Fornasier M. (Ed.) Theoretical Foundations and Numerical Methods for Sparse Recovery

Berlin: Walter de Gruyter, 2010. - 357 p.
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 exhibits 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 complexity. 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 numerical 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 – September 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 textbook 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.