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Parameter Estimation and Inverse Problems

Richard C. Aster, Brian Borchers, and Clifford H. Thurber
Publication Date: 
Number of Pages: 
[Reviewed by
Paul Phillips
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This is a nice introduction to the topic of parameter estimation in applied math models.  Aimed at first and second year graduate students in the mathematical and physical science, this book presupposes a natural list of undergraduate courses:  Calculus, Differential Equations, Linear Algebra, Probability and Statistics.  Three appendices supply a review for the main points of Linear Algebra, Statistics and Vector Calculus, which can be covered in the first few weeks of the course as a review.  The book also comes with a CD containing MATLAB tools and materials augmenting many of the examples and homework problems throughout the book. Overall, this is a well designed textbook with a very clean approach.

The basic approach in each chapter is to lay out a concept and then explore it through a series of examples.  For example, in Chapter 4 ("Rank Deficiency and Ill-Conditioning'') a straight ray path tomography example is used to demonstrate several properties of the solutions of rank deficient problems.  Computations are performed and comments are made concerning the particular features of the example which are generally present in all examples.  For instance, information from one part of the model will smear into some, but not necessarily all, of the nearby parts, even for noise-free data (see page 72).  Examples such as this provides an excellent picture for the student to hold onto in trying to remember the various effects of rank deficiency when solving inverse problems.

Each chapter ends with a small handful of homework problems, some dealing with the conceptual issues, but most providing practice with the computational techniques. Many of these come with multiple parts and should give the student plenty to do.  Each chapter typically employs fewer than half a dozen examples as well, some of which enjoy a recurring role in other chapters.  This recurrence should be helpful to the student both by returning to something familiar as well as by showing how the current topic compares in various aspects with previous topics.

One of my favorite features in this book is the extensive list of references provided.  The expository style is kept smooth by citing sources for most of the claims made, thereby keeping the computations and proofs focused on the topic at hand.  In addition, at the end of each chapter the authors provide a fine list of additional sources for further reading, together with a description of the information each source provides, be it foundational work or extensions of the topics just presented.

Again, this is a fine introduction for inverse problems in applied fields.  While of necessity, many topics are  presented without going into great depth, the numerous references offered provide an avenue for further investigation.  The tone of the writing is conversational in a way that allows the ideas to come across clearly, while the content is mathematically rigorous, presenting details relevent to the topic and providing references for the rest.

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

1. Introduction
2. Linear Regression
3. Discretizing Continuous Inverse Problems
4. Rank Deficiency and Ill-Conditioning
5. Tikhonov Regularization
6. Iterative Methods
7. Other Regularization Techniques
8. Fourier Techniques
9. Nonlinear Regression
10. Nonlinear Inverse Problems
11. Bayesian Methods
Appendix A: Review of Linear Algebra
Appendix B: Review of Probability and Statistics
Appendix C: Glossary of Notation