A Two- -Dimensional Bisection Dimensional Bisection A Two - - PowerPoint PPT Presentation

4/24/2007 School of Computing EunGyoung Han 1

A Two A Two-

• Dimensional Bisection

Dimensional Bisection Envelope Algorithm Envelope Algorithm for Fixed Points for Fixed Points

Kris Sikorski and Spencer Shellman

From published Journal of Complexity 18, 641-659(2002)

EunGyoung Han

4/24/2007 School of Computing EunGyoung Han 2

Introduction Introduction

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 ~

x x f

• =

) (

ε ≤ −

∞

x x f ~ ) ~ (

5 . < ε

 

. 1 ) / 1 ( log 2

2

+ ε

4/24/2007 School of Computing EunGyoung Han 3

History History

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 , ) ( ) ( > − ≤ −

∞ ∞

q y x q y f x f

4/24/2007 School of Computing EunGyoung Han 4

Problem Formulation Problem Formulation

Class of Lipschitz continuous functions By the Brouwer fixed point theorem

{ }

∞ ∞

− ≤ − ∈ ∀ → = y x y f x f D y x D D f

b a b a b a b a

F

) ( ) ( , | :

, , , ,

. ) ( such that , into maps

* * , * , , ,

x x f D x D D F f

b a b a b a b a

= ∈ ∃ ∈ ∀

4/24/2007 School of Computing EunGyoung Han 5

Problem Formulation Problem Formulation

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 ≤ −

∞

x x f

. ~ ε α ≤ −

∞

x

4/24/2007 School of Computing EunGyoung Han 6

Problem Formulation Problem Formulation

To find the fixed point using the Bisection

Envelope Algorithm, we are required n function evaluations of f, where

. 1 1 log 2 1

2

+             ≤ ≤ ε n

4/24/2007 School of Computing EunGyoung Han 7

Envelope Theorem Envelope Theorem

Define the fixed point sets for

b a

F f

,

∈

) ( ) ( }, ) ( | { }, ) ( | {

2 1 2 2 , 2 1 1 , 1

f f x x f D x x x f D x

b a b a

F F F F F ∩ = = ∈ = = ∈ =

4/24/2007 School of Computing EunGyoung Han 8

Theorem 3.5 Theorem 3.5

…

. satifies ~ then ,

• f

point fixed a contains addition, in If, . satifies ) ( ~ Then . intersect ) ( and ) ( both and 2 ) ( ) ( such that Let

2 1 1 1 ,

reterian absolute c y f R reterian residual c R c y R f f R l R l D R

b a

= ≤ + ⊂

−

F F ε

4/24/2007 School of Computing EunGyoung Han 9

Theorem 3.5 Theorem 3.5

ε ≤

ε ≤

) (R c

) (

2

f F

) (

1 f

F

) (

1 R

l

) (

1 R

l−

ε. x )- x f( c(R) ε (R) l (R) l R ≤ ≤ +

∞ −

~ ~ satifies so 2 satifies

1 1

4/24/2007 School of Computing EunGyoung Han 10

The The BEFix BEFix Algorithm: Definition Algorithm: Definition

.. ..

}. 1 ) 5 . , 5 . ( : { and domains the Define

1 2 5 . 1 , 5 .

≤ − ℜ ∈ = =

−

x x D D D

points. fixed all contains

• .

D within exists point fixed

• ne

least At

• n

) 1 ( continuous Lipschitz is , : If D D q f D D f = →

4/24/2007 School of Computing EunGyoung Han 11

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

5 . 1

D D D D

D D D D

D

5 . 1 5 . −

5 . −

4/24/2007 School of Computing EunGyoung Han 12

The The BEFix BEFix Algorithm: Projection Algorithm: Projection

Projection

– .. – ..

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

2 1

x x x P =

. where ,

• nto

project Let D D D D P ⊇

. ~ ) ~ ( and ~ where , for solution residual a is ) ~ ( ~ then , for solution residual a is ~ If ε ≤ − ∈ = y y f D y f y P x f y

4/24/2007 School of Computing EunGyoung Han 13

The The BEFix BEFix Algorithm: Description Algorithm: Description

• .

~ criterion absolute satifies ~ if

• nly

true is which variable logical returns Algorithm ε α ≤ − y y abs

. ~ ) ~ (

• solution t

a as ) ~ ( ~ takes Algorithm ε ≤ − = x x f y P x

4/24/2007 School of Computing EunGyoung Han 14

The The BEFix BEFix Algorithm: Construction Algorithm: Construction

Constructs a algorithm

– ε 2 ) ( ) ( if

• r
• at

satified is criterion residual a If

• :

when step at s terminate Algorithm

• 1)
• r

1 (slope rectangle closed a is Each

• .

through 1

• r

1 slope with lines along bisecting by rectangle a Constructs

• step
• n

at Evaluates

• 1

1 1 1

≤ + = ⊂

− − − k k k k k k k k k

D l D l x k D x D D D k x f

4/24/2007 School of Computing EunGyoung Han 15

Barycentric Barycentric Coordinate System Coordinate System

Find the next centroid

by using Barycentric coordinate system at .

• )

(D C

k

• b

a x x b a D C x l b l a

k k k k

• β

α β α + + =                     = =      − =       =

− − − 1 1 1 1

1 ) ( 1 1 2 2 , 1 1 2 2

b

• a
• 1

2 −

= l b

• 1

2

l a =

• )

(D C

1

• k

4/24/2007 School of Computing EunGyoung Han 16

Barycentric Barycentric coordinate system coordinate 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.

1

l

1 −

l b

• a

4/24/2007 School of Computing EunGyoung Han 17

Algorithm Analysis Algorithm Analysis

3

x

4

x

5

x

1

x

2

x

) ( ) ( ) (

1 1 1 1

D C D C x x f V > ⇒ > − =

5 =

V

) ( D C

) ( ) (

1 2 2

D C D C V > ⇒ <

x z =

) (x f z =

) (

1 x

f

) (

1

D C

1 = x

axis

• x

1 = z

      =       z x

4/24/2007 School of Computing EunGyoung Han 18

Fixed Point Fixed Point

4/24/2007 School of Computing EunGyoung Han 19

3D intersection of Pyramid function 3D intersection of Pyramid function

4/24/2007 School of Computing EunGyoung Han 20

Visualize intersection Visualize intersection

4/24/2007 School of Computing EunGyoung Han 21

Algorithm Analysis: Convergence Algorithm Analysis: Convergence

b

• a
• 1

2 −

= l b

• 1

2

l a =

4/24/2007 School of Computing EunGyoung Han 22

Algorithm Analysis: Convergence Algorithm Analysis: Convergence

Exponential

decay of infinity norm residuals.

4/24/2007 School of Computing EunGyoung Han 23

Complexity Complexity

   

. 1 ) / 1 ( log 2 1 ) / 2 ( log 2

2 2

+ = − ≤ ε ε k

where , 2 ) ( ) ( since , 2 ) ( ) ( and , 2 2 ) ( satisfy

1 1 1 1 max

= = ≤ + ≤

− −

D l D l D l D l D l D

k k k k

ε ε

4/24/2007 School of Computing EunGyoung Han 24

Numerical Tests: Numerical Tests: Pyramid basis function

Pyramid basis function

Tests Pyramid Function defined as

] 1 , [ and for ] 1 , [ : function basis Pyramid where )), , max( , 1 min( ) ( ∈ ∈ → − − =

∞

h D b D P b x h x P

h b h b

4/24/2007 School of Computing EunGyoung Han 25

Numerical Tests: Numerical Tests: Pyramid basis function

Pyramid basis function

Plots of for several values of b and h

h b

P

4/24/2007 School of Computing EunGyoung Han 26

Numerical Tests: Numerical Tests: Pyramid basis function

Pyramid basis function

4/24/2007 School of Computing EunGyoung Han 27

Numerical Tests: 3DPyramid Tests Numerical Tests: 3DPyramid Tests

Tests 3-Dimensional Pyramid function

{ }

. 4 1 , 13 , , 1

• f

and subsets empty non

• f

pairs all for )) ( ), ( ( ) ( functions

• n the

algorithm the Tested . 13 1 , 13 1 , , , integers distinct given the where )), ( , ), ( max( ) (

2 1 2 1 1

1 1 , , 1

− = = − ∀ ≤ ≤ ≤ ≤ = e S S x P x P x f j i k i i x P x P x P

s s j k h b h b i i

k i k i i i k

ε

4/24/2007 School of Computing EunGyoung Han 28

Numerical Tests: 4DPyramid Tests Numerical Tests: 4DPyramid Tests

4/24/2007 School of Computing EunGyoung Han 29

Complex Complex abs(C abs(C) )

4/24/2007 School of Computing EunGyoung Han 30

Complex angle Complex angle

4/24/2007 School of Computing EunGyoung Han 31

Numerical Tests Numerical 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 . : 29 1 ) / 1 ( log 2

2

= + ε

 

29 1 ) / 1 ( log 2

2

= + ε

4/24/2007 School of Computing EunGyoung Han 32

Future work Future 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 )), / 1 log( ) ( ( d d c d c O ε

4/24/2007 School of Computing EunGyoung Han 33

Thank You Questions ?