approximation of set-valued functions (adaptation of classical approximation operators) || front...

APPROXIMATION OF SET-VALUED FUNCTIONSAdaptation of Classical Approximation Operators

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May 2, 2013 14:6 BC: 8831 - Probability and Statistical Theory PST˙ws

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Tel Aviv University, Israel

APPROXIMATION OF SET-VALUED FUNCTIONSAdaptation of Classical Approximation Operators

Nira DynElza Farkhi

Alona Mokhov

ICP Imperial College Press

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Published by

Imperial College Press57 Shelton StreetCovent GardenLondon WC2H 9HE

Distributed by

World Scientific Publishing Co. Pte. Ltd.5 Toh Tuck Link, Singapore 596224USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication DataDyn, N. (Nira), author. Approximation of set-valued functions : adaptation of classical approximation operators / Nira Dyn, Tel Aviv University, Israel, Elza Farkhi, Tel Aviv University, Israel, Alona Mokhov, Tel Aviv University, Israel. pages cm Includes bibliographical references and index. ISBN 978-1-78326-302-8 (hardcover : alk. paper) 1. Approximation theory. 2. Linear operators. 3. Function spaces. I. Farkhi, Elza, author. II. Mokhov, Alona, author. III. Title. QA221.D94 2014 515'.8--dc23 2014023451

British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.

Copyright © 2014 by Imperial College Press

All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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October 8, 2014 10:23 9in x 6in Approximation of Set-Valued Functions:. . . b1776-fm

Preface

This book is concerned with the approximation of set-valued functions.It mainly presents our work on the design and analysis of approximationmethods for functions mapping the points of a closed real interval to generalcompact sets in R

n. Most previous results on approximation of set-valuedfunctions were confined to the special case of functions with compact convexsets in R

n as their values.We present approximation methods together with bounds on the

approximation error, measured in the Hausdorff metric. The error boundsare given in terms of the regularity of the approximated set-valuedfunction. The regularity properties used are mainly of low order, suchas Holder continuity and bounded variation. This facilitates the analysisof approximation methods for non-smooth set-valued functions, which arecommon in areas such as optimization and control. The obtained errorestimates are of similar quality to those for real-valued functions.

Our work was motivated by the need to approximate a set-valuedfunction from a finite number of its samples. Such a need arises in theproblem of “reconstruction” of a 3D object from its parallel cross-sections,which are compact 2D sets, and also in the numerical solution of non-lineardifferential inclusions. In the latter problem the set-valued solution has tobe approximated from a discrete collection of its computed values, whichare not necessarily convex sets.

The approach taken in this book is to adapt classical linear approx-imation operators for real-valued functions to set-valued functions. Forsample-based operators, the main method of adaptation is to replaceoperations between numbers by operations between sets. The main difficultyin this approach is the design of set operations, which yield operatorswith approximation properties. A second method is based on represen-tations of set-valued functions by collections of real-valued functions.Having such a representation at hand, the approximation of the set-valued

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vi Approximation of Set-Valued Functions

function is reduced to the approximation of the corresponding collection ofrepresenting real-valued functions. The main effort in this approach is thedesign of an appropriate representation consisting of real-valued functionswith regularity properties “inherited” from those of the approximated set-valued function.

The book consists of three parts. The first presents basic notionsand results needed to establish the adapted approximation methods, andto carry out their analysis. The second part is concerned with severalapproximation methods for set-valued functions with compact sets in R

n

as their values. The third part is devoted to the simpler case n = 1,where special representations of such set-valued functions are designed, andapproximation methods based on these representations are discussed.

The subject of the book is on the border of the two fields Set-ValuedAnalysis and Approximation Theory. The panoramic view, given in thebook can attract researchers from both fields to this intriguing subject. Inaddition, the book will be useful for researchers working in related fieldssuch as control and game theory, mathematical economics, optimizationand geometric modeling.

The bibliography covers various related topics. To improve the read-ability of the book, references to the bibliography do not appear in the text,but are deferred to special sections, mostly at the end of chapters.

We would like to thank the School of Mathematical Sciences at Tel-AvivUniversity for giving us a supporting and stimulating environment forcarrying out our research, and for presenting it in this book.

Tel-Aviv, May 2013The authors

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Contents

Preface v

Notations x

I Scientific Background 1

1. On Functions with Values in Metric Spaces 3

1.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Basic Approximation Methods . . . . . . . . . . . . . . 61.3 Classical Approximation Operators . . . . . . . . . . . . 7

1.3.1 Positive operators . . . . . . . . . . . . . . . . . 81.3.2 Interpolation operators . . . . . . . . . . . . . . 121.3.3 Spline subdivision schemes . . . . . . . . . . . . 13

1.4 Bibliographical Notes . . . . . . . . . . . . . . . . . . . 15

2. On Sets 17

2.1 Sets and Operations Between Sets . . . . . . . . . . . . 172.1.1 Definitions and notation . . . . . . . . . . . . . 172.1.2 Minkowski linear combination . . . . . . . . . . 182.1.3 Metric average . . . . . . . . . . . . . . . . . . . 192.1.4 Metric linear combination . . . . . . . . . . . . 21

2.2 Parametrizations of Sets . . . . . . . . . . . . . . . . . . 232.2.1 Induced metrics and operations . . . . . . . . . 232.2.2 Convex sets by support functions . . . . . . . . 242.2.3 Parametrization of sets in R . . . . . . . . . . . 252.2.4 Star-shaped sets by radial functions . . . . . . . 27

vii

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viii Approximation of Set-Valued Functions

2.2.5 General sets by signed distance functions . . . . 282.3 Bibliographical Notes . . . . . . . . . . . . . . . . . . . 29

3. On Set-Valued Functions (SVFs) 31

3.1 Definitions and Examples . . . . . . . . . . . . . . . . . 313.2 Representations of SVFs . . . . . . . . . . . . . . . . . . 323.3 Regularity Based on Representations . . . . . . . . . . . 353.4 Bibliographical Notes . . . . . . . . . . . . . . . . . . . 37

II Approximation of SVFs with Images in Rn 39

4. Methods Based on Canonical Representations 41

4.1 Induced Operators . . . . . . . . . . . . . . . . . . . . . 414.2 Approximation Results . . . . . . . . . . . . . . . . . . 434.3 Application to SVFs with Convex Images . . . . . . . . 454.4 Examples and Conclusions . . . . . . . . . . . . . . . . 484.5 Bibliographical Notes . . . . . . . . . . . . . . . . . . . 51

5. Methods Based on Minkowski Convex Combinations 53

5.1 Spline Subdivision Schemes for Convex Sets . . . . . . . 545.2 Non-Convexity Measures of a Compact Set . . . . . . . 575.3 Convexification of Sequences of Sample-Based

Positive Operators . . . . . . . . . . . . . . . . . . . . . 595.4 Convexification by Spline Subdivision Schemes . . . . . 615.5 Bibliographical Notes . . . . . . . . . . . . . . . . . . . 63

6. Methods Based on the Metric Average 65

6.1 Schoenberg Spline Operators . . . . . . . . . . . . . . . 666.2 Spline Subdivision Schemes . . . . . . . . . . . . . . . . 716.3 Bernstein Polynomial Operators . . . . . . . . . . . . . 766.4 Bibliographical Notes . . . . . . . . . . . . . . . . . . . 82

7. Methods Based on Metric Linear Combinations 85

7.1 Metric Piecewise Linear Interpolation . . . . . . . . . . 867.2 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . 917.3 Multifunctions with Convex Images . . . . . . . . . . . 947.4 Specific Metric Operators . . . . . . . . . . . . . . . . . 95

7.4.1 Metric Bernstein operators . . . . . . . . . . . . 95

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Contents ix

7.4.2 Metric Schoenberg operators . . . . . . . . . . . 967.4.3 Metric polynomial interpolation . . . . . . . . . 97

7.5 Bibliographical Notes . . . . . . . . . . . . . . . . . . . 99

8. Methods Based on Metric Selections 101

8.1 Metric Selections . . . . . . . . . . . . . . . . . . . . . . 1018.2 Approximation Results . . . . . . . . . . . . . . . . . . 1048.3 Bibliographical Notes . . . . . . . . . . . . . . . . . . . 106

III Approximation of SVFs with Images in R 107

9. SVFs with Images in R 109

9.1 Preliminaries on the Graphs of SVFs . . . . . . . . . . . 1109.2 Continuity of the Boundaries of a CBV

Multifunction . . . . . . . . . . . . . . . . . . . . . . . 1129.3 Regularity Properties of the Boundaries . . . . . . . . . 116

10. Multi-Segmental and Topological Representations 121

10.1 Multi-Segmental Representations (MSRs) . . . . . . . . 12110.2 Topological MSRs . . . . . . . . . . . . . . . . . . . . . 126

10.2.1 Existence of a topological MSR . . . . . . . . . 12710.2.2 Conditions for uniqueness of a TMSR . . . . . . 130

10.3 Representation by Topological Selections . . . . . . . . 13410.4 Regularity of SVFs Based on MSRs . . . . . . . . . . . 135

11. Methods Based on Topological Representation 137

11.1 Positive Linear Operators Based on TMSRs . . . . . . . 13711.1.1 Bernstein polynomial operators . . . . . . . . . 13911.1.2 Schoenberg operators . . . . . . . . . . . . . . . 141

11.2 General Operators Based on Topological Selections . . . 14211.3 Bibliographical Notes to Part III . . . . . . . . . . . . . 144

Bibliography 145

Index 151

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Notations

(X, ρ) metric space 3χ partition of interval 4|χ| maximal step size of χ 4χN uniform partition of interval 4ω[a,b](f, δ) modulus of continuity 4C[a, b] collection of continuous functions 4Lip([a, b], L) Lipschitz functions with constant L 4V (f, χ) variation on partition 4V b

a (f) total variation 4vf (x) variation function 4CBV [a, b] continuous functions of bounded variation 5ωk,[a,b](f, δ) modulus of smoothness of order k 6Ck[a, b] functions with continuous k-th derivatives 6| · | Euclidean norm 6||f ||∞ infinity norm 6Aχ sample-based operator 8Aδ operator depending on parameter 8φ(x, δ) positive non-decreasing in δ, φ(x, 0) = 0 8Pn space of polynomials of degree up to n 9BN Bernstein operator 9Sm,N Schoenberg operator 11

Sm,N symmetric Schoenberg operator 12Pχ polynomial interpolation operator 12K(Rn) collection of compact sets in R

n 17

x

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Notations xi

A \B set-difference 17µ(A) Lebesgue measure of A 17A×B Cartesian product 17Co(Rn) convex compact sets in R

n 17co(A) convex hull of A 17cl(A) closure of A 17[a, b] segment between two points 17〈·, ·〉 Euclidean inner product 17Sn−1 unit sphere in R

n 17haus(A, B) Hausdorff distance between two sets 18dist(a, B) distance from point to set 18‖A‖ norm of set 18ΠB(a) projection of point on set 18ΠB(A) projection of set on set 18A + B Minkowski sum of two sets 18∑N

i=1 λiAi Minkowski linear combination 18Π(A, B) collection of metric pairs 19A⊕t B metric average 19CH(A0, . . . , AN ) collection of metric chains 21⊕N

i=0λiAi metric linear combination 21A0 ⊕B1 metric sum 22A0 �B1 metric difference 22G parametrization 23gA parametrizing function 23dG(A, B) induced metric 23⊎k

i=1 tiAi induced convex combination 23

δ∗(A, ξ) support function 24

SVF set-valued function (multifunction) 31coF convex hull of SVF 31Graph(F ) graph of SVF 31

RΞ(F ) representation of SVF 33Ξ index set 33RG(F ) G-based representation of SVF 33

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xii Approximation of Set-Valued Functions

ωR[a,b](F, δ) R-based modulus of continuity 35

ωRk,[a,b](F, δ) R-based modulus of smoothness of order k 35

ωG[a,b] induced modulus of continuity 36

ωGk,[a,b] induced modulus of smoothness of order k 36

G-Ck G-smoothness of order k 36

AGχ induced sample-based operator 42

BGχ induced Bernstein operator 42

AMinkχ operator based on Minkowski sum 45

SGm,N induced symmetric Schoenberg operator 45

BMinkN Bernstein operator based on Minkowski sum 46

SMinkm,N Schoenberg operator based on Minkowski sum 46

rad(A) radius of set 57ρ(A) non-convexity measure of set 57SMA

m,N Schoenberg operator based on metric average 66

BMAN Bernstein operator based on metric average 77

s(A) separation measure of set 78A ∼ B metrically equivalent sets 78AM

χ metric sample-based operator 85

SMχ metric piecewise linear interpolant 86

CH(F |χ) metric chains of SVF at partition 86SMA

χ piecewise linear interpolant based on metric 87average

BMN metric Bernstein operator 95

SMm,N metric Schoenberg operator 96

PMχ metric polynomial interpolation operator 97

AMS operator based on metric selections 104H(F ) collection of holes of SVF 110|H(F )| number of holes of SVF 110∂H boundary of hole in graph of SVF 110∆H = [xl

H , xrH ] domain of hole 110

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Notations xiii

H(x) cross-section of hole 110PCT point of change of topology 110B(p, ε) open ball with center p and radius ε 111f low lower boundary of SVF 112fup upper boundary of SVF 112blowH lower boundary of hole 112bupH upper boundary of hole 112∂F collection of boundaries of SVF 112Df domain of f ∈ ∂F 112F [a, b] collection of CBV multifunctions with finite number

of holes121

MSR multi-segmental representation 122B(R, F ) MS-boundaries 122χF natural partition determined by SVF 123TMSR topological multi-segmental representation 127tupH , tlow

H pair of ST-selections 127R∗ least TMSR 133MSR − Ck MSR-based smoothness of order k 136AR MSR-based positive operator 138AR∗

operator based on least TMSR 143

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