Inverse problem theory and methods for model parameter estimation.

*(English)*Zbl 1074.65013
Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (ISBN 0-89871-572-5/pbk; 978-0-89871-792-1/ebook). xii, 342 p. (2005).

Physical theories allow us to make prediction: given a complete description of a physical system, we can predict the outcome of some measurements. This problem of predicting is called the forward problem. The inverse problem consists of using the actual result of some measurements to infer the values of the parameters that characterize the system. The most general theory is obtained when using a probabilistic point of view, where the a priori information on the model parameters is represented by a probability distribution over the “model space”. The theory developed in the book explains how this a priori probability distribution is transformed into the a posteriori probability distribution, by incorporating a physical theory relating the model parameters to some observable parameters and the actual result of the observations together with their uncertainties.

Instead of using conditional probabilities and the Bayes theorem the author uses a more general notion, i.e. a combination of states of information. The first part of this very interesting book deals exclusively with discrete inverse problems with a finite number of parameters. The second part of the book deals with general inverse problems, which may contain such functions as data or unknowns. As this general approach contains the discrete case in particular, the separation into two parts corresponds only to a didactical purpose.

Although this book contains a lot of mathematics, it is not a mathematical book. It tries to explain how a method of acquisition of information can be applied to the actual world, and many of the arguments are heuristic. The book is directed to all scientists, including applied mathematicians, facing the problem of quantitative interpretation of experimental data in fields such as physics, chemistry, biology, image processing, and information sciences. Considerable effort has been made so that this book can serve either as a reference manual for researchers or as a textbook in a course for students.

Contents: Preface; 1. The general discrete inverse problem; 2. Monte Carlo methods; 3. The least squares criterion; 4. Least absolute values criterion and minimax criterion; 5. Functional inverse problem; 6. Appendices; 7. Problems; References and references for reading; Index.

Instead of using conditional probabilities and the Bayes theorem the author uses a more general notion, i.e. a combination of states of information. The first part of this very interesting book deals exclusively with discrete inverse problems with a finite number of parameters. The second part of the book deals with general inverse problems, which may contain such functions as data or unknowns. As this general approach contains the discrete case in particular, the separation into two parts corresponds only to a didactical purpose.

Although this book contains a lot of mathematics, it is not a mathematical book. It tries to explain how a method of acquisition of information can be applied to the actual world, and many of the arguments are heuristic. The book is directed to all scientists, including applied mathematicians, facing the problem of quantitative interpretation of experimental data in fields such as physics, chemistry, biology, image processing, and information sciences. Considerable effort has been made so that this book can serve either as a reference manual for researchers or as a textbook in a course for students.

Contents: Preface; 1. The general discrete inverse problem; 2. Monte Carlo methods; 3. The least squares criterion; 4. Least absolute values criterion and minimax criterion; 5. Functional inverse problem; 6. Appendices; 7. Problems; References and references for reading; Index.

Reviewer: Jaromir Antoch (Praha)

##### MSC:

65C60 | Computational problems in statistics (MSC2010) |

62-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics |

62F10 | Point estimation |

62F15 | Bayesian inference |

65C05 | Monte Carlo methods |