SIAM Review Vol. 58, Issue 4 (December 2016) Book Reviews Introduction, 793 Featured Review: A Graduate Introduction to Numerical Methods: From the Viewpoint of Backward Error Analysis (Robert M. Corless and Nicolas Fillion), Alex Townsend, 795 An Introduction to Polynomial and Semi-Algebraic Optimization (Jean Bernard Lasserre), Brian Borchers, 799
Variational Analysis in Sobolev and BV Spaces. Applications to PDEs and Optimization. Second Edition (Hedy Attouch, Giuseppe Buttazzo, and Gérard Michaille), Antonin Chambolle, 800
Plasticity: Mathematical Theory and Numerical Analysis. Second Edition (Weimin Han and B. Daya Reddy), Anurag Gupta, 802
Spline Functions: Computational Methods (Larry L. Schumaker), Tatyana Sorokina, 803
Computational Mathematical Modeling: An Integrated Approach Across Scales (Daniela Calvetti and Erkki Somersalo), Martin O. Steinhauser, 805
Dirichlet–Dirichlet Domain Decomposition Methods for Elliptic Problems: h and hp Finite Element Discretizations (Vadim Glebovich Korneev and Ulrich Langer), David S. Watkins, 806
The Computing Universe: A Journey through a Revolution (Tony Hey and Gyuri Pápay), David S. Watkins, 806
There are many textbooks on numerical analysis, but I have always found it diffi-cult to find one that both satisfies me and is at the same time accessible to the studentsI am trying to teach. Some are too much like a cookbook, while others are too tech-nical. Moreover, like everything else, the field continues to evolve steadily, so newentries are always welcome. This issue’s featured review, by Alex Townsend, informsus about a recent effort intended for the graduate student market, A Graduate Introduc-tion to Numerical Methods, by Robert Corless and Nicolas Fillion. It’s a huge (thoughreasonably priced) book that covers lots of ground and could serve as the text fora variety of graduate numerical analysis courses. As a subtitle to the book indicates,backward error analysis plays a big role, and so does the barycentric formula for La-grangian interpolation. I hope you enjoy Alex’s very positive review of this interestingbook.
In addition we have reviews of books on polynomial and semialgebraic optimiza-tion, variational analysis in Sobolev and BV spaces, plasticity, mathematical theory andnumerical analysis, spline functions, and computational mathematical modeling.
Finally, there are two reviews written by me, one on an extremely technical,specialized book on domain decomposition, and the other on a book about computerscience meant for the general reader.
SIAM REVIEW c⃝ 2016 Society for Industrial and Applied MathematicsVol. 58, No. 4, pp. 795–807
Book Reviews
Edited by David S. Watkins
Featured Review: A Graduate Introduction to Numerical Methods: From theViewpoint of Backward Error Analysis. By Robert M. Corless and Nicolas Fillion. Springer,New York, Heidelberg, 2013. $99.00. xxxix+869 pp., hardcover. ISBN 978-1-4614-8452-3.
I have just finished the 1.7 kilogram book A Graduate Introduction to NumericalMethods by Corless and Fillion. I read it chapter by chapter, covering about a chaptera week, taking it with me as I traveled to three different countries.1 Every instructorof a graduate or advanced undergraduate numerical analysis course should considerteaching from this book. It deservedly made the ACM Computing Reviews’ list ofnotable books and articles in 2013 [2].
What is numerical analysis? This is a difficult question, and your answer islikely to be a personal interpretation of the fuzzy interplay between “numerics” and“analysis.” The book’s 869-page answer is a delight. It is an inclusive celebration ofMaple’s symbolic manipulations and MATLAB’s numerical computations, Turing’smathematical modeling [8] and Strang’s linear algebra [6], Higham’s backward erroranalysis [4] and Trefethen’s practical approximation theory [7]. All the usual intro-ductory numerical analysis topics are included such as interpolation, floating-pointarithmetic, numerical linear algebra, and differential equations, as well as less com-mon topics such as the barycentric formula, resultants, automatic differentiation, andthe Lambert W function. The authors are experts on these special topics; see [1, 3].The broad range of topics in this book shows that numerical analysis is an impressivelyrich and mature subject.
The book comes in four parts: Part I on error analysis, polynomials, and rootfind-ing, Part II on numerical linear algebra, Part III on interpolation, differentiation, andquadrature, and Part IV on differential equations. There are also helpful appendiceson floating-point arithmetic, complex numbers, and introductory linear algebra. Tosummarize the book I made a word cloud of the text (see Figure 1), in which the sizeof a word indicates the frequency of its occurrence. One can also use this word cloudas a visual blurb or as a possible answer to “What is numerical analysis?”
It is a pleasure to see that the three large words are “problem,” “method,” and“compute”: the key schema of computational mathematics! This schema is repeatedhundreds of times in the book. The authors are often in investigative mode, usingMaple or MATLAB or formulas to discover something interesting. A broad selectionof numerical methods are described during the numerous journeys of discovery, ina manner suitable for a textbook and in a way that is very different to NumericalRecipes [5], which is better designed for quick reference. The unwavering andragogicalstyle to Corless and Fillion’s writing is masterful.
1Security control in the United States will search a backpack containing a hardback copy of Corlessand Fillion’s book because it is 3 inches thick. Each time one must explain the unusual carry-on item.
Publishers are invited to send books for review to Book Reviews Editor, SIAM, 3600 Market St.,6th Floor, Philadelphia, PA 19104-2688.
Fig. 1 A word cloud of the text and one possible answer to “What is numerical analysis?”
Next in the word cloud I notice the common words “polynomial” and “interpo-late.” This highlights one of the unique features of the book: the extensive use ofpolynomial interpolants and the barycentric formula, both for practical computationsand for theoretical derivations. For example, how many textbooks use the barycen-tric formula for Hermite interpolants to derive cubic splines or to develop multistepmethods? How many start by interpreting the discrete Fourier transform as a poly-nomial in the monomial basis evaluated at the roots-of-unity? The use of polynomialinterpolants and the barycentric formula feels surprisingly fresh and modern.
Another common word is “example,” which appears on average about once perpage. We know that numerical analysis is useful for a diverse range of applications,and the book considers examples on the orbit of three bodies under gravity, thespread of a measles epidemic, the flight path of a spinning golf ball, and others. Theexample-oriented narrative is informal (see the xkcd comic on p. 543 and footnotes)and well illustrated with nearly 200 figures. This style makes the textbook engagingand friendly. An instructor could easily cover the examples given in the book duringclass; preferably, in a similar investigative mode.
Two other keywords that cannot be missed are “condition (number)” and “error.”The book uses backward error analysis and condition numbers throughout, not onlyin the numerical linear algebra chapters, but also for quadrature, rootfinding, and thenumerical solution of differential equations. The error analysis is professionally satis-fying and will be reassuring to conscientious learners. In the preface and afterword,the authors present a thought-provoking discussion on the importance of backwarderror in numerical analysis, which is of independent interest and deserves attention.
There are so many intriguing features of the book that can be seen directly fromthe word cloud. For example, the word “residue” appears nearly 800 times becausecomplex analysis is used extensively (another unique feature of the book), “MATLAB”is two-times larger than “Maple,” and for every occurrence of “bound” there is oneof “round” (suggesting a delightful balance between error analysis and floating-pointarithmetic).
Aside from common words, there are also amusing maxims scattered throughoutthe book. Here is a selection.
It’s What a Good Numerical Analyst Does Anyway. The book excels at teachingsomeone how to be a “good” numerical analyst. A student will get into the habit ofusing a computer to conjure up a theorem, a theorem to certify numerical computa-tions, and a diagram to illustrate a concept. Each chapter also has a welcome “Notesand References” section that details places in the literature to find out more.
Which Method Is the Best Method? It’s Like a Game of Rock-Paper-Scissors. Inthe game of rock-paper-scissors neither rock, paper, nor scissors always trumps theopponent’s gesture. An analogous situation occurs for many classes of numericaltechniques. The book has an unbiased didactic tone that is particularly prominentwhen describing quadrature rules and the numerical methods for solving differentialequations. This style will keep the material relevant for several decades.
A Pupil from Whom Nothing Is Ever Demanded which (S)he Cannot Do, NeverDoes All (S)he Can. Each chapter has a good selection of problems that examine boththe practice and the theory. One could easily set homework containing questionsdirectly from the book, assuming the solutions are not already online. The book’scharming and informative language carries over to the problems, too. For example,Problem 13.22 is on the so-called Obsessive-Compulsive Disorder Euler method. Eachchapter also has a section called “Investigations and Projects” that contains moreinvolved and open-ended questions. These will challenge the more advanced students.
I Don’t Care How Quickly You Give Me the Wrong Answer. The book’s centralphilosophy on numerical algorithms is reliability first, cost last. This makes erroranalysis and conditioning, though not necessarily algorithmic complexity, a centralactivity. I think this is a shame. Numerical analysts care about reliability and speed.As just one example, Chapter 6 on structured matrices spends the majority of timeon structured backward error analysis and only briefly mentions fast matrix-vectorproducts and matrix factorizations. Researchers usually exploit structure in matricesfor computational cost, storage, and scalability to real-world problems, not for animproved theoretical error bound. I did not enjoy the cost last philosophy of thebook.
A Problem Is “Stiff” If, in Comparison, ode15s Mops the Floor with ode45 on It.This is one of the more memorable informal definitions of a stiff differential equation,and highlights once again the conversational style of the book. Numerical methods forsolving boundary value problems, delayed differential equations, and partial differen-tial equations as well as Runge–Kutta and multistep methods are expertly described.There is even a detailed discussion on event handling and the role of numerical meth-ods for chaotic systems.
One particularly illustrative example is the classic Henon–Heiles equation, whichis a nonlinear nonintegrable Hamiltonian system given by
(1) x(t) = −∂V
∂x, y(t) = −∂V
∂y, V (x, y) =
1
2
!x2 + y2 + 2x2y − 2
3y3".
The energy E(t) = V (x(t), y(t)) + 12 (x(t)
2 + y(t)2) is conserved during motion andserves as a way to gauge the numerical error in a method. The book gives inlineMATLAB code to solve (1) for 0 < t ≤ 105 using ode113, as well as code to plot a
Fig. 2 Poincare map of the Henon–Heiles system for energy 32/375 with x(0) = x(0) = y(0) =y(0) = 0.2 (left) and energy 0.124416 with x(0) = x(0) = y(0) = y(0) = 0.24 (right;see Fig. 12.12 in the book). A red dot is plotted in the phase space (y(t), y(t)) wheneverx(t) = 0 and x(t) > 0. On the right one observes chaotic behavior with “islands” of non-chaotic phase space.
Poincare map of the phase space using event handling. I could not resist trying it outfor myself. Figure 2 shows Poincare maps for two different energies, and on the rightwe see a beautiful figure showing chaotic behavior with “islands” of nonchaotic phasespace. The energy was conserved up to eight digits. The inline codes in the book arean excellent starting point for students to begin further experimentation.
Corless and Fillion have clearly dedicated several years of their lives to this book.It has been worth it! The result is a carefully judged all-round numerical analysistextbook that expertly serves students and instructors. The book stands out in acrowded market. It is a textbook that numerical analysis can be very proud of.
REFERENCES
[1] D. A. Aruliah, R. M. Corless, L. Gonzalez-Vega, and A. Shakoori, Geometric applicationsof the Bezout matrix in the Lagrange basis, in Proceedings of the International Workshopon Symbolic-Numeric Computation, ACM Press, New York, 2007, pp. 55–64.
[3] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, On theLambert W function, Adv. Comput. Math., 5 (1996), pp. 329–359.
[4] N. J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, Philadelphia,2002.
[5] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes:The Art of Scientific Computing, 3rd ed., Cambridge University Press, Cambridge, UK,2007.
[6] G. Strang, Introduction to Linear Algebra, 4th ed., Wellesley–Cambridge Press, Wellesley, MA,2009.
[7] L. N. Trefethen, Approximation Theory and Approximation Practice, SIAM, Philadelphia,2013.
[8] A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. B, 237 (1952),pp. 37–72.
An Introduction to Polynomial and Semi-Algebraic Optimization. By Jean Bernard Las-serre. Cambridge University Press, Cambridge,UK, 2015. $60.00. xiv+339 pp., softcover.ISBN 978-1-10763-069-7.
Consider an optimization problem of theform
min f0(x)
subject to
fi(x) ≥ 0, i = 1, 2, . . . ,m.
This optimization problem can be ex-tremely hard to solve, but it has variousrestricted forms that are more easily solv-able. This book focuses on the special caseswhere the functions fi(x) are polynomials.This class of problems is surprisingly broadand includes many computationally chal-lenging problems. For example, integer pro-gramming problems involving 0–1 variablescan be formulated by imposing constraintsx2j − xj = 0.Recent developments in algebraic geom-
etry and semidefinite programming havemade the solution of small to medium sizedpolynomial optimization problems using thetechniques described in this book compu-tationally feasible. These techniques havealready been applied in a variety of applica-tions, and as modeling systems and semidef-inite programming software improve, we canexpect to see wider application of polyno-mial optimization.
The first part of the book is concernedwith nonnegative polynomials and sums ofsquares. A sum of squares polynomial is apolynomial q(x) that can be written in theform
q(x) =k∑
i=1
qi(x)2.
Clearly, any sum of squares polynomial isnonnegative for all x. For univariate poly-nomials, the converse is also true. Further-more, we can use semidefinite programmingto determine whether q(x) can be writtenas a sum of squares. A univariate polyno-mial q(x) of degree 2d can be written asa sum of squares if and only if there is asymmetric and positive semidefinite matrix
Q such that
q(x) =[1 x . . . xd
]Q
⎡
⎢⎢⎢⎣
1x...xd
⎤
⎥⎥⎥⎦ .
If q(x) can be written in sum of squaresform, then we can write
⎡
⎢⎢⎢⎣
q1(x)q2(x)
...qk(x)
⎤
⎥⎥⎥⎦= M
⎡
⎢⎢⎢⎣
1x...xd
⎤
⎥⎥⎥⎦
and let Q = MTM . Conversely, if such amatrix Q exists, then we can find a sumof squares representation of q(x) by using aCholesky factorization Q = RTR.
The semidefinite programming approachto determining whether a polynomial canbe written in sum of squares form is easilyextended to multivariate polynomials. Un-fortunately, it is not true for multivariatepolynomials that a polynomial is nonnega-tive if and only if it can be written as a sumof squares. However, it can be shown thatin general a multivariate polynomial g(x) isnonnegative if and only if it can be writtenas the sum of squares of rational functions.Equivalently, g(x) is nonnegative if and onlyif there are sum of squares polynomials f(x)and h(x) such that f(x) = g(x)h(x). Again,we can solve a semidefinite programmingproblem to determine whether or not g(x)is nonnegative.
Using results from algebraic geometry,many semidefinite programming formula-tions of polynomial optimization prob-lems and optimization problems over semi-algebraic sets are possible. The second partof this book is concerned with develop-ing these formulations and analyzing theirproperties. An important result is that a hi-erarchy of semidefinite programming relax-ation of successively larger sizes convergesfinitely to an exact solution of the polyno-mial optimization problem.
In the third part of the book, the au-thor discusses some specialized topics andapplications, including convexity in poly-nomial optimization, parametric polyno-mial optimization, convex underestimators
of polynomials, and inverse polynomial op-timization. Appendices cover semidefiniteprogramming and the author’s GloptiPolysoftware package for polynomial optimiza-tion.
Although this book has been written inthe form of a textbook with exercises atthe end of each chapter, the presentation isat an advanced level and depends on pre-requisite knowledge of algebraic geometryand semidefinite programming. Relativelyfew students are likely to have the back-ground to study this book without addi-tional support. It is likely to be most usefulas a reference for advanced graduate stu-dents and researchers working in this area.As a reference the coverage of the topic isauthoritative and thorough.
The author’s previous book focused onthe generalized moment problem, which isin a sense dual to the problem of poly-nomial optimization [1]. The generalizedmoment problem has applications in prob-ability theory, analysis of Markov chains,mathematical finance, and control systems.Although the current book discusses the du-ality between moment problems and poly-nomial optimization, readers interested inapplications will find more material on ap-plications in the earlier volume. Anotheruseful feature of the earlier book is an ap-pendix covering the required background inalgebraic geometry. Many readers will findit useful to refer to both books.
REFERENCE
[1] J. B. Lasserre, Moments, Positive Poly-nomials and Their Applications, Vol. 1,World Scientific, 2009.
BRIAN BORCHERS
New Mexico Tech
Variational Analysis in Sobolev and BVSpaces. Applications to PDEs and Opti-mization. Second Edition. By Hedy Attouch,Giuseppe Buttazzo, and Gerard Michaille. SIAM,Philadelphia, PA, 2014. $141.00. xii+793 pp.,hardcover. ISBN 978-1-611973-47-1.
This is the second edition of a book firstpublished in 2006. It is very impressive
and contains much more than the titlewould suggest. The first version was al-ready a very rich textbook in analysis, go-ing from the basics (topologies and con-vergences, calculus of variations, measuretheory, Sobolev spaces and capacities, con-vex analysis) to more involved topics (BVand SBV functions, lower semicontinu-ity and relaxation, Young measures, Γ-convergence), illustrated by the detailedanalysis of a few simple variational prob-lems. It also contained a complete chapteron the finite element method. The secondedition has been substantially enriched withmore examples, new theoretical tools suchas an introduction to (variational) stochas-tic homogenization and to optimal trans-portation, and with a new chapter (of morethan 100 pages!) on gradient flows.
Like its previous edition, this book isdivided into two parts, titled “Basic Vari-ational Principles” and “Advanced Varia-tional Analysis.” The first part containsnine chapters, starting with the fundamen-tals of analysis (distributions, topologies,lower-semicontinuity, including an originalapproach to Ekeland’s variational princi-ple), a selection of some well-presented mea-sure theoretic tools (Caratheodory’s con-struction and Hausdorff measures, Youngmeasures, capacity theory), followed bythe analysis of some variational PDEsand systems in various settings (Dirich-let, Neumann, or mixed boundary condi-tions and transmission conditions, nonlin-earities, Lame system, critical points, ob-stacles) with, in fact, many more examplesthan in the previous edition.
This part ends with a chapter on thebasics of the finite element method andan introduction to standard error estimates(note that Galerkin’s approximation has al-ready been introduced in the third section,where it is used to give a simple proofof the Lax–Milgram theorem), a detailedchapter on the spectral decomposition ofthe Laplace operator, and a 55-page intro-duction to convex duality, in Chapter 9,which could be seen as a minitextbook initself. Indeed, this last section is very com-plete and nicely introduces theoretical toolssuch as inf-convolutions, Legendre–Fenchelduality, subdifferential calculus, and prac-tical optimization (multipliers, KKT con-
ditions, dual problems, linear programs).It ends with a detailed introduction toconvex-concave saddle points problems andFenchel–Rockafellar duality.
The second part of the new edition nowcontains eight chapters. The first one, as be-fore, is devoted to BV and SBV functions.Many important technical points, such asthe rectifiability of the level sets, are proven,at least partially. SBV functions are intro-duced to pave the way for the models of im-age segmentation and fracture growth whichare introduced further on, in Chapter 14.Then follows a long and detailed chapter onlower semicontinuous relaxation, in partic-ular, in BV or measure spaces. A short the-oretical introduction is followed by a sectionon integral functionals with p-growth, whereimportant concepts such as Morrey’s quasi-convexity are discussed (in fact, part of thisdiscussion is spread between this chapterand a further one on lower semicontinuity);the Young measure approach to relaxationis further discussed in great detail in a quitesubstantial section. The case of relaxationfor problems with growth 1, in the spaceBVof functions with bounded variation, whichrequires more advancedmeasure-theoreticaltools, is also briefly considered (and reap-pears two chapters later). An addition tothe second edition is a very brief section onmass transportation, which introduces someessential notions and describes the Monge–Kantorovich duality. (This part might havebeen more appropriate as an illustration ofconvex duality in Chapter 9, or as a separateand more complete 18th chapter at the endof the book, whereWasserstein flows or Bre-nier’s theorem could have been discussed.)
The next chapter introduces the notionof Γ-convergence (in metrizable spaces) andthen quickly switches to useful applicationssuch as 3D-2D limits or (variational) ho-mogenization. Quite interesting in this newedition are the almost 30 pages on stochas-tic homogenization of minimization prob-lems (with growth p > 1), which covera topic rarely found in textbooks. Thischapter ends with a brief description ofModica–Mortola and Ambrosio–Tortorelliapproximations of perimeter/free disconti-nuity problems. Chapter 13 is devoted tothe lower semicontinuity of integral func-tionals in the scalar and vectorial cases,
and refines some of the results of Chap-ter 11 (maybe these parts could have beenmerged together, as there is some redun-dancy). It also addresses the issue of SBVfunctions, which are used in some of theapplications studied in the next chapter.Indeed, this next part, which is very inter-esting, shows how the previously introducedtools are used in practical examples suchas (Hencky) plasticity, fracture mechanics,and the Mumford–Shah functional. Chap-ter 15 is quite original. It addresses theissue of coercivity and introduces tools forthe study of noncoercive variational prob-lems. In particular, it contains a very de-tailed analysis of the properties of reces-sion functions of convex functions, whichare not easily found elsewhere. Next comesan introduction to some shape optimiza-tion problems. A few interesting examplesare given and the most useful technicaltools to deal with a few fundamental op-timization problems (with respect to a do-main or a potential in an elliptic PDE)are described. Finally, Chapter 17 is en-tirely new and is an important addition tothe book. It is devoted to gradient flows,mostly in the convex case, and containsfundamental notions as well as quite orig-inal material. Four important subjects aredeveloped. It starts with classical results(Cauchy–Lipschitz theorem, asymptotics),followed by a quite complete descriptionof convex gradient flows, with importanttools from the theory of maximal mono-tone operators such as Moreau–Yosida ap-proximation, Chernoff’s lemma, a versionof Opial’s lemma, etc. This is illustratedby PDE examples such as the Stefan prob-lem. A following section is devoted to theasymptotics of descent trajectories of real-analytic functions, based on the Kurdyka–#Lojasiewicz inequality; the recent extensionto the nonsmooth case (using semialgebraicfunctions) is also described. A third partstudies limits of sequences of gradient flowproblems, in the convex case (hence, basedon Mosco-convergence of functionals), withan interesting application to stochastic ho-mogenization in diffusion equations. Even-tually, gradient flows in metric spaces arerapidly introduced together with the mini-mizing movement approach of De Giorgi, ina very short section which refers primarily
to the well-known monograph of Ambrosio,Gigli, and Savare.
All in all, this long textbook is verycomplete and pleasant to read, with a pro-gressive level of difficulty and complexityand many nice examples which illustratethe theoretical results. It contains deepand precise information on many impor-tant tools in variational analysis (functionalanalysis, convex analysis) and many ad-vanced methods, together with a generaloverview of most of the modern techniques.It should be useful for both students andresearchers, whether they need to learn orreview some advanced techniques in anal-ysis or are looking for an introduction tomore recent theories.
ANTONIN CHAMBOLLE
CNRSEcole Polytechnique
Plasticity: Mathematical Theory and Nu-merical Analysis. Second Edition. By Wei-min Han and B. Daya Reddy. Springer, New York,2013. $149. xvi+424 pp., hardcover. ISBN 978-1-4614-5939-2.
The discipline of plasticity is concernedwith the study of irreversible deforma-tions in solids. The governing equations forplastic evolution can take widely differentforms depending on the phenomenologicalmodel and loading conditions considered.This renders plasticity a challenging pur-suit for both the solid mechanician and themathematician. Moreover, due to the rapidprogress made in plasticity research, themechanician has been left rather unfamiliarwith the developments in the mathematicaltheory and likewise the mathematician withthe physical background of the recent plas-ticity models. The book under review is anambitious effort to fill this gap by makingrecent mathematical research in plasticityaccessible to the nonmathematician withoutlosing sight of both the rigor as well as thephysical basis for plasticity theories. Thebook is unique in its broad consideration ofanalytical and numerical aspects of plastic-ity. In this second edition, important ma-terial relevant to strain gradient and singlecrystal plasticity theories has been added.
The book is divided into three parts.The first part, consisting of four chap-ters, provides a quick summary of the rel-evant notions from continuum mechanicsand introduces several formulations of rate-independent elastoplasticity. The latter ispresented first in a classical framework andthen in a convex-analytic setting. After fix-ing the notation in Chapter 1, conceptsfrom continuum mechanics, including kine-matics, balance laws, dissipation, and lin-earized elasticity, are collected in Chap-ter 2. Next, in Chapter 3, this is followedby a survey of rate-independent elastoplas-tic theories within a classical framework,with an emphasis on isotropic strain gra-dient models of plasticity (particularly theGurtin–Anand model) and small deforma-tion single crystal plasticity. Chapters 2 and3 provide only aminimalistic, although well-written, exposure of the physical aspects ofplasticity to a nonspecialist, who will haveto look elsewhere for more comprehensivetreatments. The elastoplastic problems arerestated again in Chapter 4, now withina convex-analytic framework. In this form,the problems can be formulated in moregenerality, such as allowing for nonsmoothyield loci, in addition to being amenable forfurther mathematical analysis.
The second part of the book seeks toresolve the well-posedness of the initial-boundary value problems of elastoplastic-ity introduced in Chapters 3 and 4. Tothis end, the problems are posited in termsof variational inequalities which are sub-sequently used to demonstrate existenceand uniqueness of the solutions. The pre-requisite notions from functional analysisand variational inequalities are collected inChapters 5 and 6, respectively. An inter-ested reader, looking for details, would haveto look at several excellent texts availableon these subjects. Chapters 7 and 8, com-bined with their numerical counterparts inChapters 12 and 13, form the core of thebook. They contain a detailed mathemat-ical analysis of the primal and the dualvariational problems of elastoplasticity, re-spectively. Whereas the former has dis-placement, plastic strain, and hardeningparameters as unknowns, the latter solvesfor generalized stress as the unknown vari-able. In both the cases, the emphasis is
on proving existence and uniqueness of so-lution to the weak formulation of classicaland strain-gradient plasticity problems forpolycrystalline and single-crystalline mate-rials. Drawing heavily from their researchpapers, the authors have successfully man-aged to present a developed picture of themathematical theory. This should be ex-tremely valuable for both the mechanicianand the mathematician.
The final part of the book is concernedwith the numerical analysis of computa-tional algorithms for solving elastoplasticityproblems. The temporal and spatial varia-tions are approximated using the finite dif-ference and finite element methods, respec-tively. After succinctly recalling pertinentaspects of finite element analysis in Chapter9, approximation of variational equationsand inequalities using the finite elementmethod is discussed in Chapter 10. The ma-jority of the discussion is on obtaining rea-sonable error estimates for the finite elementsolution of the variational problems. Chap-ter 11 takes another step toward that goalby introducing semidiscrete and fully dis-crete approximations and establishing therelevant error estimates. Convergence un-der minimal regularity assumptions is alsoestablished. Finally, in Chapters 12 and13, we come back to elastoplasticity. Chap-ter 12 focuses on the implementation ofnumerical schemes for the primal elasto-plastic problem. The error estimates andsolution algorithms are derived for vari-ous plasticity models and convergence ofthe algorithms is rigorously established. InChapter 13, numerical analysis of the dualvariational problem is undertaken. Again,several numerical schemes are introducedand analyzed. The focus of this chapter ison implementing time-discrete schemes forclassical problems in plasticity.
The book is well written and carefullypresented overall. It presents a wealth ofuseful material otherwise absent from otherplasticity books. It exposes both the in-terested mechanician and the interestedmathematician to mathematical problemsin plasticity in a unified manner, albeit tobe pursued in their own way. My only dis-appointment with the book is the absence ofbibliographic notes at the end of each chap-ter (aside from one in Chapter 3). Their
inclusion would have not only provided fur-ther reading directions and open problems,but also given an overall perspective on anactive research discipline within which thecontents of the book are placed.
ANURAG GUPTA
Indian Institute of Technology Kanpur
Spline Functions: Computational Meth-ods. By Larry L. Schumaker. SIAM, Philadelphia,PA, 2015. $83.00. xii+412 pp., hardcover. ISBN978-1-611973-89-1.
The book is a long-awaited sequel to twoprevious books on splines, [2] and [1],by the same author. The new book isa product of over fifty years of research,teaching, and collaboration with numerousscientists within and outside the field ofsplines. The complete bibliography com-prised of over 100 pages was too long forthe hard copy and has been put online;see [3]. A very valuable part of the bookis a MATLAB package, SplinePak, whichis freely available both on [3] and on LarryL. Schumaker’s website [4]. For the impa-tient reader, I would recommend download-ing the package and going straight to theexamples in the book, which, admittedly,is exactly what I did. Within minutes, Ihad a beautiful picture of a minimal energyquadratic smooth spherical spline interpo-lating f(x, y, z) = x4 + 1.1y4 + 1.3z4 at 42vertices of a triangulation of the unit sphere;see Figure 3. Each numerical example in thetext describes a problem and has a referenceto the code that provides a solution. Theonly drawback is that the MATLAB func-tions in SplinePak are currently p-files, andthus cannot be modified. The script filesare the usual m-files. A complete list of thescripts and the functions in the package isincluded at the end of the book.
For the more patient reader, I recom-mend downloading the package and gettinghold of both of Larry Schumaker’s previousbooks on splines, Spline Functions: BasicTheory [2] and Spline Functions on Trian-gulations [1] (coauthored with M. J. Lai).This takes us to the next truly valuablefeature of the book: it is equally useful tothe reader looking for algorithms to solve
practical problems and to the reader in-terested in rigorous foundations for suchalgorithms. While the book itself includesfew proofs, it contains rigorous statementsof all theorems used, and full referencesto their proofs elsewhere. Most of the ref-erenced proofs can be found in [2] and[1]. I will demonstrate this approach bysummarizing the content of one section—section 5.2—titled “The C1 Powell–SabinInterpolant.” It begins by introducing theso-called Powell–Sabin refinement of a tri-angulation. The following theorem statesinterpolating conditions sufficient to definea unique interpolating C1 quadratic splineon the Powell–Sabin refinement. Immedi-ately after the theorem we find a referenceto section 6.3 of [1], where the proof can befound. Next, we discover all the necessaryformulae to define the coefficients of the in-terpolating spline, followed by a referenceto a code in SplinePak that returns a vectorof the coefficients of the spline along with afigure of the spline surface. Two numericalexamples that are discussed next includethe maximum and root mean square errorsand the convergence rate analysis. The twoexamples treat an interpolation problem ofFranke’s function on a triangulation with 36vertices, and on several refinements of type-I triangulation. The section concludes withthe statement of a rigorous error bound inSobolev norm followed by the general ideaof the proof and a reference in [1], wherethe proof can be found.
The material covered by the book isbroad. Piecewise polynomials (splines) in
one or two variables can be used to solveapproximation, interpolation, data fitting,numerical quadrature, and ODE and PDEproblems, and this is exactly what the bookshows the reader how to do. Chapter 1 dealswith univariate splines, their evaluation,and their use in interpolation, approxima-tion, and solving two-point boundary-valueproblems. Chapter 2 treats similar aspectsof bivariate tensor-product splines. The-oretical foundations are covered in [2]. InChapter 3 we learn how to deal with triangu-lations computationally. Chapter 4 is essen-tial for understanding the rest of the book.It covers foundations of the Bernstein–Bezier representation of bivariate polyno-mials and splines. The first application ofbivariate splines—Hermite interpolation—is demonstrated in Chapter 5, followed byscattered data fitting in Chapters 6, 7, and8. Elliptic PDEs of orders two and four arethe focus of Chapter 9, where the Ritz–Galerkin method is implemented with bi-variate macroelements. Chapters 9 and 10cover spherical splines and their applica-tions. Theoretical foundations for Chap-ters 3–5, 10, and 11 can be found in [1].The bibliography included at the end of thebook contains referenced books only, whilethe online one contains research papers andother resources.
Yet another aspect of this book thatmakes it very attractive is the fact thatthe research content is fully up to date. Forexample, the classical subject of comput-ing triangulations includes recent develop-ments on triangulations with hanging ver-
tices. Solving a classical PDE problem usingthe finite element method bypasses the ref-erence triangle by means of the Bernstein–Bezier representation and uses finite ele-ments of higher degree and smoothness.Each chapter ends with remarks and his-torical notes, where all original sources arecited and additional research articles arereferenced.
The book is an invaluable resource formany different types of readers, includingresearchers in the field of numerical analy-sis, applied mathematicians, computer sci-entists and engineers, and graduate stu-dents, as well as computational specialistsfrom other sciences who are willing to applysplines to their fields.
REFERENCES
[1] M.-J. Lai and L. L. Schumaker, SplineFunctions on Triangulations, Cam-bridge University Press, Cambridge,UK, 2007.
[2] L. L. Schumaker, Spline Functions: Ba-sic Theory, Wiley Interscience, NewYork, 1981.
[3] www.siam.org/books/ot142.
[4] www.math.vanderbilt.edu/∼schumake/.
TATYANA SOROKINA
Towson University
Computational Mathematical Modeling:An Integrated Approach Across Scales.By Daniela Calvetti and Erkki Somersalo. SIAM,Philadelphia, PA, 2013. $71.50. x+224 pp., soft-cover. ISBN 978-1-611972-47-4.
The book Computational MathematicalModeling: An Integrated Approach AcrossScales is divided into nine chapters, wherethe first four are devoted to deterministicmodels which actually boil down to solv-ing ordinary differential equations. The re-maining five chapters deal with mathemat-ical models that include noise, i.e., randomnumbers as a stochastic element.
The material is presented in a writingstyle typical of mathematical textbooks,with many theorems and formal definitionsthat will probably be easy to grasp for mathmajors. Each topic is presented so as to con-vey the basic principles without going into
too much detail. For example, Chapter 5on random variables and distributions couldeasily fill an entire book. Hence, in generalthe surface of each topic is merely scratchedand, after an introduction of the govern-ing equations, examples are provided forpossible solution strategies. Some readerswill find the MATLAB exercises, which arescattered throughout the book, useful eventhough their solutions are only provided ascode snippets, so the student will have to fillin the gaps. No introduction to MATLAB isprovided. This comes as somewhat of a dis-appointment as the word “computational”in the book title suggests more than justa bunch of MATLAB snippets. In termsof modeling, the authors stay almost com-pletely in the realm of pure mathematicsand only rarely are examples given fromother areas. For example, in Chapter 7 thestochastic simulation of chemical reactionsis discussed though this, however—to thereviewer’s knowledge—is not really used forcalculating reaction kinetics in chemistry.Chemists use the Arrhenius equation forthat. Markov processes and the standardpredator-prey model used in biology arecovered to some extent in Chapter 8.
Each chapter ends with a number ofexercises which are almost all purely math-ematical. Some of them involve writingMATLAB code. Considering that neitherhints nor solutions are provided to the exer-cises, I have doubts that they will be of muchuse to the average student, because most ofthem are hard to carry out after having readonly the preceding chapter. There is a use-ful subject index and a good bibliography atthe end of the book. Each chapter addition-ally has its own bibliography, which is quiteuseful. However, the chapter bibliographiesare not very extensive: most referencescite other books and there are hardly anyreferences to primary source papers.
All in all, this reviewer thinks that thisbook is a good read, is technically sound,and can be recommended for beginning toadvanced graduate students who want tobecome acquainted with several basic ideasin mathematical modeling. The book willprobably be most useful to math majorsdue to its presentation style. I doubt thatthe book will be of much use for studentsmajoring in subjects other than math, de-
spite what the back cover suggests, becausevery few applications from other fields arediscussed. Students fluent in the MATLABscripting language will find the correspond-ing exercises and examples helpful.
MARTIN O. STEINHAUSER
University of Basel
Dirichlet–Dirichlet Domain Decomposi-tion Methods for Elliptic Problems: hand hp Finite Element Discretizations. ByVadim Glebovich Korneev and Ulrich Langer. WorldScientific, Hackensack, NJ, 2015. $128.00.xx+463 pp., hardcover. ISBN 978-981-4578-45-5.
The finite element method for solving el-liptic boundary value problems first gainedpopularity some fifty years ago. The useof iterative methods to solve the result-ing systems became imperative as their sizegrew over time. In order to solve the hugesystems in parallel, domain decomposition(DD) techniques were created. These areiterative methods that proceed by solvingsmaller problems on subdomains. Althoughthey can stand on their own as iterativemethods, they are more effective when usedas preconditioners for the conjugate gradi-ent method. For maximum efficiency, thesubdomain problems are also solved itera-tively, typically by a preconditioned conju-gate gradient method. A major objectiveis to achieve optimal computational com-plexity by producing a method for whichthe number of iterations is bounded as theelement size h → 0. (In the hp version,the polynomial degree p → ∞ as well.) Forthis it suffices to keep the condition num-ber of the preconditioned system boundedas h → 0 (and, in the hp version, p → ∞).The right choice of preconditioners for thesubdomain problems is crucial.
The DD literature is huge, having ex-ploded in the past thirty years. DD tech-niques come in a variety of flavors; for onething, the subdomains can be overlappingor not. The overlapping methods are eas-ier to understand and implement, but thenonoverlapping methods may be more use-ful. For example, if the coefficients of thedifferential operator have jump discontinu-
ities, it is better to have subdomain bound-aries at the jumps and not have any overlap.
Within the nonoverlapping methodsthere are many variants. The book un-der review focuses on a specific class ofDD methods called Dirichlet–Dirichlet af-ter the type of boundary conditions thatare used in the subdomains. This is a bookfor experts only. The entire jargon of finiteelement and iterative solver theory is usedwith little or no explanation. The authorsexpect that the reader knows what thesethings mean.
The book has one chapter on overlappingmethods, but the main focus is on nonover-lapping Dirichlet–Dirichlet DD techniques.There are separate chapters on the 2D and3D cases, as there are significant differencesin the way they are organized. A major ob-jective of the authors was to present the hptheory, which is more recent and more com-plicated, along with the older and better-established h theory. Thus, there is onechapter each on h version in 2D, h versionin 3D, hp version in 2D, and hp versionin 3D. By now I have mentioned most ofthe book’s chapters, and this is the mainstructure of the book, but there are a fewother supporting chapters as well.
This book will certainly be useful to indi-viduals pursuing research on DD methods.It could have benefited from some aggres-sive copyediting by a person whose nativelanguage is English.
DAVID S. WATKINS
Washington State University
The Computing Universe: A Journeythrough a Revolution. By Tony Hey and GyuriPapay. Cambridge University Press, Cambridge,UK, 2015. $85.00. xvi+397 pp., hardcover.ISBN 978-052-1766-45-6.
This is a popular book on computer sciencemeant to be intelligible to high school anduniversity students, as well as general read-ers. The goal is to get young people excitedabout the field. Technicalities are kept toa minimum, and almost all of the chapterscould be read by just about anybody.
The book covers a lot of ground, from thebeginning of the electronic computing era
right up to the present. Here is a sampleof chapter titles: “The Software is in theHoles,” “Mr. Turing’s Amazing Machines,”“Computing gets Personal,” “Licklider’s In-tergalactic Computer Network,” “The DarkSide of the Web,” “The End of Moore’sLaw,” “The Third Age of Computing.”There are 17 chapters in all. They do notneed to be read in order: The reader canturn to a random chapter, or even a randompage, and find something interesting. On al-most every page there is at least one boxor sidebar with information about a famousperson in the field, or an interesting anec-dote, or a cartoon. On pages 82–83 you canread about three space launch catastrophesthat were caused by software errors. Onpage 117 you will learn about Kurt Godel’sinterview for U.S. citizenship, accompanied
by character witnesses Einstein and Mor-genstern. Fortunately, the interview wassuccessful even though Godel had noticedinconsistencies in the constitution. On page359 is a photograph of Turing’s work areaat Bletchley Park with his tea mug chainedto the radiator so that it won’t be stolen. Iguess these old items stand out to this oldreader, but there is lots of newer materialas well.
I’ve found this book to be interesting andinformative, and I’m still reading. WhenI’m done, I’ll leave it lying around some-where where the 16-year-old in my house-hold might pick it up and start reading it.This assumes she can tear hers eyes awayfrom her smart phone.
