Approximation of Set-Valued Functions: Adaptation of Classical Approximation Operators

168 Pages·2014·1.41 MB·English

by  Nira Dyn

APPROXIMATION OF
SET-VALUED FUNCTIONS
Adaptation of Classical Approximation Operators
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APPROXIMATION OF
SET-VALUED FUNCTIONS
Adaptation of Classical Approximation Operators
Nira Dyn
Elza Farkhi
Alona Mokhov
Tel Aviv University, Israel
Imperial College Press
ICP
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Published by
Imperial College Press
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Distributed by
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Library of Congress Cataloging-in-Publication Data
Dyn, 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 Data
A 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,
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Preface
This book is concerned with the approximation of set-valued functions.
It mainly presents our work on the design and analysis of approximation
methodsforfunctionsmappingthepointsofaclosedrealintervaltogeneral
compact sets in Rn. Most previous results on approximation of set-valued
functionswereconfinedtothespecialcaseoffunctionswithcompactconvex
sets in Rn as their values.
We present approximation methods together with bounds on the
approximation error, measured in the Hausdorff metric. The error bounds
are given in terms of the regularity of the approximated set-valued
function. The regularity properties used are mainly of low order, such
as H¨older continuity and bounded variation. This facilitates the analysis
of approximation methods for non-smooth set-valued functions, which are
common in areas such as optimization and control. The obtained error
estimates are of similar quality to those for real-valued functions.
Our work was motivated by the need to approximate a set-valued
function from a finite number of its samples. Such a need arises in the
problem of “reconstruction” of a 3D object from its parallel cross-sections,
whicharecompact2Dsets,andalsointhenumericalsolutionofnon-linear
differential inclusions. In the latter problem the set-valued solution has to
be approximated from a discrete collection of its computed values, which
are 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. For
sample-based operators, the main method of adaptation is to replace
operationsbetweennumbersbyoperationsbetweensets.Themaindifficulty
in this approach is the design of set operations, which yield operators
with 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
v
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vi Approximation of Set-Valued Functions
function is reduced to the approximation of the corresponding collection of
representing real-valued functions. The main effort in this approach is the
design of an appropriate representation consisting of real-valued functions
with regularity properties “inherited” from those of the approximated set-
valued function.
The book consists of three parts. The first presents basic notions
and results needed to establish the adapted approximation methods, and
to carry out their analysis. The second part is concerned with several
approximation methods for set-valued functions with compact sets in Rn
as their values. The third part is devoted to the simpler case n = 1,
wherespecialrepresentationsofsuchset-valuedfunctionsaredesigned,and
approximation methods based on these representations are discussed.
The subject of the book is on the border of the two fields Set-Valued
Analysis and Approximation Theory. The panoramic view, given in the
book can attract researchers from both fields to this intriguing subject. In
addition, the book will be useful for researchers working in related fields
such as control and game theory, mathematical economics, optimization
and geometric modeling.
The bibliography covers various related topics. To improve the read-
abilityofthebook,referencestothebibliographydonotappearinthetext,
but are deferred to special sections, mostly at the end of chapters.
WewouldliketothanktheSchoolofMathematicalSciencesatTel-Aviv
University for giving us a supporting and stimulating environment for
carrying out our research, and for presenting it in this book.
Tel-Aviv, May 2013
The 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 . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Basic Approximation Methods . . . . . . . . . . . . . . 6
1.3 Classical Approximation Operators . . . . . . . . . . . . 7
1.3.1 Positive operators . . . . . . . . . . . . . . . . . 8
1.3.2 Interpolation operators . . . . . . . . . . . . . . 12
1.3.3 Spline subdivision schemes . . . . . . . . . . . . 13
1.4 Bibliographical Notes . . . . . . . . . . . . . . . . . . . 15
2. On Sets 17
2.1 Sets and Operations Between Sets . . . . . . . . . . . . 17
2.1.1 Definitions and notation . . . . . . . . . . . . . 17
2.1.2 Minkowski linear combination . . . . . . . . . . 18
2.1.3 Metric average . . . . . . . . . . . . . . . . . . . 19
2.1.4 Metric linear combination . . . . . . . . . . . . 21
2.2 Parametrizations of Sets . . . . . . . . . . . . . . . . . . 23
2.2.1 Induced metrics and operations . . . . . . . . . 23
2.2.2 Convex sets by support functions . . . . . . . . 24
2.2.3 Parametrization of sets in R . . . . . . . . . . . 25
2.2.4 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 . . . . 28
2.3 Bibliographical Notes . . . . . . . . . . . . . . . . . . . 29
3. On Set-Valued Functions (SVFs) 31
3.1 Definitions and Examples . . . . . . . . . . . . . . . . . 31
3.2 Representations of SVFs . . . . . . . . . . . . . . . . . . 32
3.3 Regularity Based on Representations . . . . . . . . . . . 35
3.4 Bibliographical Notes . . . . . . . . . . . . . . . . . . . 37
II Approximation of SVFs with Images in Rn 39
4. Methods Based on Canonical Representations 41
4.1 Induced Operators . . . . . . . . . . . . . . . . . . . . . 41
4.2 Approximation Results . . . . . . . . . . . . . . . . . . 43
4.3 Application to SVFs with Convex Images . . . . . . . . 45
4.4 Examples and Conclusions . . . . . . . . . . . . . . . . 48
4.5 Bibliographical Notes . . . . . . . . . . . . . . . . . . . 51
5. Methods Based on Minkowski Convex Combinations 53
5.1 Spline Subdivision Schemes for Convex Sets . . . . . . . 54
5.2 Non-Convexity Measures of a Compact Set . . . . . . . 57
5.3 Convexification of Sequences of Sample-Based
Positive Operators . . . . . . . . . . . . . . . . . . . . . 59
5.4 Convexification by Spline Subdivision Schemes . . . . . 61
5.5 Bibliographical Notes . . . . . . . . . . . . . . . . . . . 63
6. Methods Based on the Metric Average 65
6.1 Schoenberg Spline Operators . . . . . . . . . . . . . . . 66
6.2 Spline Subdivision Schemes . . . . . . . . . . . . . . . . 71
6.3 Bernstein Polynomial Operators . . . . . . . . . . . . . 76
6.4 Bibliographical Notes . . . . . . . . . . . . . . . . . . . 82
7. Methods Based on Metric Linear Combinations 85
7.1 Metric Piecewise Linear Interpolation . . . . . . . . . . 86
7.2 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . 91
7.3 Multifunctions with Convex Images . . . . . . . . . . . 94
7.4 Specific Metric Operators . . . . . . . . . . . . . . . . . 95
7.4.1 Metric Bernstein operators . . . . . . . . . . . . 95
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Contents ix
7.4.2 Metric Schoenberg operators . . . . . . . . . . . 96
7.4.3 Metric polynomial interpolation . . . . . . . . . 97
7.5 Bibliographical Notes . . . . . . . . . . . . . . . . . . . 99
8. Methods Based on Metric Selections 101
8.1 Metric Selections . . . . . . . . . . . . . . . . . . . . . . 101
8.2 Approximation Results . . . . . . . . . . . . . . . . . . 104
8.3 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 . . . . . . . . . . . 110
9.2 Continuity of the Boundaries of a CBV
Multifunction . . . . . . . . . . . . . . . . . . . . . . . 112
9.3 Regularity Properties of the Boundaries . . . . . . . . . 116
10. Multi-Segmental and Topological Representations 121
10.1 Multi-Segmental Representations (MSRs) . . . . . . . . 121
10.2 Topological MSRs . . . . . . . . . . . . . . . . . . . . . 126
10.2.1 Existence of a topological MSR . . . . . . . . . 127
10.2.2 Conditions for uniqueness of a TMSR . . . . . . 130
10.3 Representation by Topological Selections . . . . . . . . 134
10.4 Regularity of SVFs Based on MSRs . . . . . . . . . . . 135
11. Methods Based on Topological Representation 137
11.1 Positive Linear Operators Based on TMSRs . . . . . . . 137
11.1.1 Bernstein polynomial operators . . . . . . . . . 139
11.1.2 Schoenberg operators . . . . . . . . . . . . . . . 141
11.2 General Operators Based on Topological Selections . . . 142
11.3 Bibliographical Notes to Part III . . . . . . . . . . . . . 144
Bibliography 145
Index 151