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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