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Theoretical Foundations and Numerical Methods for Sparse Recovery

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Radon Series on Computational and Applied Mathematics 9

Managing Editor

Heinz W. Engl (Linz/Vienna)

Editorial Board

Hansjörg Albrecher (Lausanne)

Ronald H. W. Hoppe (Augsburg/Houston)

Karl Kunisch (Graz)

Ulrich Langer (Linz)

Harald Niederreiter (Linz)

Christian Schmeiser (Vienna)

Radon Series on Computational and Applied Mathematics

1 Lectures on Advanced Computational Methods in Mechanics

Johannes Kraus and Ulrich Langer (eds.), 2007

2 Gröbner Bases in Symbolic Analysis

Markus Rosenkranz and Dongming Wang (eds.), 2007

3 Gröbner Bases in Control Theory and Signal Processing

Hyungju Park and Georg Regensburger (eds.), 2007

4 A Posteriori Estimates for Partial Differential Equations

Sergey Repin, 2008

5 Robust Algebraic Multilevel Methods and Algorithms

Johannes Kraus and Svetozar Margenov, 2009

6 Iterative Regularization Methods for Nonlinear Ill-Posed Problems

Barbara Kaltenbacher, Andreas Neubauer and Otmar Scherzer, 2008

7 Robust Static Super-Replication of Barrier Options

Jan H. Maruhn, 2009

8 Advanced Financial Modelling

Hansjörg Albrecher, Wolfgang J. Runggaldier and Walter Schachermayer

(eds.), 2009

9 Theoretical Foundations and Numerical Methods for Sparse Recovery

Massimo Fornasier (ed.), 2010

Theoretical Foundations

and Numerical Methods

for Sparse Recovery

Edited by

Massimo Fornasier

De Gruyter

Mathematics Subject Classification 2010

49-01, 65-01, 15B52, 26B30, 42A61, 49M29, 49N45, 65K10, 65K15, 90C90, 90C06

ISBN 978-3-11-022614-0

e-ISBN 978-3-11-022615-7

ISSN 1865-3707

Library of Congress Cataloging-in-Publication Data

Theoretical foundations and numerical methods for sparse recovery /

edited by Massimo Fornasier.

p. cm. ⫺ (Radon series on computational and applied mathe-

matics ; 9)

Includes bibliographical references and index.

ISBN 978-3-11-022614-0 (alk. paper)

1. Sparse matrices. 2. Equations ⫺ Numerical solutions. 3. Dif-

ferential equations, Partial ⫺ Numerical solutions. I. Fornasier,

Massimo.

QA297.T486 2010

512.91434⫺dc22

2010018230

Bibliographic information published by the Deutsche Nationalbibliothek

The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie;

detailed bibliographic data are available in the Internet at http://dnb.d-nb.de.

” 2010 Walter de Gruyter GmbH & Co. KG, Berlin/New York

Typesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.de

Printing: Hubert & Co. GmbH & Co. KG, Göttingen

⬁ Printed on acid-free paper

Printed in Germany

www.degruyter.com

Preface

Sparsity has become an important concept in recent years in applied mathematics,

especially in mathematical signal and image processing, in the numerical treatment

of partial differential equations, and in inverse problems. The key idea is that many

types of functions arising naturally in these contexts can be described by only a small

number of signiﬁcant degrees of freedom. This feature allows the exact recovery of

solutions from a minimal amount of information. The theory of sparse recovery ex-

hibits fundamental and intriguing connections with several mathematical ﬁelds, such

as probability, geometry of Banach spaces, harmonic analysis, calculus of variations

and geometric measure theory, theory of computability, and information-based com-

plexity. The link to convex optimization and the development of efﬁcient and robust

numerical methods make sparsity a concept concretely useful in a broad spectrum of

natural science and engineering applications.

The present collection of four lecture notes is the very ﬁrst contribution of this

type in the ﬁeld of sparse recovery and aims at describing the novel ideas that have

emerged in the last few years. Emphasis is put on theoretical foundations and nu-

merical methodologies. The lecture notes have been prepared by the authors on the

occasion of the Summer School “Theoretical Foundations and Numerical Methods for

Sparse Recovery” held at the Johann Radon Institute for Computational and Applied

Mathematics (RICAM) of the Austrian Academy of Sciences on August 31 – Septem-

ber 4, 2009. The aim of organizing the school and editing this book was to provide a

systematic and self-contained presentation of the recent developments. Indeed, there

seemed to be a high demand of a friendly guide to this rapidly emerging ﬁeld. In

particular, our intention is to provide a useful reference which may serve as a text-

book for graduate courses in applied mathematics and engineering. Differently from a

unique monograph, the chapters of this book are already in the form of self-contained

lecture notes and collect a selection of topics on speciﬁc facets of the ﬁeld. We tried

to keep the presentation simple, and always start from basic facts. However, we did

not neglect to present also more advanced techniques which are at the core of sparse

recovery from probability, nonlinear approximation, and geometric measure theory as

well as tools from nonsmooth convex optimization for the design of efﬁcient recovery

algorithms. Part of the material presented in the book comes from the research work

of the authors. Hence, it might also be of interest for advanced researchers who may

ﬁnd useful details and use the book as a reference for their work. An outline of the

content of the book is as follows.

The ﬁrst chapter by Holger Rauhut introduces the theoretical foundations of com-

pressive sensing. It focuses on `

1

-minimization as a recovery method and on struc-

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