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Publisher:

SIAM

Publication Date:

2004

Number of Pages:

342

Format:

Paperback

Price:

85.00

ISBN:

0-89871-572-5

Category:

Monograph

[Reviewed by , on ]

Paul Phillips

06/3/2006

In my experience, math texts usually come in two basic varieties: theory-oriented texts that develop a topic in detail providing lots of proofs along the way, and problem-oriented texts that give a large number of practice problems and just enough theory to work them. This text doesn't fall into either category. Instead it is a book that outlines a conceptual approach to the solution of inverse problems and provides a rich heuristic understanding of the topic.

Billed as a reference manual for researchers or as a text for undergraduate and graduate students, this is one of those rare books that is extremely useful to the right sort of audience: those who can benefit from the trade secrets and insights divulged to them from an advisor or expert colleague. The author not only provides techniques but also the intuitive principles for the proper use of those techniques. This book could well be used in a course for advanced students who have a good background in multivariable calculus, linear algebra, and probability.

The entire book is written with a particular thematic goal in mind: considering the solution to an inverse problem to be a probability distribution in the space of possible parameters. The enlightening exposition provides a persuasive argument for such an approach in practical problems. This presentation provides a general conceptual framework for understanding inverse problems, which by their very nature have problematic (usually, non-unique) solutions. The statistical approach sidesteps many of the difficult aspects of defining a solution, while retaining and even highlighting all the information that can be extracted from a given situation.

While many techniques are discussed, this is not a text with endless assignments for learning and refining one's technique. In fact, this book contains no problem sets whatsoever. Rather, each chapter contains plentiful examples that amplify the message of the text and demonstrate the techniques in action. The last chapter is a true pearl, being nothing other than a list of problems together with well explained solutions, where all the themes are brought to bear.

Finally, the layout of the book is quite good. Formulas are easy to read and the figures are helpful, well placed and usefully commented. The organization of the chapters is well planned and executed. While not itself proof oriented, the text provides an extensive bibliography for those who would like to investigate other topics in more depth. All in all a very good book, and one that I thoroughly enjoyed reading and working through.

Paul Phillips is Associate Professor of Mathematics at the University of Dallas.

Chapter 1: The General Discrete Inverse Problem; Chapter 2: Monte Carol Methods; Chapter 3: The Least-Squares Criterion; Chapter 4: Least-Absolute Values Criterion and Minimax Criterion; Chapter 5: Functional Inverse Problems; Chapter 6: Appendices; Chapter 7: Problems; References; Index.

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