = vl l isin Zk sub int
is refinable relative to Note that in this construction of discontinuouswavelets we restrict the choice of the points in the set V0 to interior pointsof
As in [200 201] we choose a refinable curve f =[fl l isin Zk] rarrRk
which satisfies a refinement equation
f φi = Aif i isin Zμ (459)
for some prescribed k times k matrices Ai i isin Zμ We remark that if there isg rarr Rk and a k times k nonsingular matrix B such that f = Bg then g is alsoa refinable curve We let
F0 = spanfl l isin Zkand suppose that dimF0 = k Furthermore we require that for any b = [bl l isin Zk] isin Rk there exists a unique element f isin F0 such that f (vi) = bii isin Zk In other words there exist k elements in F0 which we also denoteby f0 f1 fkminus1 such that fi(vj) = δij i j isin Zk Refinable sets that admit
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45 Multiscale interpolating bases 185
a unique Lagrange interpolating polynomial were constructed in [198] Whenthis condition holds we say fi i isin Zk interpolates on the set V0 and thatfj interpolates at vj j isin Zk Under this condition any element f isin F0 has arepresentation of the form
f =sumiisinZk
f (vi)fi
A set V0 sube X is called (Lagrange) admissible relative to (F0) if it isrefinable relative to and there is a basis of functions fi i isin Zk for F0 whichinterpolate on the set V0 In this subsection we shall always assume that V0
is (Lagrange) admissible We record in the next proposition the simple fact ofthe Lagrange admissibility of any set of cardinality k for the special case when = defined by (439) = [0 1] and F0 = Pkminus1 the space of polynomialsof degree le k minus 1
Proposition 431 If V0 sub [0 1] is refinable relative to and has cardinalityk then V0 is Lagrange admissible relative to (Pkminus1)
Proof It is a well-known fact that the polynomial basis functions satisfythe refinement equation (459) with φi = ψi for some matrices Ai Hencethis result follows immediately from the unique solvability of the univariateLagrange interpolation
In a manner similar to the construction of orthogonal wavelets in Section 43we define linear operators Tε Linfin()rarr Linfin() ε isin Zμ by
(Tεx)(t) =
x(φminus1ε (t)) t isin φε()
0 t isin φε()and set
Fi+1 =oplusεisinZμ
TεFi i isin N0
This sequence of spaces is nested that is Fi sube Fi+1 i isin N0 and dimFi = kμii isin N0
We next construct a convenient basis for each of the spaces Fi For thispurpose we let F0 = fj j isin Zk where fj j isin Zk interpolate the set V0 and
Fi =⋃εisinZμperpTεFiminus1
= Tε0 middot middot middot Tεiminus1 fj j isin Zk ε isin Zμ isin Zi i isin N(460)
Since the functions fj j isin Zk interpolate the set V0 we conclude that theelements in Fi interpolate the set Vi In other words the functions in the set Fi
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186 Multiscale basis functions
satisfy the condition
(Tε0 middot middot middot Tεiminus1 fj)(φεprime0 middot middot middot φεprimeiminus1(vjprime)) = δ(ε0εiminus1 j)(εprime0εprimeiminus1 jprime) (461)
where we use the notation
δaaprime =
1 a = aprime0 a = aprime a aprime isin Ni
0 i isin N
For ease of notation we let ei = [εj j isin Zi] and Tei fj = Tε0 middot middot middot Tεiminus1 fjBy equation (461) this function interpolates at φei(vj) Moreover the relation
Fi = span Fi i isin N0 (462)
holds Now for each n isin N0 we decompose the space Fn+1 as the direct sumof the space Fn and its complement space Gn+1 which consists of the elementsin Fn+1 vanishing at all points in Vn that is
Fn+1 = Fn oplusGn+1 n isin N0 (463)
This decomposition is analogous to the orthogonal decomposition in theconstruction of the orthogonal wavelet-like basis in Section 43 and can beviewed as an interpolatory decomposition in the sense that we describe below
We first label the points in the set Vn according to the set wavelet decom-position for Vn given in Section 44 We assume that the initial set wavelet isgiven by W1 = wj j isin Zr with r = k(μminus 1) and we let
t0j = vj j isin Zk t1j = wj j isin Zr
tij = φew j = μ(e)r + e isin Ziminus1μ isin Zr i = 2 3 n
Then we conclude that Vn = tij (i j) isin Un where Un = (i j) i isinZn+1 j isin Zw(i) with
w(i) =
k i = 0k(μminus 1)μiminus1 i ge 1
The Lagrange interpolation problem for Fn relative to Vn is to find for avector b = [bij (i j) isin Un] an element f isin Fn such that
f (tij) = bij (i j) isin Un (464)
The following fact is useful in this regard
Lemma 432 If V0 is Lagrange admissible relative to (F0) then for eachn isin N0 the set Vn is also Lagrange admissible relative to (Fn)
Proof This result follows immediately from equations (461)
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45 Multiscale interpolating bases 187
Lemma 432 insures that each f isin Fn+1 has the representation f = Pn f+gnwhere Pn f is the Lagrange interpolant to f from Fn relative to Vn and gn =f minus Pn f is the error of the interpolation Therefore we have that
Gn+1 = gn gn = f minus Pn f f isin Fn+1 (465)
The fact that the subspace decomposition (463) is a direct sum also followsfrom equation (465) and the unique solvability of the Lagrange interpolationproblem (464) For this reason the spaces Gn are called the interpolatingwavelet spaces and in particular the space G1 is called the initial interpolatingwavelet space Direct computation yields the dimension of the wavelet spaceGn dimGn = kμnminus1(μ minus 1) Also we have an interpolating waveletdecomposition for Fn+1
Fn+1 = F0 oplusG1 oplus middot middot middot oplusGn+1 (466)
In the next theorem we describe a recursive construction for the waveletspaces Gn To establish the theorem we need the following lemma regardingthe distributivity of the linear operators Tε ε isin Zμ relative to a direct sum oftwo subspaces of Linfin()
Lemma 433 Let BC sub Linfin() be two subspaces If BoplusC is a direct sumthen for each ε isin Zμ Tε(Boplus C) = (TεB)oplus (TεC)
Proof It is clear that Tε(B oplus C) = (TεB) + (TεC) Therefore it remains toverify that the sum on the right-hand side is a direct sum To this end we letx isin (TεB) cap (TεC) and observe that there exist f isin B and g isin C such that
x = Tε f = Tεg (467)
By the definition of the operators Tε we have that x(t) = 0 for t isin φε()Now for each t isin φε() there exists τ isin such that t = φε(τ ) and thususing equation (467) we observe that
x(t) = f (φminus1ε (t)) = f (τ ) isin B x(t) = g(φminus1
ε (t)) = g(τ ) isin C
Since B oplus C is a direct sum we conclude that x(t) = 0 for t isin φε() Itfollows that x = 0
We also need the following fact for the proof of our main theorem
Lemma 434 If Y sube Linfin() then there holds
TεY cap TεprimeY = 0 ε εprime isin Zμ ε = εprime
Proof Let x isin TεY capTεprimeY There exist y1 y2 isin Y such that x = Tεy1 = Tεprimey2By the definition of the operators Tε we conclude from the first equality that
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188 Multiscale basis functions
x(t) = 0 for t isin φε() and from the second that x(t) = 0 for t isin φεprime()Since ε = εprime we have that meas (φε()capφεprime()) = 0 This implies that x = 0ae in and therefore establishes the result in this lemma
We are now ready to prove the main result of this section
Theorem 435 Let V0 be Lagrange admissible relative to (F0) and Wnn isin N be the set wavelets generated from V0 Then
G1 = spanTε fj ε isin Zμ j isin Zk Tε fj interpolates at φε(vj) isin W1
Gn+1 =oplusεisinZμ
TεGn n isin N (468)
and Gn = span Gn where
Gn = Ten fj en isin Znμ j isin Zk Ten fj interpolates at φen(vj) isin Wn
Proof Let Tε fj interpolate at φε(vj) isin W1 Then Tε fj has the property
(Tε fj)(φεprime(vjprime)) = 0 ε εprime isin Zμ εprime = ε or j jprime isin Zk jprime = j
By the definition of the set wavelet W1=V1 V0 we conclude for all vjprime isinV0
that we have (Tε fj)(vjprime) = 0 Thus by the definition of G1 we have thatcorresponding to each point φε(vj) isin W1 the basis function Tε fj is in G1Note that the cardinality of W1 is given by the formula card W1 = card V1 minuscard V0 = k(μminus1) It follows that the number of basis functions for which Tε fjinterpolate at φε(vj) in W1 is r the dimension of G1 Because these r functionsare linearly independent they constitute a basis for G1
We next prove equation (468) by induction on n For this purpose weassume that equation (468) holds for n le m and consider the case whenn = m+ 1 By the definition of Fm+1 and Gm we have that
Fm+1 =oplusεisinZμ
TεFm =oplusεisinZμ
Tε(Fmminus1 oplusGm)
Using Lemma 433 we obtain that
Fm+1 =oplusεisinZμ[(TεFmminus1)oplus (TεGm)] =
⎛⎝oplusεisinZμ
TεFmminus1
⎞⎠oplus⎛⎝oplusεisinZμ
TεGm
⎞⎠
It then follows from the definition of Fm that
Fm+1 = Fm oplus⎛⎝oplusεisinZμ
TεGm
⎞⎠
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45 Multiscale interpolating bases 189
Let
G =oplusεisinZμ
TεGm
and assume that f isin G Then there exist g0 gμminus1 isin Gm such that
f =sumεisinZμ
Tεgε
For each v isin Vm there exist vprime isin Vmminus1 and εprime isin Zμ such that v = φεprime(vprime)By the definition of the linear operators Tε ε isin Zμ and the fact that gε isin Gmε isin Zμ direct computation leads to the condition for each v isin Vm that
f (v) =sumεisinZμ
Tεgε(φεprime(vprime)) = gεprime(φminus1εprime φεprime(vprime)) = gεprime(v
prime) = 0
Hence G sube Gm+1 Moreover it is easy to see that dimG = dimGm+1 whichimplies that G = Gm+1
To prove the second part of the theorem it suffices to establish the recursion
Gn+1 =⋃εisinZμ
perpTεGn
The ldquoperprdquo on the right-hand side of this equation is justified by Lemma 434 Toestablish its validity we let
G =⋃εisinZμ
perpTεGn
Hence the set G consists of the elements TεnTen fj where Ten fj interpolates atφen(vj) isin Wn εn isin Zμ By Theorem 428 we have that
φen+1(vj) = φεn φen(vj) εn isin Zμ φen(vj) isin Wn sube φen+1(vj) isin Wn+1Hence G sube Gn+1 Since card G = card Gn+1 = card Wn+1 we conclude thatG = Gn+1
Theorem 436 It holds that
L2() =⋃
nisinN0
Fn = F0 oplus(oplus
nisinNGn
)
Proof Since the mappings φε ε isin Zμ are contractive the condition ofTheorem 47 of [201] is satisfied The finite-dimensional spaces Fn appearinghere are the same as those generated by the family of mutually orthogonalisometries in [201] if we begin with the same initial space F1 Therefore the
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190 Multiscale basis functions
first equality holds An examination of the proof for Theorem 47 of [201]shows that the same proof proves the second equality
As a result of the decomposition obtained in Theorems 435 and 436 wepresent a multiscale algorithm for the Lagrange interpolation To this end welet gj j isin Zr be a basis for G1 so that
gj(t0i) = 0 i isin Zk j isin Zr gj(t1jprime) = δj jprime j jprime isin Zr
We label those functions according to points in Vn in the following way Let
g0j = fj j isin Zk g1j = gj j isin Zr
gij = Teg j = μ(e)r + e isin Ziminus1μ isin Zr i = 2 3 n
With this labeling we see that gij(tiprimejprime) = δiiprimeδjjprime (i j) (iprime jprime) isin Un i ge iprime and
Fn = span gij (i j) isin UnFunctions gij (i j) isin U are also called interpolating wavelets Now we expressthe interpolation projection in terms of this basis For each x isin C() theinterpolation projection Pnx of x is given by
Pnx =sum
(i j)isinUn
xijgij (469)
The coefficients xij in (469) can be obtained from the recursive formula
x0j = x(t0j) j isin Zk
xij = x(tij)minussum
(iprime jprime)isinUiminus1
xiprimejprimegiprimejprime(tij) (i j) isin Un
This recursive formula allows us to interpolate a given function efficiently byfunctions in Fn When we increase the level from n to n+ 1 we do not need torecompute the coefficients xij for 0 le i le n We describe this important pointwith the formula Pn+1x = Pnx+Qn+1x where Qn+1x isin Gn+1 and
Qn+1x =sum
jisinZJ(n+1)
xn+1 jgn+1 j
The coefficients xn+1 j are computed by the previous recursive formula usingthe coefficients obtained for the previous levels that is
xn+1 j = x(tn+1 j)minussum
(iprime jprime)isinUn
xiprimejprimegiprimejprime(tn+1 j)
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45 Multiscale interpolating bases 191
452 Multiscale Hermite interpolation
In the last section we showed how refinable sets lead to a multiresolutionstructure and result in what we call set wavelets In the last subsection wethen used this recursive structure of the points to construct the Lagrange inter-polation that has a much desired multiscale structure In this subsection wedescribe a similar construction for Hermite piecewise polynomial interpolationon invariant sets
Let X be Euclidean space Rd and = φε ε isin Zμ be a family ofcontractive mappings on X is the invariant set relative to the family ofcontractive mappings Let V0 be a nonempty finite subset of distinct pointsin X and recursively define
Vi = (Viminus1) i isin N
It was shown in the last section that the collection of sets Vi i isin N0 isstrictly nested if and only if the set V0 is refinable relative to Denote
Wi = Vi Viminus1 i isin N
When the contractive mappings have a continuous inverse on X
φε(int) cap φεprime(int) = empty ε εprime isin Zμ ε = εprime
and W1 is chosen to be a subset of int then the sets Wi+1 i isin N can begenerated recursively from W1 by the formula
Wi+1 = perp(Wi) =⋃εisinZμ
perpφε(Wi) i isin N
and the invariant set has the decomposition
= V0 cupperp(⋃
nisinNperpWn
)
In the following we first describe a construction of multiscale discontinuousHermite interpolation and then a construction of multiscale smooth Hermiteinterpolation on the interval [0 1]1 Multiscale discontinuous Hermite interpolationWe start with nonempty finite sets V sub int and U sub Nd
0 For u = [ui i isinZd] isin U we set
Du = part |u|
parttu01 middot middot middot parttudminus1
d
where |u| =sumiisinZdui
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192 Multiscale basis functions
Let P be a linear space of functions on We say that (P U V) is Hermiteadmissible provided that V is refinable and for any given real numbers cr r =(u v) isin U times V there exists a unique element p isin P such that
Dr(p) = (Dup)(v) = cr (470)
When this is the case the dimension of P is (card V)(card U) and there existsa basis pr r isin U times V for P such that for every r = (u v) isin U times V andrprime = (uprime vprime) isin U times V
(Duprimepr)(vprime) = δrrprime (471)
Moreover for any function p isin P the representation
p =sum
risinUtimesV
Dr( p)pr
holds We call pr r isin U times V a Hermite basis for P relative to U times V Toproceed further we must restrict the family to have the form
φε(t) = aε t + bε t isin (472)
where aε t is the vector formed by the componentwise product of the vectorsaε and t We also require linear operators Tε Linfin() rarr Linfin() ε isin Zμ
defined by
Tε f = f φminus1ε χφε()
and for every en = [εj j isin Zn] isin Znμ we define constants
aminusuen
= aminusuε0
aminusuε1middot middot middot aminusu
εnminus1
Lemma 437 If is a family of contractive mappings of the form (472) thenfor all en isin Zn
μ and u isin Nd0 the following formula holds
DuTen = aminusuen
Ten Du n isin N
Proof We prove this lemma by induction on n and the proof begins by firstverifying the case when n = 1 by the chain rule The induction hypothesis isthen advanced by again using the chain rule and the case n = 1
We suppose that (P U V0) is admissible P is a subspace of polynomialswith Hermite basis
F0 = pr r isin U times V0relative to U times V0 and use the operators Tε ε isin Zμ to recursively define thesets
Fn = auenTen pr r = (u v) isin U times V0 en isin Zn
μ
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45 Multiscale interpolating bases 193
Since the polynomials pr r isin UtimesV0 were chosen to be a Hermite basis for Prelative to Utimes V0 the functions in Fn form a Hermite basis for Fn = span Fn
relative to U times Vn That is the function auenTen pr satisfies the condition
Duprime(auenTen pr)(φeprimen(v
prime)) = δ(ren)(rprimeeprimen) (473)
where rprime = (uprime vprime) isin U times V0Since Fn sube Fn+1 we can decompose Fn+1 as the direct sum of the space
Fn and Gn+1 defined to be the elements in Fn+1 whose uth derivatives foru isin U vanish at all points in Vn We let Pn f isin Fn be uniquely defined by theconditions
(DuPn f )(v) = Duf (v) v isin Vn u isin U (474)
Hence each f isin Fn+1 has the representation f = Pn f + gn where gn isin Gn+1
with Gn+1 = f minus Pn f f isin Fn+1 and we have the decomposition
Fn = F0 oplusG1 oplus middot middot middot oplusGn
Most importantly the spaces Gn can be generated recursively the proof ofwhich follows the pattern of those given for Theorems 435 and 436 To statethe next result we make use of the following notation For each n isin N0 and asubset A sube Vn we let Zn
μ(A) denote the subset of Znμ consisting of the indices
en isin Znμ such that there exists r = (u v) isin U times V0 for which equation (473)
holds and φen(v) isin A
Theorem 438 If P is a subspace of polynomials (P U V0) is admissibleand Wn n isin N is the set wavelets generated by V0 then
G1 = spanauεTεpr r = (u v) isin U times V0 ε isin Zμ(W1)
and
Gn+1 =oplusεisinZμ
TεGn n isin N
Moreover we have that Gn = span Gn where for n isin N
Gn = auenTen pr r = (u v) isin U times V0 en isin Zn
μ(Wn)and the formula
L2() =⋃
nisinN0
Fn = F0 oplus(oplus
nisinNGn
)
holds
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194 Multiscale basis functions
2 Multiscale smooth Hermite interpolation on [0 1]In the following we focus on a construction of smooth multiscale Hermite inter-polating polynomials on the interval [0 1] which generate finite-dimensionalspaces dense in the Sobolev space Wmp[0 1] where m is a positive integer and1 le p ltinfin To this end we choose affine mappings = φε ε isin Zμ
φε(t) = (tε+1 minus tε)t + tε ε isin Zμ
where 0 = t0 lt t1 lt middot middot middot lt tμminus1 lt tμ = 1 and μ gt 1 The invariantset of is [0 1] and this family of mappings has all the properties delineatedearlier We let V0 be a refinable set containing the endpoints of [0 1] that isV0 = v0 v1 vkminus1 where 0 = v0 lt v1 lt middot middot middot lt vkminus2 lt vkminus1 = 1 Sincethe endpoints are the fixed points of the first and last mappings respectivelyW1 = V1 V0 sub (0 1)
We let Fn be the space of piecewise polynomials of degree le km minus 1 inWmp[0 1] with knots at φen(0 1) en isin Zn
μ In particular F0 is the spaceof polynomials of degree le kmminus 1 on [0 1] and
dim Fn = μn(k minus 1)m+ m Fn sube Fn+1 n isin N0
This sequence of spaces is dense in Wmp[0 1] for 1 le p ltinfinWe construct multiscale bases for these spaces Fn using the solution of the
Hermite interpolation problem
p(i)(φen(v)) = f (i)(φen(v)) en isin Znμ v isin V0 i isin Zm (475)
which has a unique solution p isin Fn for any f isin Wmp[0 1] Hence in thisspecial case the refinability of V0 insures that (FnZm Vn) is admissible
Let Gn+1 be the space of all functions in Fn+1 such that g(i) i isin Zm vanishat all points in Vn A basis for the space Gn can be constructed recursivelystarting with interpolating bases for F0 and F1 To this end for r = (i j) isinZm times Zk we let pr isin F0 satisfy the conditions
p(iprime)
r (vjprime) = δrrprime rprime = (iprime jprime) isin Zm times Zk
Then the set of functions
F0 = pr r isin Zm times Zkconstitutes a Hermite basis for the space F0 relative to Zm times Zk To constructa basis for F1 we recall the linear operators Tε which in this case have thespecial forms
(Tε f )(t) = f
(t minus tε
tε+1 minus tε
)χ[tε tε+1](t) t isin [0 1]
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45 Multiscale interpolating bases 195
We remark that in general the range of operators Tε is not contained in theSobolev space Wmp[0 1] However we do have the following fact whosestatement uses for 1 le p ltinfin the spaces
Wmp0minus[0 1] = f isin Wmp[0 1] f (i)(0) = 0 i isin Zm+1
Wmp0+[0 1] = f isin Wmp[0 1] f (i)(1) = 0 i isin Zm+1
and
Wmp0 [0 1] = Wmp
0minus[0 1] capWmp0+[0 1]
Lemma 439 The following inclusions hold
Tμminus1(Wmp0minus[0 1]) sube Wmp
0minus[0 1]T0(W
mp0+[0 1]) sube Wmp
0+[0 1]and
Tε(Wmp0 [0 1]) sube Wmp
0 [0 1] ε isin Zμ
Next we show how to use the functions pr in F0 and the operators Tε toconstruct a basis of F1 The operators Tε may introduce discontinuities whenapplied to a function in F0 thereby leading to an unacceptable basis for F1Lemma 439 reveals exactly what happens when we apply Tε to pr Using theoperators Tε ε isin Zμ we define for r = (i ) isin ZmtimesZk and j = ε(kminus1)+
qij = (tε+1 minus tε)iTεpr
when ε = 0 = 0 or ε isin Zμ minus 1 isin Zkminus2 or ε = μminus 1 = k minus 1 and
qij = (tε minus tεminus1)iTεminus1p(ikminus1) + (tε+1 minus tε)
iTεp(i0)
when ε minus 1 isin Zμminus1 = 0 Set
F1 = qij i isin Zm j isin Zμ(kminus1)+1In the next lemma we state some properties of this set of functions To thisend we number the points in V1 according to the scheme
zj = φε(v) j = ε(k minus 1)+ ε isin Zμ minus 1 isin Zkminus1
Lemma 440 The set F1 forms a basis for F1 such that
q(iprime)
ij (zjprime) = δrrprime r = (i j) rprime = (iprime jprime) isin Zm times Zμ(kminus1)+1 (476)
Proof Lemma 439 insures that these functions are in Wmp[0 1] Hence it isclear that they are elements in F1 Moreover a direct verification leads to theconclusion that these functions satisfy the conditions (476)
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196 Multiscale basis functions
For a general n we follow the same process as described earlier to constructa basis for space Fn from the basis for Fnminus1 At each level it requiresa procedure of eliminating discontinuities introduced by the operators Tε However we do not construct the basis for Fn directly for n gt 1 Insteadwe turn our attention to the construction of bases for the complement spacesG1G2 Gn Surprisingly the next theorem shows that we can choose(μ minus 1)(k minus 1)m functions from the set F1 to form a basis for G1 andrecursively generate bases for the spaces Gn from this basis of G1 by applyingthe operators Tε We see that the construction of bases for Gn for n ge 2 doesnot require the process of eliminating discontinuities which is required for thedirect construction of bases for Fn because G1 sube Wmp
0 [0 1]Theorem 441 If V0 is refinable relative to the affine mappings and
G1 = qij j isin Zμ(kminus1)+1(W1) i isin Zmthen
G1 = span G1
Moreover if
Gn+1 = aienTen qij qij isin G1 i isin Zm en isin Zn
μwhere
aen =prodiisinZn
(tε+1 minus tε)
then
Gn+1 = span Gn+1 n isin N
Proof Since the cardinality of W1 equals the dimension of the space G1 theset G1 consists of (μminus 1)(kminus 1) linearly independent functions It remains toshow that G1 sub G1 To this end we prove that the functions in G1 vanish at allpoints in V0 Let qij isin G1 By (476) we obtain that q(i)ij (zj) = 1 for zj isin W1
and q(iprime)
ij (zjprime) = 0 for (iprime jprime) = (i j) Since V0 sub V1 and zj isin V0 we conclude
that q(i)ij i isin Zm vanish at all points in V0 and thus qij isin G1We now prove the second statement of the theorem It follows from
Lemma 439 that the functions aienTen qij isin Wmp
0 [0 1] since qij isin Wmp0 [0 1]
In addition we have that
(aienTen qij)
(iprime)(φeprimen(zjprime))= aiminusiprimeen
Ten q(iprime)
ij (φeprimen(zjprime))
= aiminusiprimeen
δeneprimen q(iprime)
ij (zjprime)
= δeneprimenδiiprimeδjjprime
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46 Bibliographical remarks 197
This equation implies that these functions are linearly independent in Gn+1Next we observe that the cardinality of the set Gn+1 is equal to the
dimension of Gn+1 Consequently we conclude that Gn+1 = span Gn+1
Since
Fn = F0 oplusG1 oplus middot middot middot oplusGn
and the sequence of spaces Fn n isin N0 is dense in the space Wmp[0 1] for1 le p ltinfin we obtain the following result
Theorem 442 The equation
F0 oplus(oplus
nisinNGn
)= Wmp[0 1]
holds for 1 le p ltinfin
In the finite element method the space Wmp0 [0 1] has a special importance
For this reason we define
F00 = f isin F0 f (i)(0) = f (i)(1) = 0 i isin Zm
and observe that dim F00 = (k minus 2)m where F0
0 = span F00 and
F00 = pi
j i isin Zm jminus 1 isin Zkminus1Corollary 443 The equation
F00 oplus
(oplusnisinN
Gn
)= Wmp
0 [0 1]
holds for 1 le p ltinfin
46 Bibliographical remarks
The material presented in this chapter regarding the multiscale bases wasmainly taken from [65 196 200 201] The construction of orthogonalwavelets on invariant sets was originally introduced in [200] Then theconstruction was extended in [201] to a general bounded domain and bi-orthogonal wavelets In particular the construction of the initial wavelet spacewas formulated in [201] in terms of a general solution of a matrix completionproblem Later [65] gave a construction of interpolating wavelets on invariantsets The concept of a refinable set relative to a family of contractivemappings on a metric space which define the invariant set is introducedin [65] and a recursive structure was explored in the paper for multiscale
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198 Multiscale basis functions
function representation and approximation constructed by interpolation oninvariant sets For the notion of invariant sets the reader is referred to [148]The material about multiscale partitions of a multidimensional simplex wasoriginally developed in [74] Paper [198] constructed refinable sets that admit aunique Lagrange interpolating polynomial (see also [199]) The description formultiscale Hermite interpolation in Section 452 follows [66] Moreover [69]presented a construction of multiscale basis functions and the correspondingmultiscale collocation functionals both having vanishing moments (see alsoSection 71)
For wavelets on an unbounded domain the reader is referred to [43 48 8284 92 93 97 98 100 101 232] and the references cited therein
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5
Multiscale Galerkin methods
The main purpose of this chapter is to present fast multiscale Galerkin methodsfor solving the second-kind Fredholm integral equations
uminusKu = f (51)
defined on a compact domain in Rd The classical Galerkin method usingthe piecewise polynomial applied to equation (51) leads to a linear systemof equations with a dense coefficient matrix Hence the numerical solutionof this equation is computationally costly The multiscale Galerkin methodto be described in this chapter makes use of the multiscale feature and thevanishing moment property of the multiscale piecewise polynomial basis andresults in a linear system with a numerically sparse coefficient matrix As aresult fast algorithms may be designed based on a truncation of the coefficientmatrix Specifically the multiscale Galerkin method uses the L2-orthogonalprojection for a discretization principle with the multiscale basis functionswhose construction is described in Chapter 4
The fast multiscale Galerkin method is based on a matrix compressionscheme We show that the matrix compression scheme preserves almostthe optimal convergence order of the standard Galerkin method while itreduces the number of nonzero entries of its coefficient matrix from O(N2)
to O(N logσ N) where N is the size of the matrix and σ may be 1 or 2 Wealso prove that the condition number of the compressed matrix is uniformlybounded independent of the size of the matrix
The kernels of the integral operators in which we are interested in thischapter are weakly singular or smooth We present theoretical results for theweakly singular case in detail and only give comments for the smooth case
199
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200 Multiscale Galerkin methods
51 The multiscale Galerkin method
In this section we present the multiscale Galerkin method for solving equa-tion (51) For this purpose we first describe the properties of multiscalebases required necessarily for developing the multiscale Galerkin methodThese properties are satisfied for the multiscale bases constructed in Chapter 3However the multiscale bases constructed in Chapter 3 have other propertiesthat are not essential for developing the multiscale Galerkin method
511 Multiscale bases
The multiscale basis requires a multiscale partition of the domain Weassume that there is a family of partitions i i isin N0 such that for eachscale i isin N0 i = ij j isin Ze(i) where e(i) denotes the cardinality of ihas the properties
(1)⋃
jisinZe(i)ij =
(2) meas(ij⋂ijprime) = 0 j jprime isin Ze(i) j = jprime
(3) meas(ij) sim ddi for all j isin Ze(i)
where di = maxd(ij) j isin Ze(i) Here the notation ai sim bi for i isin N0
means that there are positive constants c1 and c2 such that c1ai le bi le c2ai forall i isin N0
In addition we assume that(4) the sets ij j isin Ze(i) are star-shapedWe remark that a set A sub Rd is called a star-shaped set if it contains a point
for which the line segment connecting this point and any other point in the setis contained in the set Such a point is called a center of the set
We further suppose that there is a nested sequence of finite-dimensionalsubspaces Xn n isin N0 of X that is
Xnminus1 sub Xn n isin N
Thus for each n isin N0 a subspace Wn sub Xn can be defined such that Xn is anorthogonal direct sum of Xnminus1 and Wn Moreover we assume that Xn n isin N0
is ultimately dense in L2() in the sense that⋃nisinN0
Xn = L2()
We then have an orthogonal decomposition of space L2()
L2() =oplusnisinN0
perpWn (52)
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51 The multiscale Galerkin method 201
where we have used the notation W0 = X0 Set w(n) = dim Wn and s(n) =dim Xn for n isin N0 It follows that
s(n) =sum
iisinZn+1
w(i)
For each n isin N0 we introduce an index set Un = (i j) i isin Zn+1 j isinZw(i) We also use the notation U = (i j) i isin N0 j isin Zw(i) We assumethat there is a family of basis functions wij (i j) isin U sub X such that
Wn = spanwnj j isin Zw(n) n isin N0
Thus
Xn = spanwij (i j) isin UnWe require that the following multiscale properties hold for the partitions andthe basis functions
(I) There is a positive integer μ gt 1 such that for i isin N0
di sim μminusid w(i) sim μi and s(i) sim μi
(II) There exist positive integers ρ and γ such that for every i gt γ and j isinZw(i) written in the form j = νρ + s where s isin Zρ and ν isin N0
wij(t) = 0 t isin iminusγ ν
Setting Sij = iminusγ ν we see that the support of wij is contained in Sij Itcan easily be verified that
di sim maxd(Sij) j isin Ze(i)Because of this property we shall not distinguish di from the right-handside of the above equation
(III) For any (i j) isin U with i ge 1 and polynomial p of total degree less than apositive integer k
(p wij) = 0
where (middot middot) denotes the L2-inner product(IV) There is a constant θ0 such that for any (i j) isin U
wij = 1 and wijinfin le θ0μi2
where middot and middotinfin denote the L2-norm and the Linfin-norm respectively
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202 Multiscale Galerkin methods
(V) There is a positive constant θ1 such that for all n isin N0 v =sum(ij)isinUn
vijwij
Env2 sim v2 and v2 le θ1vwhere v = [vij (i j) isin Un] En =
[(wiprimejprime wij) (iprime jprime) (i j) isin Un
]and
the notation xp 1 le p le infin for a vector x = [xj j isin Zn] denotes thep-norm defined by
xp = (sum
jisinZn|xj|p
)1p 1 le p ltinfin
max|xj| j isin Zn p = infin
(VI) If Pn is the orthogonal projection from X onto Xn then there exists apositive constant c such that for any u isin Hk()
uminus Pnu le cdknuHk
All of these properties are fulfilled by the multiscale basis functionsconstructed in Chapter 3 In general the matrix En is a block diagonal matrixMoreover if wij (i j) isin U is a sequence of orthonormal basis functions thenEn is the identity matrix and property (V) holds with v2 = v Furthermoreif Xn n isin N0 are spaces of piecewise polynomials of total degree less than kthen the vanishing moment property (III) and the approximation property (VI)hold naturally
512 Formulation of the multiscale Galerkin method
As we have discussed in Section 31 the Galerkin method for equation (51) isto find un isin Xn that satisfies the operator equation
(I minusKn)un = Pnf (53)
where Kn = PnK|Xn It is clear that the following theoretical results hold forthe Galerkin method (see Section 33)
Theorem 51 Let K be a linear compact operator not having one as itseigenvalue Then there exists N gt 0 such that for all n ge N the Galerkinscheme (53) has a unique solution un isin Xn and there is a constant c gt 0such that for all n ge N
(I minusKn)minus1 le c
Moreover if the solution u of equation (51) satisfies u isin Hk() then thereexists a positive constant c such that for all n ge N
uminus un le cμminusknduHk
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51 The multiscale Galerkin method 203
Using the multiscale bases for spaces Xn described in the last section theabove Galerkin method (53) is to seek
un =sum
(ij)isinUn
uijwij isin Xn
such that sum(ij)isinUn
uij(wiprimejprime wij minusKwij) = (wiprimejprime f ) (iprime jprime) isin Un (54)
Because the multiscale basis is used in order to distinguish it from thetraditional Galerkin method we call (54) the multiscale Galerkin method
To write (54) in a matrix form we use the lexicographic ordering on Zn+1timesZn+1 and define the matrix
Kn = [(wiprimejprime Kwij) (iprime jprime) (i j) isin Un]and vectors
fn = [(wiprimejprime f ) (iprime jprime) isin Un] un = [uij (i j) isin Un]Note that these vectors have length s(n) With these notations equation (54)takes the equivalent matrix form
(En minusKn)un = fn (55)
Even though the coefficient matrix Kn is a full matrix it differs significantlyfrom the matrix Kn in Section 331 The use of multiscale basis functionsmakes the matrix Kn numerically sparse By the numerically sparse matrixwe mean a matrix with significantly large number of entries being very smallin magnitude This forms a base for developing the fast multiscale Galerkinmethod We illustrate this observation by the following example
Example 52 Consider = [0 1] and the compact integral operator withkernel
K(s t) = log |sminus t| s t isin [0 1]We choose Xn as the piecewise linear functions with knots j2n j isin N2nminus1In this case k = 2 The Galerkin matrix of this operator with respect to theLagrange interpolating basis is illustrated in Figure 51 with n = 6
We can see that generating the full matrix and then solving the correspond-ing linear system requires large computational cost when its order is large Theidea to overcome this computational deficiency is to change the basis for thepiecewise polynomial space so that the projection of the integral operator K tothe space has a numerically sparse Galerkin matrix under the new basis
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204 Multiscale Galerkin methods
0
20
40
60
80
020
4060
80
0
1
2
3
4
5
6
x 10minus3
Figure 51 The Galerkin matrix with respect to the piecewise linear polynomialbasis
020
4060
80
020
4060
80
0
05
1
15
2
Figure 52 The Galerkin matrix with respect to the piecewise linear polynomialmultiscale basis
The Galerkin matrix of this operator with respect to the piecewise linearpolynomial multiscale basis described in the last section is illustrated inFigure 52 with n = 6 It can be seen that the absolute value of the entries offthe diagonals of the blocks corresponding to different scales of spaces is very
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52 The fast multiscale Galerkin method 205
small We can set the entries small in magnitude to zero and obtain a sparsematrix which leads to a fast Galerkin method We present this fast method andits analysis in the next several sections
52 The fast multiscale Galerkin method
In this section we develop the fast multiscale Galerkin method based on amatrix truncation strategy We consider two classes of kernels Class oneconsists of kernels having weak singularity along the diagonal Specificallyfor σ isin [0 d) and integer k ge 1 we define Sσ k We say K isin Sσ k if fors t isin s = t K has continuous partial derivatives Dα
s Dβt K(s t) for |α| le k
|β| le k and there exists a constant c gt 0 such that for |α| = |β| = k
|Dαs Dβ
t K(s t)| le c
|sminus t|σ+2k s t isin (56)
Related to the kernel on the right-hand side of (56) we remark that whenσ = 0 the function 1xσ is understood as log x Class two consists of kernelsK isin Ck(times) Kernels in this class are smooth
Set
Kiprimejprimeij = (wiprimejprime Kwij) (i j) (iprime jprime) isin Un
and observe that Kiprimejprimeij are entries of matrix Kn In the next lemma we estimatethe bound of Kiprimejprimeij
Lemma 53 Suppose that conditions (I)ndash(IV) hold
(1) If K isin Sσ k for some σ isin [0 d) and a positive integer k and there is aconstant r gt 1 such that
dist(Sij Siprimejprime) ge maxrdi rdiprime then there exists a positive constant c such that for all (i j) (iprime jprime) isin Ui iprime isin N
|Kiprimejprimeij| le c(didiprime)kminus d
2
min
dd
iprime maxsisinSiprime jprime
intSij
dt
|sminus t|2k+σ ddi max
tisinSij
intSiprimejprime
ds
|sminus t|2k+σ
(2) If K isin Ck(times) then there exists a positive constant c such that for alli iprime isin N
|Kiprimejprimeij| le cdk+d2iprime dk+d2
i
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206 Multiscale Galerkin methods
Proof We present a proof for part (1) only since the proof for part (2) issimilar This is done by using the Taylor theorem By hypothesis for each(i j) isin U the set Sij is star-shaped Let s0 and t0 be centers of the sets Siprimejprime andSij respectively It follows from the Taylor theorem that
K(s t) = p(s t)+ q(s t)+sum|α|=k
sum|β|=k
(sminus s0)α(t minus t0)β
αβ rαβ(s t)
where p(s middot) and q(middot t) are polynomials of total degree less than k in t and ins respectively and
rαβ(s t) =int 1
0
int 1
0Dα
s Dβt K(s0 + θ1(sminus s0) t0
+ θ2(t minus t0))(1minus θ1)kminus1(1minus θ2)
kminus1dθ1dθ2
By conditions (II) and (III) we have that
Kiprimejprimeij =sum|α|=k
sum|β|=k
intSiprimejprime
intSij
(sminus s0)α(t minus t0)β
αβ rαβ(s t)wiprimejprime(s)wij(t)dsdt
This with conditions (I) and (IV) yields the bound
|Kiprimejprimeij| le cdkminus d
2i d
kminus d2
iprimesum|α|=k
sum|β|=k
1
αβint
Siprimejprime
intSij
|rαβ(s t)|dsdt (57)
We conclude from the mean-value theorem and the hypothesis K isin Sσ k thatthere exist sprime isin Siprimejprime and tprime isin Sij such that
|rαβ(s t)| = kminus2|Dαs Dβ
t K(sprime tprime)| le c
|sprime minus tprime|2k+σ
The assumption of this lemma yields
|sprime minus tprime| ge |sprime minus t| minus di ge (1minus rminus1)|sprime minus t|Thus for a new constant c
|rαβ(s t)| le c
|sprime minus t|2k+σ
This inequality with (57) and the relationship meas(Siprimejprime) sim ddiprime leads to the
desired estimate
The above lemma shows that most of the entries are so small that theycan be neglected without affecting the overall accuracy of the approximationscheme This observation leads to a matrix truncation strategy To present itwe partition matrix Kn into a block matrix
Kn = [Kiprimei iprime i isin Zn+1] with Kiprimei = [Kiprimejprimeij jprime isin Zw(iprime) j isin Zw(i)]
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52 The fast multiscale Galerkin method 207
For each iprime i isin Zn+1 we choose a truncation parameter δniprimei which will be
specified later We define for the weakly singular case
Kiprimejprimeij =
Kiprimejprimeij dist(Siprimejprime Sij) le δniprimei
0 otherwise(58)
and obtain a truncation matrix
Kn = [Kiprimei iprime i isin Zn+1]where
Kiprimei = K(δniprimei)iprimei = [Kiprimejprimeij jprime isin Zw(iprime) j isin Zw(i)]
Likewise for the smooth case we define for each iprime i isin N
Kiprimei =
Kiprimei i+ iprime le n0 otherwise
(59)
This truncation strategy leads to the fast multiscale Galerkin method which isto find un = [uij (i j) isin Un] isin Rs(n) such that
(En minus Kn)un = fn (510)
Example 54 We again consider the compact integral operator with the kernel
K(s t) = log |sminus t| s t isin [0 1]and choose Xn as the piecewise linear functions (k = 2) with knots j2nj isin N2nminus1 The truncated Galerkin matrix of this operator with respect to thepiecewise linear polynomial multiscale basis is illustrated in Figure 53 withn = 6
The analysis of the fast multiscale Galerkin method requires the availabilityof an operator form of equation (510) To this end we first introduce theconcept of the matrix representation of an operator
Definition 55 The matrix B is said to be the matrix representation of thelinear operator A relative to the basis = φj j isin Nn if
TB = A(T)
Proposition 56 The matrix representation of the operator K relative to thebasis Wn = wij (i j) isin Un is Bn = Eminus1
n Kn
Proof Let Bn = [biprimejprimeij (i j) isin Un] be the matrix representation of theoperator K relative to the basis Wn According to Definition 55 we have that
Kwij =sum
(kl)isinUn
bklijwkl for all (i j) isin Un
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208 Multiscale Galerkin methods
010
2030
4050
60
010
2030
4050
60
0
05
1
15
0 10 20 30 40 50 60
0
10
20
30
40
50
60
nz = 1638
(a) (b)
Figure 53 (a) The truncated Galerkin matrix with respect to the piecewise linearpolynomial multiscale basis (b) The figure of nonzero entries of the truncatedmatrix
This leads to
(wiprimejprime Kwij) =sum
(kl)isinUn
bklij(wiprimejprime wkl) for all (i j) (iprime jprime) isin Un
which means Kn = EnBn and completes the proof
We next convert the linear system (510) to an abstract operator equationform Let βiprimejprimeij (i j) (iprime jprime) isin Un denote the entries of matrix Eminus1
n KnEminus1n
and let
Kn(s t) =sum
(ij)(iprimejprime)isinUn
βiprimejprimeijwiprimejprime(s)wij(t)
We denote by Kn the integral operator defined by the kernel Kn(s t)
Proposition 57 Solving the linear system (510) is equivalent to finding
un =sum
(ij)isinUn
uijwij isin Xn
such that
(I minus Kn)un = Pnf
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53 Theoretical analysis 209
Proof It follows for (i j) (iprime jprime) isin Un that
(wiprimejprime Knwij) =(
wiprimejprime int
Kn(middot t)wij(t)dt
)=
sum(kl)(kprimelprime)isinUn
βkprimelprimekl(wkl wij)(wiprimejprime wkprimelprime)
=sum
(kl)(kprimelprime)isinUn
(En)iprimejprimekprimelprimeβkprimelprimekl(En)klij
= (Kn)iprimejprimeij (511)
which means
Kn = [(wiprimejprime Knwij) (i j) (iprime jprime) isin Un]and leads to the desired result of this proposition
The analysis of the fast multiscale Galerkin method with an appropriatechoice of the truncation parameters δn
iprimei will be discussed in the next section
53 Theoretical analysis
In this section we analyze the fast multiscale Galerkin method Specificallywe show that the number of nonzero entries of the truncated matrix is of linearorder up to a logarithm factor prove that the method is stable and that itgives almost optimal order of convergence We also prove that the conditionnumber of the truncated matrix is uniformly bounded We consider the weaklysingular case in Sections 531ndash533 Special results for the smooth case willbe presented in the last subsection without proof
531 Computational complexity
The computational complexity of the fast multiscale Galerkin method ismeasured in terms of the number of nonzero entries of the truncated matrix Inthis subsection we estimate the number of nonzero entries of matrix Kn For amatrix A we denote by N (A) the number of its nonzero entries
Lemma 58 If conditions (I) and (II) hold then there exists a constant c gt 0such that for all iprime i isin N0 and for all n isin N
N (Kiprimei) le cμi+iprime(
ddi + dd
iprime + (δniprimei)
d)
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210 Multiscale Galerkin methods
Proof For fixed i iprime jprime and an arbitrarily fixed point s0 in Siprimejprime we let
S(i iprime jprime) = s isin Rd |sminus s0| le di + diprime + δniprimei
If Kiprimejprimeij = 0 then dist(Siprimejprime Sij) le δniprimei Thus Sij sub S(i iprime jprime) Let Niiprimejprime denote
the number of indices (i j) such that Sij is contained in S(i iprime jprime) Property (3)of the partition i and condition (I) imply that there exists a constant c gt 0such that
Niiprimejprime le meas(S(i iprime jprime))minmeas(Sij) Sij sub S(i iprime jprime) le cμi(di + diprime + δn
iprimei)d
It follows from condition (II) that the number of functions wij having supportscontained in Sij is bounded by ρ Since w(iprime) sim μiprime
N (Kiprimei) le ρsum
jprimeisinZw(iprime)
Niiprimejprime le cμi+iprime(di + diprime + δniprimei)
d
proving the desired result
To continue estimating the number of nonzero entries of matrix Kn we nowspecify choices of the truncation parameters δn
iprimei Specifically for each i iprime isinZn+1 and for arbitrarily chosen constants a gt 0 and r gt 1 we choose thetruncation parameter δn
iprimei such that
δniprimei le max
aμ[minusn+α(nminusi)+αprime(nminusiprime)]d rdi rdiprime
(512)
where α and αprime are any numbers in (minusinfin 1] The lemma above and thechoice of truncation parameters lead to the following estimate of the numberof nonzero entries of matrix Kn
Theorem 59 If the truncation parameters δniprimei are chosen according to (512)
and if conditions (I) and (II) hold then
N (Kn) =
O(s(n) log2 s(n)) α = αprime = 1O(s(n) log s(n)) otherwise
Proof Because
N (Kn) =sum
iprimeisinZn+1
sumiisinZn+1
N (Kiprimei) (513)
we use Lemma 58 to estimate N (Kn) The choice (512) of truncationparameters ensures that
δniprimei le aμ[minusn+α(nminusi)+αprime(nminusiprime)]d + rdi + rdiprime
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53 Theoretical analysis 211
Using (513) and substituting the above estimate into the inequality inLemma 58 we have that
N (Kn) le csum
iisinZn+1
sumiprimeisinZn+1
μi+iprime(
2μminusi + 2μminusiprime + adμminusn+α(nminusi)+αprime(nminusiprime))
= c
⎡⎣4(n+ 1)sum
iisinZn+1
μi + adμn
⎛⎝ sumiisinZn+1
μ(αminus1)(nminusi)
⎞⎠times⎛⎝ sum
iprimeisinZn+1
μ(αprimeminus1)(nminusiprime)
⎞⎠⎤⎦=
O(μn(n+ 1)2) α = αprime = 1O(μn(n+ 1)) otherwise
as nrarrinfin This leads to the desired result of this theorem
532 Stability and convergence
In this subsection we show that the fast multiscale Galerkin method is stableand it has an almost optimal convergence order
The first lemma that we present here gives an estimate for the discrepancybetween the block Kiprimei and Kiprimei = K(δ)iprimei where the latter is obtained by usingthe truncation strategy with parameter δ = δn
iprimei
Lemma 510 Suppose that Kiprimei is obtained from the truncation strategy (58)with truncation parameter δ If conditions (I)ndash(IV) hold and K isin Sσ k for someσ isin [0 d) and a positive integer k then for any r gt 1 and δ gt 0 there existsa constant c such that when δ ge maxrdi rdiprime
Kiprimei minus Kiprimei2 le c(didiprime)kδminusη
where η = 2k minus d + σ gt 0
Proof By the definition of Kiprimei we have that
Kiprimei minus Kiprimeiinfin = maxjprimeisinZw(iprime)
sumjisinZδ|Kiprimejprimeij|
where
Zδ = j j isin Zw(i) dist(Sij Siprimejprime) gt δ
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212 Multiscale Galerkin methods
It follows from Lemma 53 that∥∥∥Kiprimei minus Kiprimei∥∥∥infin le c(didiprime)
kminus d2 dd
iprime maxjprimeisinZw(iprime)
maxsisinSiprimejprime
sumjisinZδ
intSij
dt
|sminus t|2k+σ
le c(didiprime)kminus d
2 ddiprime
int|t|gtδ
dt
|t|2k+σ
le c(didiprime)kminus d
2 ddiprimeδminusη
Likewise we have that∥∥∥Kiprimei minus Kiprimei∥∥∥
1le c(didiprime)
kminus d2 dd
i δminusη
Since the spectral radius of a matrix A is less than or equal to any of its matrixnorms
A22 = ρ(ATA) le ATAinfin le ATinfinAinfin = A1Ainfin
Using the above inequality we have that∥∥∥Kiprimei minus Kiprimei∥∥∥2
2le∥∥∥Kiprimei minus Kiprimei
∥∥∥1
∥∥∥Kiprimei minus Kiprimei∥∥∥infin
Substituting both estimates obtained earlier into the right-hand side of theabove inequality proves the desired result
We now describe a second criterion for the choice of truncation parametersδn
iprimei For each i iprime isin Zn+1 and for arbitrarily chosen constants a gt 0 and r gt 1we choose the truncation parameter δn
iprimei such that
δniprimei ge max
aμ[minusn+α(nminusi)+αprime(nminusiprime)]d rdi rdiprime
(514)
where α and αprime are any numbers in (minusinfin 1] For real numbers a and b we set
μ[a b n] =sum
iisinZn+1
μaidsum
iprimeisinZn+1
μbiprimed
We next estimate the error Rn = Kn minus Kn of the truncation operator in termsof the function μ[middot middot n]Lemma 511 Let u isin Hm() with 0 le m le k and K isin Sσ k for someσ isin [0 d) and a positive integer k If the truncation parameters δn
iprimei are chosenaccording to (514) and conditions (I)ndash(VI) hold then there exists a positiveconstant c such that for all n isin N0
(Kn minus Kn)Pnu le cμ[k + mminus αη k minus αprimeη n]μminus(m+dminusσ)nduHm
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53 Theoretical analysis 213
Proof For any u v isin X we project them into the subspace Xn Hence
Pnu =sum
iisinZn+1
(Pi minus Piminus1)u =sum
(ij)isinUn
uijwij
for some constants uij and
Pnv =sum
iisinZn+1
(Pi minus Piminus1)v =sum
(ij)isinUn
vijwij
for some constants vij where Pminus1 = 0 By the definitions of operators Kn andKn we have that((Kn minus Kn)PnuPnv
)=
sumiiprimeisinZn+1
((Kn minus Kn)(Pi minus Piminus1)u (Piprime minus Piprimeminus1)v
)=
sumiiprimeisinZn+1
sumjisinZw(i)
sumjprimeisinZw(iprime)
(Kiprimejprimeij minus Kiprimejprimeij)uijviprimejprime
Set
en =∣∣∣((Kn minus Kn)PnuPnv
)∣∣∣ Using the CauchyndashSchwarz inequality and condition (V) we conclude that
en le csum
iiprimeisinZn+1
Kiprimei minus Kiprimei2(Pi minus Piminus1)u(Piprime minus Piprimeminus1)v
It follows from condition (VI) that for u isin Hm() with 0 le m le k
(Pi minus Piminus1)u le cdmiminus1uHm
Combining the above estimates and using Lemma 510 we have for u isinHm() and v isin Hmprime() with 0 le m mprime le k that
en le csum
iiprimeisinZn+1
(didiprime)k(δn
iprimei)minusηdm
iminus1dmprimeiprimeminus1uHmvHmprime
Using di sim μminusid and the choice of δniprimei we conclude that
en le caminusηsum
iiprimeisinZn+1
μ(k+mminusαη)(nminusi)d+(k+mprimeminusαprimeη)(nminusiprime)d
μminus(m+mprime+dminusσ)nduHmvHmprime
= caminusημ[k + mminus αη k + mprime minus αprimeη n] middot μminus(m+mprime+dminusσ) nd uHmvHmprime
Since (Kn minus Kn) = Pn(Kn minus Kn) we have for u isin X that∥∥∥(Kn minus Kn)Pnu∥∥∥ = sup
visinXv=0
∣∣∣((Kn minus Kn)PnuPnv)∣∣∣ v2
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214 Multiscale Galerkin methods
Combining this equation with the inequality above with mprime = 0 yields thedesired result of this lemma
The next theorem provides a stability estimate for operator I minus Kn Recallthat for the standard Galerkin method there exist positive constants c0 and N0
such that for all n gt N0
(I minusKn)v ge c0v for all v isin Xn (515)
Theorem 512 Let K isin Sσ k for some σ isin [0 d) and a positive integer kSuppose that the truncation parameters δn
iprimei are chosen according to (514) with
α gt1
2minus d minus σ
2η αprime gt 1
2minus d minus σ
2η α + αprime gt 1
If conditions (I)ndash(VI) hold then there exist a positive constant c and a positiveinteger N such that for all n ge N and v isin Xn
(I minus Kn)v ge cvProof Note that for any real numbers a b and e
limnrarrinfinμ[a b n]μminusend = 0
when e gt max0 a b a+ b Thus the choice of δniprimei ensures that there exists
a positive integer N such that for all n ge N
cμ[k minus αη k minus αprimeη n]μminus(dminusσ)nd le c02
This with the estimate in Lemma 511 leads to
(Kn minus Kn)v le c0
2v for all v isin Xn (516)
Combining (516) and the stability estimate (515) of the standard Galerkinmethod yields
(I minus Kn)v ge (I minusKn)v minus (Kn minus Kn)v ge c0
2v
for any v isin Xn This completes the proof
The above stability estimate ensures that (Iminus Kn)minus1 exists and is uniformly
bounded As a result the fast multiscale Galerkin method (510) has a uniquesolution for a sufficiently large n
Theorem 513 Let u isin Hk() and K isin Sσ k for some σ isin [0 d) anda positive integer k Suppose that the truncation parameters δn
iprimei are chosenaccording to (514) with α and αprime satisfying one of the following conditions
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53 Theoretical analysis 215
(i) α ge 1αprime gt 12 minus dminusσ
2η α + αprime gt 1+ kη
or α gt 1αprime ge 12 minus dminusσ
2η α + αprime gt1+ k
ηor α gt 1αprime gt 1
2 minus dminusσ2η α + αprime ge 1+ k
η
(ii) α = 1αprime = kη
or α = 2kη
αprime = 12 minus dminusσ
2η
If conditions (I)ndash(VI) hold then there exist a positive constant c and a positiveinteger N such that for all n ge N
uminus un le cs(n)minuskd(log s(n))τuHk()
where τ = 0 in case (i) and τ = 1 in case (ii)
Proof It follows from Theorem 512 that there exist a positive constant c anda positive integer N such that for all n ge N
Pnuminus un le c(I minus Kn)(Pnuminus un) (517)
Since
Pn(I minusK)u = (I minus Kn)un = Pnf
we have that
(I minus Kn)(Pnuminus un) = Pn(I minusK)(Pnuminus u)+ (Kn minus Kn)Pnu (518)
Now by the triangle inequality we have that
uminus un le uminus Pnu + Pnuminus un (519)
Using inequality (517) and equation (518) we obtain that
Pnuminus un le cI minusKPnuminus u + c(Kn minus Kn)PnuSubstituting this estimate into the right-hand side of (519) yields
uminus un le (1+ cI minusK)Pnuminus u + c(Kn minus Kn)PnuIt follows from Lemma 511 that
(Kn minus Kn)Pnu le cμ[2k minus αη k minus αprimeη n]μminus(dminusσ)ndμminusknduHk
Observing that
μ[a b n]μminusend =
⎧⎪⎪⎨⎪⎪⎩O(1) if e ge a e gt b e gt a+ b
or e gt a e ge b e gt a+ bor e gt a e gt b e ge a+ b
O(n) if e = a b = 0 or e = b a = 0
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216 Multiscale Galerkin methods
as nrarrinfin we obtain that
μ[2k minus αη k minus αprimeη n]μminus(dminusσ)nd =
O(1) in case (i)O(n) in case (ii)
with a = 2k minus αη b = k minus αprimeη and e = d minus σ This yields the desiredresult
533 The condition number of the truncated matrix
We show in this subsection that the condition number of the truncated matrixis uniformly bounded To this end we need a norm equivalence result whichis presented below
Lemma 514 If conditions (II) (IV) and (V) hold then for any n isin N andv =sum(ij)isinUn
vijwij
v sim v2
where v = [vij (i j) isin Un]Proof Since condition (V) holds it suffices to prove that there is a positiveconstant θ2 such that for all v
v le θ2v2
It follows from the orthogonal decomposition (52) that
v2 =sum
iisinZn+1
∥∥∥ sumjisinZw(i)
vijwij
∥∥∥2
According to the construction of the partition of and condition (II) for alli gt γ ∥∥∥ sum
jisinZw(i)
vijwij
∥∥∥2 =sum
νisinZe(iminusγ )
∥∥∥ sumjisinZ(ν)
vijwij
∥∥∥2
where Z(ν) = j supp wij sube Sij = iminusγ ν Using the CauchyndashSchwarzinequality and condition (II) we have that∥∥∥ sum
jisinZ(ν)vijwij
∥∥∥2 leint
sumjisinZ(ν)
v2ij
sumjisinZ(ν)
w2ij(t)dt le ρ
sumjisinZ(ν)
v2ij
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53 Theoretical analysis 217
The last inequality holds because the cardinality of Z(ν) is less than or equalto ρ and the L2-norm of wij is equal to 1 Hence we conclude that there is apositive constant θ2 such that
v2 le θ22
sum(ij)isinUn
v2ij = θ2
2v22
This completes the proof
With the help of the above lemma we are ready to show that the conditionnumber of the coefficient matrix
An = En minus Kn
is uniformly bounded
Theorem 515 Suppose that K isin Sσ k for some σ isin [0 d) and a positiveinteger k and the truncation parameters δn
iprimei are chosen according to (514)with α and αprime satisfying the following conditions
α gt1
2minus d minus σ
2η αprime gt 1
2minus d minus σ
2η α + αprime gt 1
If conditions (I)ndash(VI) hold then the condition number of the coefficient matrixof the truncated approximate equation (510) is bounded that is there exists apositive constant c such that for all n isin N
cond2(An) le c
Proof For any v = [vij (i j) isin Un] isin Rs(n) let
v =sum
(ij)isinUn
vijwij
and
g = (I minus Kn)v
Thus g isin Xn and it can be written as
g =sum
(ij)isinUn
gijwij
Set
g = [gij (i j) isin Un]It can be verified that
g = (En minus Kn)v
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218 Multiscale Galerkin methods
It follows from Theorem 512 Lemma 514 and the above equations thatthere exist a positive constant c and positive integer N such that for all n ge N
v2 le cv le c(I minus Kn)v = cg le cg2 = c(En minus Kn)v2
This means that
(En minus Kn)minus12 le c (520)
Conversely we have that
(En minus Kn)v2 = g2 le cg = c(I minus Kn)vNote that
(I minus Kn)v le (I minusKn)v + (Kn minus Kn)vThis with (516) implies that
(I minus Kn)v le (1+ K)v + c0
2v le cv2
Thus
En minus Kn2 le c (521)
The result of this theorem follows from (520) and (521)
To close this section we would like to know if we can choose appro-priate truncation parameters such that the optimal results about the orderof convergence and computational complexity can be reached CombiningTheorems 59 512 513 and 515 leads to the following
Theorem 516 Let u isin Hk() and K isin Sσ k for some σ isin [0 d) and apositive integer k If conditions (I)ndash(VI) hold and δn
iprimei are chosen as
δniprimei = max
aμ[minusn+α(nminusi)+αprime(nminusiprime)]d rdi rdiprime
with α = 1 and 1minus kηlt αprime le 1 then the following hold the stability estimate
(I minus Kn)v ge cv for all v isin Xn
the boundedness of the condition number
cond2(An) le c
the optimal convergence order
uminus un le cs(n)minuskduHk()
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53 Theoretical analysis 219
and the optimal (up to a logarithmic factor) order of the complexity
N (Kn) =
O(s(n) log2 s(n)) α = αprime = 1O(s(n) log s(n)) otherwise
534 Remarks on the smooth kernel case
In this subsection we present special results for the smooth kernel case Sincethe proofs are similar to those for the weakly singular case we omit the detailsof the proof except for Lemma 519 whose results have something differentfrom Lemma 511
Lemma 517 If conditions (I)ndash(IV) hold and K isin Ck( times ) then thereexists a positive constant c such that for i iprime isin N and for all n isin N
Kiprimei2 le cdki dk
iprime
To avoid computing the entries whose values are nearly zero we make aspecial block truncation strategy that is setting
Kiprimei =
Kiprimei i+ iprime le n0 otherwise
iprime i isin N (522)
to obtain a sparse truncation matrix
Kn = [Kiprimei iprime i isin Zn+1]The following theorems provide the computational complexity the conver-
gence estimate and the stability of the truncation scheme for integral equationswith smooth kernels
Theorem 518 Suppose that condition (I) holds and K isin Ck( times ) If thetruncated matrix Kn is chosen as (522) then
N (Kn) = O(s(n) log s(n))
Lemma 519 Suppose that conditions (I)ndash(VI) hold and K isin Ck(times) Ifthe truncated matrix Kn is chosen as (522) then there exists a constant c suchthat for all u isin Hk() and for all n isin N
(Kn minus Kn)Pnu le cμminusknduHk
and for u isin L2()
(Kn minus Kn)Pnu le c(n+ 1)μminuskndu
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220 Multiscale Galerkin methods
Proof Similar to the proof of Theorem 511 for any u v isin X
Pnu =sum
(ij)isinUn
uijwij Pnv =sum
(ij)isinUn
vijwij
we have((Kn minus Kn)Pnu v
)=
sumiiprimeisinZn+1
sumjisinZw(i)
sumjprimeisinZw(iprime)
(Kiprimejprimeij minus Kiprimejprimeij)uijviprimejprime
Using the CauchyndashSchwarz inequality and condition (V) we conclude that itsabsolute value is bounded by
csum
iiprimeisinZn+1
Kiprimei minus Kiprimei2(Pi minus Piminus1)u(Piprime minus Piprimeminus1)v
It follows from condition (VI) that for u isin Hm() with 0 le m le k
(Pi minus Piminus1)u le cdmiminus1uHm
Denote Zprime(i) = iprime isin Zn+1 iprime gt nminus i Combining the above estimates usingLemma 517 and the truncation strategy (522) we have that for u isin Hm()
and v isin L2()∣∣∣((Kn minus Kn)Pnu v)∣∣∣ le c
sumiisinZn+1
sumiprimeisinZprime
(i)
(didiprime)kdm
iminus1uHmv (523)
Since di sim μminusid a simple computation yields thatsumiisinZn+1
sumiprimeisinZprime
(i)
(didiprime)kdm
iminus1 le csum
iisinZn+1
sumiprimeisinZprime
(i)
μminusk(i+iprime)dminusm(iminus1)d
= cμminuskndsum
iisinZn+1
μminusm(iminus1)dsum
iprimeisinZprime(i)
μminusk(i+iprimeminusn)d
For any i isin Zn+1sumiprimeisinZprime
(i)
μminusk(i+iprimeminusn)d lesumlisinN
μminuskld le μminuskd
1minus μminuskd
which leads to the fact that there exists a constant c such thatsumiisinZn+1
sumiprimeisinZprime
(i)
(didiprime)kdm
iminus1 le
cμminusknd if 0 lt m le kc(n+ 1)μminusknd if m = 0
(524)
Combining the above inequalities (523) (524) and
(Kn minus Kn)Pnu = supvisinXv=0
|((Kn minus Kn)Pnu v)|v
we obtain the estimates of this lemma
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54 Bibliographical remarks 221
The compactness of K and the property of the orthogonal projection Pn
lead to the stability estimate of the operator equation This with the secondestimate of Lemma 519 yields the following theorem about the stability ofthe truncation equation
Theorem 520 Suppose that conditions (I)ndash(VI) hold and K isin Ck(times) Ifthe truncated matrix Kn is chosen as (522) then there exist a positive constantc0 and an integer N such that for all n ge N and x isin Xn
(I minus Kn)x ge c0xWe have the following convergence estimate similar to Theorem 513
Theorem 521 Suppose that conditions (I)ndash(VI) hold and K isin Ck(times) Ifthe truncated matrix Kn is chosen as (522) then there exist a positive constantc and an integer N such that for all n ge N
uminus un le cs(n)minuskduHk
We also have that the condition number of the coefficient matrix An =En minus Kn of the truncated scheme is bounded by a constant independent of n
Theorem 522 Suppose that conditions (I)ndash(VI) hold and K isin Ck( times )If the truncated matrix Kn is chosen as (522) then the condition number ofthe coefficient matrix of the truncated approximate equation is bounded thatis there exists a positive constant c such that for all n isin N
cond2(An) le c
54 Bibliographical remarks
Since the 1990s wavelet and multiscale methods have been developed forsolving the Fredholm integral equation of the second kind The history of fastmultiscale solutions of the equation began with the remarkable discovery in[28] that the matrix representation of a singular Fredholm integral operatorunder a wavelet basis is numerically sparse This fact was then used indeveloping the multiscale Galerkin (PetrovndashGalerkin) method for solvingthe Fredholm integral equation see [5 64 68 88ndash91 94 95 135 136 139140 202 251 260 261] and references cited therein Readers are referred tothe Introduction of this book for more information The multiscale piecewisepolynomial PetrovndashGalerkin discrete multiscale PetrovndashGalerkin and multi-scale collocation methods were developed in [64 68 69] We give an in-depth
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222 Multiscale Galerkin methods
discussion of these methods in the next two chapters A numerical implemen-tation issue of the multiscale Galerkin method was considered in [109]
The convergence results presented in this chapter are for the smoothsolution However solutions of the Fredholm integral equation of the secondkind with weakly singular kernels may not be smooth When the solutionis not smooth a fast singularity-preserving multiscale Galerkin method wasdeveloped in [46] for solving weakly singular Fredholm integral equations ofthe second kind This method was designed based on the singularity-preservingGalerkin method introduced originally in [41] and a matrix truncation strategysimilar to what we have discussed in Section 52
There are several fast methods in the literature for solving the Fredholmintegral equation of the second kind which are closely related to the fastmultiscale method They include the fast multipole method the panel clus-tering method and the method of sparse grids The fast multipole method[114 115 235 250] was originally introduced by V Rokhlin and L Greengardbased on the multipole expansion It effectively reduces the computationalcomplexity involving a certain type of the dense matrix which can ariseout of many physical systems The panel clustering method proposed byW Hackbusch and Z Nowak also significantly lessens the computationalcomplexity (see for example [124 125]) For the method of sparse gridsreaders are referred to [36] and the references cited therein Fast FourierndashGalerkin methods developed in [37 53 154 155 263] for solving boundaryintegral equations are special cases of the method of sparse grids Fast methodsfor solving Fredholm integral equations of the second kind in high dimensionswere developed in [272] and [102] respectively based on a combinationtechnique and the lattice integration
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6
Multiscale PetrovndashGalerkin methods
This chapter is devoted to presenting multiscale PetrovndashGalerkin methods forsolving Fredholm integral equations of the second kind In a manner similar tothe Galerkin method the PetrovndashGalerkin method also suffers from the densityof the coefficient matrix of its resulting linear system We show that with themultiscale basis the PetrovndashGalerkin method leads to a linear system havinga numerically sparse coefficient matrix We propose a matrix compressionscheme for solving the linear system and prove that it almost preserves theoptimal convergence order of the numerical solution that the original PetrovndashGalerkin method enjoys and it reduces the computational complexity from thesquare order to the quasi-linear order We also present the discrete versionof the multiscale PetrovndashGalerkin method which further treats the nonzeroentries of the compressed coefficient matrix that results from the multiscalePetrovndashGalerkin method by using the product integration method We call thismethod the discrete multiscale PetrovndashGalerkin method
In Section 61 we first present the development of the multiscale PetrovndashGalerkin method and its analysis We then discuss in Section 62 the discretemultiscale PetrovndashGalerkin method
61 Fast multiscale PetrovndashGalerkin methods
In this section we describe the construction of two sequences of multiscalebases for trial and test spaces and use them to develop multiscale PetrovndashGalerkin methods for solving the second-kind integral equations
223
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224 Multiscale PetrovndashGalerkin methods
611 Multiscale bases for PetrovndashGalerkin methods
We review first a special case of the recursive construction given in Chapter 4for piecewise polynomial spaces on = [0 1] which can be used to developa multiscale PetrovndashGalerkin scheme
We start with positive integers k kprime ν and μ which satisfy kν = kprimeμ andkprime le k We choose our initial trial space and test space to be X0 = Sk
ν andY0 = Skprime
μ and thereafter we recursively divide the corresponding subintervalsinto μ pieces to obtain two sequences of subspaces
Xn = Skνμn Yn = Skprime
μn+1 n isin N0
These spaces are referred to as the (k kprime) element spacesWe have that
dimXn = dimYn n isin N0
Xn sub Xn+1 Yn sub Yn+1 n isin N0
and ⋃nisinN0
Xn =⋃
nisinN0
Yn = L2()
Moreover XnYn forms a regular pair (see Definition 230)We use Xn = fij (i j) isin Un and Yn = hij (i j) isin Un for the associated
multiscale bases for Xn and Yn respectively where Un = (i j) i isin Zn+1 j isinZw(i) with w(0) = kν = kprimeμ w(i) = kν(μminus 1)μiminus1 = kprime(μminus 1)μi i isin Nfor given k ν kprimeμ isin N These bases can be constructed recursively by themethod described in Section 41 such that both fij (i j) isin U and hij (i j) isin U are orthonormal bases in X = L2[0 1] having some importantproperties such as vanishing moment conditionsint 1
0tfij(t)dt = 0 isin Zk j isin Zw(i) i isin Nint 1
0thij(t)dt = 0 isin Zkprime j isin Zw(i) i isin N
and compact support properties
meas(suppfij) le 1μiminus1 meas(supphij) le 1μiminus1 j isin Zw(i) i isin N
The vanishing moment conditions play an important role in developingtruncated schemes (see Chapter 5) Therefore it is expected to raise the orderof the vanishing moments of hij to k when kprime lt k This can be done as follows
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61 Fast multiscale PetrovndashGalerkin methods 225
We first choose basis g0j j isin Zw(0) for Y0 which is bi-orthogonal tof0j j isin Zw(0) that is
( f0jprime g0j) = δjjprime j jprime isin Zw(0)
Then for j isin Zw(1) we find a vector [cjs s isin Zs(1)] isin Rs(1) where s(i) =dimYi i isin N0 such that
g1j =sum
sisinZw(0)
cjsh0s +sum
sisinZw(1)
cjw(0)+sh1s j isin Zw(1)
satisfies the equations
(f0jprime g1j) = 0 jprime isin Zw(0)
and
(f1jprime g1j) = δjjprime jprime isin Zw(1)
Noting that the matrix of order s(1) for this linear system of equations is
H = [(fiprimejprime hij) (iprime jprime) (i j) isin U1]and XnYn forms a regular pair we conclude that H is nonsingular Thusthere exists a unique solution which satisfies the above equations It can easilybe verified that these functions g1j j isin Zw(1) are linearly independent andY1 = spangij (i j) isin U1 Using the isometry operator Tε (see (42)) wedefine recursively for i isin N that
gi+1 j = Tεgil
where j = εw(i) + l ε isin Zμ l isin Zw(i) Then we have that Yn = spangij (i j) isin Un for n isin N0 Defining
Wi = spanfij j isin Zw(i) and Vi = spangij j isin Zw(i) i isin N0
we have that
Xn =oplus
iisinZn+1
perpWi and Yn =
oplusiisinZn+1
Vi n isin N0
Proposition 61 The multiscale bases fij (i j) isin U and gij (i j) isin Uhave the following properties
(I) There exist positive integers ρ and r such that for every i gt r and j isinZw(i) written in the form j = νρ + s where s isin Zρ and ν isin N0
fij(x) = 0 gij(x) = 0 x isin iminusrν
Setting Sij = iminusrν the supports of fij and gij are then contained in Sij
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226 Multiscale PetrovndashGalerkin methods
(II) For any (i j) (iprime jprime) isin U
( fij fiprimejprime) = δiprimeiδjprimej
(III) For any (i j) (iprime jprime) isin U and iprime ge i
( fij giprimejprime) = δiprimeiδjprimej
(IV) For any (i j) isin U with i ge 1 and polynomial p of total degree less than k
( fij p) = 0 (gij p) = 0
(V) There is a positive constant c such that for any (i j) isin U
fijinfin le cμi2 and gijinfin le cμi2
Set
En = [(giprimejprime fij) (iprime jprime) (i j) isin Un]It is useful to make the construction of the matrix En clear
Lemma 62 For any n isin N the following statements hold
(i) The matrix En has the form
En =
⎡⎢⎢⎢⎢⎢⎢⎣
I0 G0
I1 G1
Gnminus1
In
⎤⎥⎥⎥⎥⎥⎥⎦
where Ii i isin N0 is the w(i)times w(i) identity matrix
G0 = [(g0jprime f1j) jprime isinZw(0) j isin Zw(1)]G1 = [(g1jprime f2j) jprime isinZw(1) j isin Zw(2)]
and Gi i isin N is the block diagonal matrix diag(G1 G1 G1) with μiminus1
diagonal blocks(ii) There exists a positive constant c such that
En2 le c
Proof (i) We first partition the matrix En into a block matrix
En = [Eiprimei iprime i isin Zn+1]where
Eiprimei = [(giprimejprime fij) jprime isin Zw(iprime) j isin Zw(i)]
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61 Fast multiscale PetrovndashGalerkin methods 227
It follows from property (III) that
Eiprimei =
Ii iprime = i0 iprime gt i
When i ge iprime + 2 it follows from giprimejprime isin Yiprime fij isin Wi and the fact that Yiprime subeXiprime+1 perpWi that
(giprimejprime fij) = 0
which means that
Eiprimei = 0 for all i ge iprime + 2
We finally consider the case i = iprime + 1 When iprime = 0 and i = 1 it is clearthat E01 = G0 When iprime ge 1 and i = iprime + 1 assume that giprimejprime = Teprimeg1lprime andfij = Tef2l where jprime = μ(eprime)w(1)+ lprime and j = μ(e)w(2)+ l with eprime e isin Ziprimeminus1
μ lprime isin Zw(1) and l isin Zw(2) Using Proposition 415 we conclude that
(giprimejprime fij) = δeprimee(g1lprime f2l)
This means that for iprime ge 1 Eiprimeiprime+1 = Gi is the block diagonal matrixdiag(G1 G1 G1)
(ii) It is clear from (i) that
Eninfin = maxGiinfin + 1 i isin 0 1and
En1 = maxGi1 + 1 i isin 0 1Thus we obtain that
En2 le c = maxEninfin En1= maxGil + 1 i isin 0 1 l isin 1infin
This completes the proof
To estimate the norm of an element u isin Xn or v isin Yn we introduce asequence of functions ξij (i j) isin Uwhich is bi-orthogonal to gij (i j) isin UTo obtain the sequence we can find ξij isin Xi (i j) isin U1 such that
(giprimejprime ξij) = δiprimeiδjprimej (iprime jprime) (i j) isin U1
and then set
ξij = Teξ1l j = μ(e)w(1)+ l e isin Ziminus1μ l isin Zw(1)
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228 Multiscale PetrovndashGalerkin methods
Using this sequence we have for v isin Yn that
v =sum
(ij)isinUn
vijgij
with vij =langv ξij
rang Let
n = [(ξiprimejprime ξij) (iprime jprime) (i j) isin Un]Lemma 63 There exists a positive constant c such that
n2 le c
Proof We first estimate the entries of matrix n The fact that ξij (i j)isin U is bi-orthogonal to gij (i j) isin U implies that ξij i isin N has vanishingmoments of order kprime For iprime ge i and i isin Z2 let t0 be the center of the set Siprimejprime and write ξij = summisinZk
cm(t minus t0)m on Siprimejprime There exists a positive constant csuch that
|(ξiprimejprime ξij)| le cd(Siprimejprime)kprimeint
Siprimejprime|ξiprimejprime(t)|dt le cd(Siprimejprime)
kprime+12ξiprimejprime le cμminusiprime(kprime+12)
When iprime ge i gt 1 there exist eprime e isin Ziminus1μ lprime isin Zw(iprimeminusi+1) and l isin Zw(1) such
that
jprime = μ(eprime)w(iprime minus i+ 1)+ lprime j = μ(e)w(1)+ l
and
ξiprimejprime = Teprimeξiprimeminusi+1lprime ξij = Teξ1l
Thus
|(ξiprimejprime ξij)| = δeprimee|(ξiprimeminusi+1lprime ξ1l)| le cδeprimeeμminus(iprimeminusi+1)(kprime+12)
Combining the above estimates we obtain for (i j) (iprime jprime) isin Un that
|(ξiprimejprime ξij)| le cδeprimeeμminus|iprimeminusi|(kprime+12)
where eprime e isin Z|iprimeminusi|μ
We next partition n into a block matrix
n = [iprimei iprime i isin Zn+1]with
iprimei = [(ξiprimejprime ξij) jprime isin Zw(iprime) j isin Zw(i)]
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61 Fast multiscale PetrovndashGalerkin methods 229
and estimate the norm of these blocks It can be seen that
iprimeiinfin = maxjprimeisinZw(iprime)
sumjisinZw(i)
|(ξiprimejprime ξij)| le cw(|iprime minus i|)μminus|iprimeminusi|(kprime+12)
le cμminus|iprimeminusi|(kprimeminus12)
We next estimate matrix n Using the above inequality we have that
n1 = ninfin le maxiprimeisinZn+1
sumiisinZn+1
iprimeiinfin le 2c
1minus μminus(kprimeminus12)
which leads to the desired result of this lemma
Using the above lemmas we can verify the following proposition
Proposition 64 There exist two positive constants cminus and c+ such that forall n isin N0 u isin Xn having form u = sum(ij)isinUn
uijfij = sum(ij)isinUnuijξij and
v isin Yn having form v =sum(ij)isinUnvijgij
u = u2 (61)
cminusu le u2 le c+u (62)
and
cminusv le v2 le c+v (63)
where u = [uij (i j) isin Un] u = [uij (i j) isin Un] and v = [vij (i j) isin Un]Proof Recall that fij (i j) isin U is an orthonormal basis in X and ξij (i j) isin U is bi-orthogonal to gij (i j) isin U Therefore for (i j) isin Unuij = (u fij) uij = (u gij) and vij = (v ξij) Moreover equation (61) holdsIt can easily be verified that u2 = uTnu and u = Enu Using Lemmas 6263 and (61) we have that
u le (n2u22)12 le cu2 and u2 le Enu2 le cuwhich yield (62)
Noting that Yn sube Xn+1 any v isin Yn can be expressed as
v =sum
(ij)isinUn+1
(v gij)ξij
Thus we have that
v = n+1v
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230 Multiscale PetrovndashGalerkin methods
where v = [(v ξij) (i j) isin Un+1] and v = [(v gij) (i j) isin Un+1] ByLemma 337 and (62) we conclude that
v2 le n+12v2 le cv (64)
On the contrary
v2 =⎛⎝ sum(ij)isinUn
(v ξij)gijsum
(ij)isinUn+1
(v gij)ξij
⎞⎠=
sum(ij)isinUn
(v ξij)(v gij)
le v2v2 le cv2vThis with (64) yields (63)
612 Multiscale PetrovndashGalerkin methods
We now formulate the PetrovndashGalerkin method using multiscale bases forFredholm integral equations of the second kind given in the form
uminusKu = f (65)
where
(Ku)(s) =int
K(s t)u(t)dt
the function f isin X = L2() the kernel K isin L2( times ) are given and u isin X
is the unknown function to be determinedWe assume that there are two sequences of multiscale functions fij (i j) isin
U and gij (i j) isin U where U = (i j) j isin Zw(i) i isin N0 such that thesubspaces
Xn = spanfij (i j) isin Un and Yn = spangij (i j) isin Unsatisfy condition (H) and XnYn forms a regular pair These bases may notbe those constructed in the above subsection but they are demanded to satisfythe properties listed in Propositions 61 and 64
The PetrovndashGalerkin method for solving equation (65) seeks a vector un =[uij (i j) isin Un] such that the function
un =sum
(ij)isinUn
uij fij isin Xn
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62 Discrete multiscale PetrovndashGalerkin methods 231
satisfies
(giprimejprime un minusKun) = (giprimejprime f ) (iprime jprime) isin Un (66)
Equivalently we obtain the linear system of equations
(En minusKn)un = fn
where
Kn = [(giprimejprime Kfij) (iprime jprime) (i j) isin Un]En = [(giprimejprime fij) (iprime jprime) (i j) isin Un
]and
fn = [(gij f ) (i j) isin Un]The truncated scheme and its analysis of convergence and computational
complexity are nearly the same as for the multiscale Galerkin method we leavethem to the reader Readers are also referred to the discrete version of themultiscale PetrovndashGalerkin methods in the next section
62 Discrete multiscale PetrovndashGalerkin methods
One can find that the compression strategy for the design of the fast multiscalePetrovndashGalerkin method is similar to that of the fast multiscale Galerkinmethod and the practical use of the fast multiscale method requires thenumerical computation of integrals appearing in the method Therefore in thissection we turn our attention to discrete multiscale schemes We develop adiscrete multiscale PetrovndashGalerkin (DMPG) method for integral equationsof the second kind with weakly singular kernels A compression strategy fordesigning fast algorithms is suggested Estimates for the order of convergenceand computational complexity of the method are provided
We consider in this section the following Fredholm integral equations
uminusKu = f (67)
where K is an integral operator with a weakly singular kernelThe idea that we use to develop our DMPG method is to combine the
discrete PetrovndashGalerkin (DPG) method with multiscale bases to exploit thevanishing moment property of the multiscale bases and the computationalalgorithms for computing singular integrals of the DPG method Note that theanalysis of the DPG method was done in [80] in the Linfin-norm since this normis natural for discrete methods which use interpolatory projections However
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232 Multiscale PetrovndashGalerkin methods
for our DMPG method in order to make use of the vanishing moment propertyof the multiscale bases we have to switch back and forth between the Linfin-normand the L2-norm to obtain the necessary estimates We give special attentionto this issue
621 DPG methods and Lp-stability
We review the abstract framework outlined in [80] for analysis of discretenumerical methods of Fredholm integral equations of the second kind withweakly singular kernels To this end we let X be a Banach space with norm middot and V be a subspace of X We require that K Xrarr V be a compact linearoperator and that the integral equation (14) be uniquely solvable in X for allf isin X Note that whenever f isin V the unique solution of (14) is in V LetXn n isin N be a sequence of finite-dimensional subspaces of X satisfying
V sube X =⋃nisinN
Xn sube X
Suppose that the operators K and I (the identity from X to X) are approximatedby operators Kn X rarr V and Qn X rarr Xn respectively Specificallywe assume that Kn and Qn converge pointwise to K and I respectively Anapproximation scheme for solving equation (14) is defined by the equation
(I minusQnKn)un = Qnf n isin N (68)
This approximate scheme includes the discrete and nondiscrete versions ofthe PetrovndashGalerkin method collocation method and quadrature method asspecial cases Under various conditions elucidated in [80] for n large enoughequation (68) has a unique solution We discuss this issue later Instead weturn to specifying the operators and other related quantities needed for thedefinition of the DPG method for our current context In this section we fixX = Linfin() and V = C() with = [0 1] and use the followingterminology about the singularity
Definition 65 We say a kernel K(s t) s t isin = [0 1] is quasi-weaklysingular provided that
supsisinK(s middot)1 ltinfin
and
limsprimerarrsK(s middot)minus K(sprime middot)1 = 0
where middot 1 is the L1()-norm on
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62 Discrete multiscale PetrovndashGalerkin methods 233
It can easily be verified that the weakly singular kernel in the sense ofDefinition 24 with a continuous function M is quasi-weakly singular Everyquasi-weakly singular kernel determines by the formula
(Ku)(s) =int
K(s t)u(t)dt s isin u isin Linfin() (69)
a compact operator from Linfin() into C()For n isin N we partition into N (depending on n) subintervals 0 = i
i isin ZN That is we have
=⋃
risinZN
r meas(i capj) = 0 i = j i j isin ZN
Moreover we assume that as nrarrinfin the sequence of partition lengths
h = max|i| i isin ZNgoes to zero For each i isin ZN let Fi denote the linear function that maps theinterval one to one and onto i Thus Fi has the form
Fi(t) = |i|t + bi t isin i isin ZN (610)
for some constant bi For every partition 0 of described above and anypositive integer k we let Sk(0) be the space of all functions defined on which are continuous from the right and on each subinterval i it is apolynomial of degree at most kminus1 (at the right-most endpoint of we requirethat the functions in Sk(0) are left continuous)
We use the following mechanism to refine a given fixed partition = Jj j isin Z chosen independently of n For any i isin ZN written in the form i =k+ j j isin Z k isin ZN we define the intervals
Hi = Fk(Jj)
which collectively determine the partition
0 = Hi i isin ZNThis partition consists of N ldquocopiesrdquo of each of which is put on thesubintervals i i isin ZN Given two partitions 1 and 2 of (independentof n) and positive integers k1 and k2 we introduce the following trial and testspaces
Xn = f f Fi isin Sk1(1) i isin ZN = Sk1(0 1)
and
Yn = Sk2(0 2)
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234 Multiscale PetrovndashGalerkin methods
respectively These are also spaces of piecewise polynomials of degree k1 minus 1k2 minus 1 on finer partitions induced by 1 2 and 0 respectively To insurethat the spaces Xn and Yn have the same dimension we require that
dim Sk1(1) = dim Sk2(2) = λ
We choose bases in Xn and Yn in the following manner Starting with spaces
Sk1(1) = spanξi i isin Zλand
Sk2(2) = spanηi i isin Zλfor j = λi+ where i isin ZN and isin Zλ we define functions
ξj = (ξ Fminus1i )χi and ηj = (η Fminus1
i )χi
These functions form a basis for spaces Xn and Yn respectively that is Xn =spanξj j isin ZλN and Yn = spanηj j isin ZλN
To construct a quadrature formula we introduce a third piecewise poly-nomial space Sk3(3) of dimension γ where 3 is yet another partition of (independent of n) and choose distinct points tj j isin Zγ in such thatthere exist unique functions ζi isin Sk3(3) i isin Zγ satisfying the interpolationconditions ζi(tj) = δij i j isin Zγ The functions ζi i isin Zγ form a basis for thespace Sk3(3) As above for j = γ i+ where i isin ZN and isin Zγ we definefunctions
ζj = (ζ Fminus1i )χi
and points
tj = Fi(t)
We also introduce the subspace
Qn = Sk3(0 3)
and observe that
Qn = spanζj j isin ZγNWe require the linear projection Zn Xrarr Qn by
Q = Zng =sum
jisinZγN
g(tj)ζj (611)
where for a function g isin Linfin() g(t) is defined in the sense described inSection 353 that is as any norm-preserving bounded linear functional which
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62 Discrete multiscale PetrovndashGalerkin methods 235
extends point evaluation at t from C() to Linfin() (cf [21]) For any x y isin Xwe introduce the following discrete inner product
(x y)n =sum
jisinZγN
wjx(tj)y(tj) (612)
where
wj =int
ζj(t)dt
Note that
wj = |i|int
ζ(t)dt = |i|w
where j = γ i + with i isin ZN and isin Zγ and for every isin Zγ w =intζ(t)dt Henceforth we assume that w gt 0 isin Zγ This way xn =
(x x)12n is a semi-norm on X
We now define a pair of operators using the discrete inner product Specifi-cally we define the operator Qn Linfin()rarr Xn by requiring
(Qnx y)n = (x y)n y isin Yn (613)
An element Qnx isin Xn satisfying (613) is called the discrete generalized bestapproximation (DGBA) to x from Xn with respect to Yn Similarly we letQprimen Linfin()rarr Yn be the discrete generalized best approximation projectionfrom Linfin() onto Yn with respect to Xn defined by the equation
(vQprimenx)n = (v x)n v isin Xn (614)
The following lemma proved in [80] presents a necessary and sufficientcondition for Qn and Qprimen to be well defined To state this lemma we introducea matrix notation Let
= [ξi(tj) i isin Zλ j isin Zγ ] = [ηi(tj) i isin Zλ j isin Zγ ]W = diag(wj j isin Nγ )
and define the square matrix of order λ
M = WT
Lemma 66 Let x isin Linfin() Then the following statements are equivalent
(i) The discrete generalized best approximation to x from Xn with respect toYn is well defined
(ii) The discrete generalized best approximation x from Yn with respect to Xn
is well defined
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