Recovery Methodologies: Regularization and Sampling

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Print and Digital

Willi Freeden and M. Zuhair Nashed

Publisher:

AMS

Publication Date:

2023

Number of Pages:

491

Format:

Paperback

Price:

129.00

ISBN:

978-1-4704-7345-7

Category:

Monograph

MAA Review
Table of Contents

[Reviewed by

Bill Satzer

, on

12/31/2023

]

Recovery questions arise in applied mathematics when one needs to recover something- a function, a signal, or an image – from partial or incomplete information. Such problems, often described as inverse problems, are usually ill-posed and often difficult to resolve. This book offers an approach that describes methods to use regularization and sampling to find solutions. The term “recovery problem” is never defined; indeed, at the very end of the long book, the authors say “… it was our relief to be able to finalize this work without having to define what is mean by the term …”.

Regularization is often applied to get approximate solutions of ill-posed problems in the presence of contaminated data. Sampling here refers to algorithms and procedures that produce a judicious choice of data to support solution of the inverse problem. The authors contend that their goal is to create a unifying mathematical description of the concepts constituting a common thread underlying inverse and sampling problem theory.

This is an advanced text that offers few avenues of approach for those without some expertise in the field. Its goal is to highlight the value of exploring regularization and sampling methods together to solve difficult recovery problems.

Recovery as the authors describe it requires a substantial set of machinery from functional analysis, Fourier analysis, special functions and approximation theory, numerical analysis, and lattice point theory.

The examples the authors offer of recovery problems are largely conceptual and not very detailed. The simplest is recovery of the aperture of a one-dimensional antenna from the far-field antenna data. More complex are recovery problems associated with inverse gravimetry and magnetometry in geophysics.

Bill Satzer (bsatzer@gmail.com), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications. He did his PhD work in dynamical systems and celestial mechanics.

See the publisher's website.

Tags:

Inverse Problems