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

A Two- -Dimensional Bisection Dimensional Bisection A Two 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 School of Computing 4/24/2007 1 EunGyoung Han

Introduction Introduction � = � � How we solve for two-dimensional f ( x ) x – domain: [0, 1]X[0, 1] – f : Lipschitz continuous function (q = 1). � Previous method – Time complexity was bad � Paper introduce new algorithm ~ ~ ~ − ≤ ε x f ( x ) x – Computes approximate satisfying ∞ – Tolerance ε < 0 . 5 – Upper bound on the function evaluations   ε + 2 log ( 1 / ) 1 . 2 School of Computing 4/24/2007 2 EunGyoung Han

History History � 1920s - Present – Banach’s simple iteration algorithm – Homotopy continuation – Simplicial and Newton type methods � Time complexity − ≤ − > f ( x ) f ( y ) q x y , q 1 ∞ ∞ – Lipschitz function (q>1) • Exponential in the worst case • Lower bound is also exponential (best case) School of Computing 4/24/2007 3 EunGyoung Han

Problem Formulation Problem Formulation � Class of Lipschitz continuous functions { } = → ∀ ∈ − ≤ − F f : D D | x , y D f ( x ) f ( y ) x y a , b a , b a , b a , b ∞ ∞ � By the Brouwer fixed point theorem * * * ∀ ∈ ∃ ∈ = f F maps D into D , x D such that f ( x ) x . a , b a , b a , b a , b School of Computing 4/24/2007 4 EunGyoung Han

Problem Formulation Problem Formulation � We know a solution exists, we just need a constructive algorithm… � Two different criteria to satisfy – Residual criterion ~ ~ − ≤ ε f ( x ) x . • Can always ∞ be satisfied – Absolute criterion ~ − α ≤ ε x . • Can sometimes ∞ be satisfied School of Computing 4/24/2007 5 EunGyoung Han

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 n 2  log  1 . 2     ε School of Computing 4/24/2007 6 EunGyoung Han

Envelope Theorem Envelope Theorem � Define the fixed point sets for ∈ f F a , b � � = ∈ = F { x D | f ( x ) x }, 1 a , b 1 1 � � = ∈ = { x D | f ( x ) x }, F 2 a , b 2 2 = ∩ F F ( f ) F ( f ) 1 2 School of Computing 4/24/2007 7 EunGyoung Han

Theorem 3.5 Theorem 3.5 � … ⊂ + ≤ ε Let R D such that l ( R ) l ( R ) 2 − a , b 1 1 and both ( f ) and ( f ) intersect R . F F 1 2 ~ = Then y c ( R ) satifies residual c reterian . If, in addition, R contains a fixed point ~ of f , then y satifies absolute c reterian . School of Computing 4/24/2007 8 EunGyoung Han

Theorem 3.5 Theorem 3.5 R satifies F 1 f ( ) l − 1 R ( ) ε + ≤ l (R) l (R) 2 l 1 R ( ) − 1 1 so c(R) satifies ≤ ε c ( R ) ~ ~ ε . ≤ f( x )- x ∞ ( ) F f 2 ≤ ε School of Computing 4/24/2007 9 EunGyoung Han

The BEFix BEFix Algorithm: Definition Algorithm: Definition The � .. = Define the domains D D − 0 . 5 , 1 . 5 2 = ∈ ℜ − ≤ and D { x : x ( 0 . 5 , 0 . 5 ) 1 }. 0 1 � .. → = If f : D D , f is Lipschitz continuous ( q 1 ) on D - At least one fixed point exists within D . - D contains all fixed points. School of Computing 4/24/2007 10 EunGyoung Han

The BEFix BEFix Algorithm: Figure D, et. Algorithm: Figure D, et. The 1 . 5 1 . 5 D D D 0 D D D 0 0 D D D 0 − 0 . 5 − 0 . 5 School of Computing 4/24/2007 11 EunGyoung Han

The BEFix BEFix Algorithm: Projection Algorithm: Projection The � Projection – .. ⊇ Let P project D onto D , where D D . = ( ) (max( 0 , min( 1 , )), max( 0 , min( 1 , ))) P x x x 1 2 ~ – .. If y is a residual solution for f , ~ ~ = then x P ( y ) is a residual solution for f , ~ ~ ~ ∈ − ≤ ε where y D and f ( y ) y . 0 School of Computing 4/24/2007 12 EunGyoung Han

The BEFix BEFix Algorithm: Description Algorithm: Description The ~ ~ � = Algorithm takes ( ) as a solution t o x P y ~ ~ − ≤ ε ( ) . f x x Algorithm returns logical variable abs � ~ which is true only if y satifies ~ − α ≤ ε absolute criterion y . School of Computing 4/24/2007 13 EunGyoung Han

The BEFix BEFix Algorithm: Construction Algorithm: Construction The � Constructs a algorithm k - Evaluates f at x on step k – ⊂ - Constructs a rectangle D D by − k k 1 k bisecting D along lines with slope 1 or - 1 through x . − k 1 = - Each D is a closed rectangle (slope 1 or - 1) k - Algorithm terminate s at step k when : k - If a residual criterion is satified at x + ≤ ε - or if l ( D ) l ( D ) 2 − 1 k 1 k School of Computing 4/24/2007 14 EunGyoung Han

Barycentric Coordinate System Coordinate System Barycentric � � Find the next centroid by using C (D ) k � Barycentric coordinate system at . C (D ) k - 1 �    −  � 1 1 2 2 = =     a l , b l − 1 1     1 1 2 2 � � � � = a l = l b 1 k = = − 2 1 x C ( D ) 2 k     � � α �  � �    − k 1 � � β � a b x     �     a � �     1 b � � � − k 1 = + α + β x a b School of Computing 4/24/2007 15 EunGyoung Han

Barycentric coordinate system coordinate system Barycentric � Define the basis vectors of the Barycentric coordinate system relative to the origin defined by x . � � � The vectors and point in the a b directions of the and edges of the l l − 1 1 rectangle. School of Computing 4/24/2007 16 EunGyoung Han

5 = V 0 Algorithm Analysis Algorithm Analysis = z 1 f 1 x ( ) ⇒ < 0 ( ) > ( ) V C D C D 2 2 1 z = f ( x ) 1 1 = − > V f ( x ) x 0 1 ⇒ > C ( D ) C ( D ) 1 0 z = x C ( D ) 1 x - axis C ( D ) 0 5     x x 0 = x 1 2 =     x 1 4 3 x x     x z 0 School of Computing 4/24/2007 17 EunGyoung Han

Fixed Point Fixed Point School of Computing 4/24/2007 18 EunGyoung Han

3D intersection of Pyramid function 3D intersection of Pyramid function School of Computing 4/24/2007 19 EunGyoung Han

Visualize intersection Visualize intersection School of Computing 4/24/2007 20 EunGyoung Han

Algorithm Analysis: Convergence Algorithm Analysis: Convergence � � b = l = a l − 1 1 2 2 � � b a School of Computing 4/24/2007 21 EunGyoung Han

Algorithm Analysis: Convergence Algorithm Analysis: Convergence � Exponential decay of infinity norm residuals. School of Computing 4/24/2007 22 EunGyoung Han

Complexity Complexity 2 ≤ ε D satisfy l ( D ) , and k max k 2 + ≤ ε l ( D ) l ( D ) 2 , − 1 k 1 k = = since ( ) ( ) 2 , where l D l D − 1 0 1 0     ≤ ε − = ε + k 2 log ( 2 / ) 1 2 log ( 1 / ) 1 . 2 2 School of Computing 4/24/2007 23 EunGyoung Han

Numerical Tests: Pyramid basis function Numerical Tests: Pyramid basis function � Tests Pyramid Function defined as h = − − P ( x ) min( 1 , max( h x b , 0 )), b ∞ where Pyramid basis function h → ∈ ∈ P : D [ 0 , 1 ] for b D and h [ 0 , 1 ] b School of Computing 4/24/2007 24 EunGyoung Han

Numerical Tests: Pyramid basis function Numerical Tests: Pyramid basis function h P � Plots of for several values of b and h b School of Computing 4/24/2007 25 EunGyoung Han

Numerical Tests: Pyramid basis function Numerical Tests: Pyramid basis function School of Computing 4/24/2007 26 EunGyoung Han

Numerical Tests: 3DPyramid Tests Numerical Tests: 3DPyramid Tests � Tests 3-Dimensional Pyramid function h h � = i P ( x ) max( P i ( x ), , P ( x )), 1 k i i b b � 1 , , k i i 1 k where given the distinct integers � ≤ ≤ ≤ ≤ ∀ i , , i , 1 k 13 , 1 i 13 j . 1 k j − Tested the algorithm on the functions = ( ) ( ( ), ( )) for all pairs of f x P x P x s 1 s 2 { } � ε = − non empty subsets S and S of 1 , , 13 , 1 e 4 . 1 2 School of Computing 4/24/2007 27 EunGyoung Han

Numerical Tests: 4DPyramid Tests Numerical Tests: 4DPyramid Tests School of Computing 4/24/2007 28 EunGyoung Han

Complex abs(C abs(C) ) Complex School of Computing 4/24/2007 29 EunGyoung Han

Complex angle Complex angle School of Computing 4/24/2007 30 EunGyoung Han