A Two-Dimenstional Bisection Envelope Algorithm for Fixed Pointsehan/cs7940/presentation.pdf ·...

4/24/2007School of Computing

EunGyoung Han 1

A TwoA Two--Dimensional Bisection Dimensional Bisection

Envelope Algorithm Envelope Algorithm

for Fixed Pointsfor Fixed Points

Kris Sikorski and Spencer ShellmanFrom published Journal of Complexity 18, 641-659(2002)

EunGyoung Han


EunGyoung Han 2

IntroductionIntroduction

� How we solve for two-dimensional – domain: [0, 1]X[0, 1]

– f : Lipschitz continuous function (q = 1).

� Previous method– Time complexity was bad

� Paper introduce new algorithm– Computes approximate satisfying

– Tolerance

– Upper bound on the function evaluations

x~

xxfrr

=)(

ε≤−∞

xxf ~)~(

5.0<ε

.1)/1(log2 2 +ε


EunGyoung Han 3

HistoryHistory

� 1920s - Present

– Banach’s simple iteration algorithm

– Homotopy continuation

– Simplicial and Newton type methods

� Time complexity

– Lipschitz function (q>1)

• Exponential in the worst case

• Lower bound is also exponential (best case)

1,)()( >−≤−∞∞

qyxqyfxf


EunGyoung Han 4

Problem FormulationProblem Formulation

� Class of Lipschitz continuous functions

� By the Brouwer fixed point theorem

{ }∞∞

−≤−∈∀→= yxyfxfDyxDDf babababaF )()(,|: ,,,,

.)(such that , into maps **

,

*

,,, xxfDxDDFf babababa =∈∃∈∀


EunGyoung Han 5


� We know a solution exists, we just need a

constructive algorithm…

� Two different criteria to satisfy

– Residual criterion

• Can always be satisfied

– Absolute criterion

• Can sometimes be satisfied

.~)~( ε≤−∞

xxf

.~ εα ≤−∞

x


EunGyoung Han 6


� To find the fixed point using the Bisection

Envelope Algorithm, we are required nfunction evaluations of f, where

.11

log21 2 +

≤≤

εn


EunGyoung Han 7

Envelope TheoremEnvelope Theorem

� Define the fixed point sets forbaFf ,∈

)()(

},)(|{

},)(|{

21

22,2

11,1

ff

xxfDx

xxfDx

ba

ba

FFF

F

F

∩=

=∈=

=∈=rr

rr


EunGyoung Han 8

Theorem 3.5Theorem 3.5

� …

. satifies ~ then , of

point fixed a contains addition,in If,

. satifies )(~Then

.intersect )( and )(both and

2)()(such that Let

21

11,

reterianabsolute cyf

R

reterianresidual cRcy

Rff

RlRlDR ba

=

≤+⊂ −

FF

ε


EunGyoung Han 9

Theorem 3.5Theorem 3.5

ε≤

ε≤)(Rc

)(2 fF

)(1 fF

)(1 Rl

)(1 Rl−

ε.x)-xf(

c(R)

ε(R)l(R)l

R

≤

≤+

∞

−

~~

satifies so

2

satifies

11


EunGyoung Han 10

The The BEFixBEFix Algorithm: DefinitionAlgorithm: Definition

� ..

� ..

}.1)5.0,5.0(:{ and

domains theDefine

1

2

0

5.1,5.0

≤−ℜ∈=

= −

xxD

DD

points. fixed all contains -

.D within existspoint fixed oneleast At -

on )1( continuous Lipschitz is ,: If

D

DqfDDf =→


EunGyoung Han 11

The The BEFixBEFix Algorithm: Figure D, et.Algorithm: Figure D, et.

5.1

D D

DD

0D

0D

0D

0DD

5.1

5.0−

5.0−


EunGyoung Han 12

The The BEFixBEFix Algorithm: ProjectionAlgorithm: Projection

� Projection

– ..

– ..

))),1min(,0max()),,1min(,0(max()( 21 xxxP =

. where, onto project Let DDDDP ⊇

.~)~( and ~ where

,for solution residual a is)~(~then

,for solution residual a is ~ If

0 ε≤−∈

=

yyfDy

fyPx

fy


EunGyoung Han 13

The The BEFixBEFix Algorithm: Description Algorithm: Description

�

�

.~criterion absolute

satifies ~ ifonly trueiswhich

variablelogical returns Algorithm

εα ≤−y

y

abs

.~)~(

osolution t a as )~(~ takesAlgorithm

ε≤−

=

xxf

yPx


EunGyoung Han 14

The The BEFixBEFix Algorithm: ConstructionAlgorithm: Construction

� Constructs a algorithm

–

ε2)()( ifor -

at satified iscriterion residual a If-

: when stepat s terminateAlgorithm -

1)-or 1 (slope rectangle closed a is Each -

. through 1-or 1 slope with lines along bisecting

by rectangle a Constructs -

stepon at Evaluates -

11

1

1

≤+

=

⊂

−

−

−

kk

k

k

k

k

kk

k

DlDl

x

k

D

xD

DD

kxf


EunGyoung Han 15

BarycentricBarycentric Coordinate SystemCoordinate System

� Find the next centroid by using Barycentric coordinate system at .

�

)(DC k

r

bax

xba

DCx

lbla

k

k

k

k

rrr

MM

rM

rM

rMM

r

rr

βα

β

α

++=

==

−=

=

−

−

−

1

1

11

1

)(

1

1

2

2,

1

1

2

2

br

ar

12

−= lbr

12la =

r

)(DC 1-k

r


EunGyoung Han 16

BarycentricBarycentric coordinate systemcoordinate system

� Define the basis vectors of the

Barycentric coordinate system

relative to the origin defined by x.

� The vectors and point in the

directions of the and edges of the rectangle.

1l 1−l

br

ar


EunGyoung Han 17

Algorithm AnalysisAlgorithm Analysis

3x

4x

5x

1x2

x

)()(

0)(

01

11

1

DCDC

xxfV

>⇒

>−=

05 =V

)( 0DC

)()(0 122 DCDCV >⇒<

xz =

)(xfz =

)(1 xf

)( 1DC

1=x

axis- x

1=z

=

0

0

z

x


EunGyoung Han 18

Fixed PointFixed Point


EunGyoung Han 19

3D intersection of Pyramid function3D intersection of Pyramid function


EunGyoung Han 20

Visualize intersectionVisualize intersection


EunGyoung Han 21

Algorithm Analysis: ConvergenceAlgorithm Analysis: Convergence

br

ar

12

−= lbr

12la =

r


EunGyoung Han 22

Algorithm Analysis: ConvergenceAlgorithm Analysis: Convergence

� Exponential decay of infinity norm residuals.


EunGyoung Han 23

ComplexityComplexity

.1)/1(log21)/2(log2 22 +=−≤ εεk

where,2)()( since

,2)()(

and ,2

2)(satisfy

0101

11

max

==

≤+

≤

−

−

DlDl

DlDl

DlD

kk

kk

ε

ε


EunGyoung Han 24

Numerical Tests: Numerical Tests: Pyramid basis functionPyramid basis function

� Tests Pyramid Function defined as

]1,0[ and for ]1,0[:

function basis Pyramid where

)),0,max(,1min()(

∈∈→

−−=∞

hDbDP

bxhxP

h

b

h

b


EunGyoung Han 25


� Plots of for several values of b and hh

bP


EunGyoung Han 26



EunGyoung Han 27

Numerical Tests: 3DPyramid TestsNumerical Tests: 3DPyramid Tests

� Tests 3-Dimensional Pyramid function

{ } .41,13,,1 of and subsetsempty non

of pairs allfor ))(),(()(

functions on the algorithm theTested

.131,131,,,

integersdistinct given the where

)),(,),(max()(

21

21

1

1

1,,1

−=

=

−

∀≤≤≤≤

=

eSS

xPxPxf

jikii

xPxPxP

ss

jk

h

b

h

biiki

ki

i

ik

εL

L

LL


EunGyoung Han 28

Numerical Tests: 4DPyramid TestsNumerical Tests: 4DPyramid Tests


EunGyoung Han 29

Complex Complex abs(Cabs(C))


EunGyoung Han 30

Complex angleComplex angle


EunGyoung Han 31

Numerical TestsNumerical Tests

� Tests algorithm on the functions

– Average ratio of a test’s function evaluations to

– Total number of tests satisfying the absolute error criterion: 21,776.

– Average ratio of a test’s function evaluations to

, for satisfying the absolute error criterion:0.522.

– Minimum number of function evaluations achieved by a test: 1.

.759.0:291)/1(log2 2 =+ε

291)/1(log2 2 =+ε


EunGyoung Han 32

Future workFuture work

� Plan algorithm works for any dimension

– Complexity in lower bound

� Investigate the restricted function class may have finite complexity in the absolute

criterion.

.2≥d

.in polynomial is )( where

)),/1log()((

ddc

dcO ε


EunGyoung Han 33

� Thank You

� Questions ?

Date post:	28-Nov-2020
Category:	Documents
Upload:	others
View:	1 times
Download:	0 times

A Two-Dimenstional Bisection Envelope Algorithm for Fixed Pointsehan/cs7940/presentation.pdf ·...

Documents