The use of actual observations to infer the properties of a model is an inverse problem, which are often difficult as they may not have a unique solution. This book proposes a general approach that is valid for linear as well as for nonlinear problems. The philosophy is essentially probabilistic and allows the reader to understand the basic difficulties appearing in the resolution of inverse problems.
The first part of this book deals exclusively with discrete inverse problems with a finite number of parameters. Some real problems are naturally discrete, while others contain functions of a continuous variable and can be discretized if the functions under consideration are smooth enough compared to the sampling length, or if the functions can conveniently be described by their development on a truncated basis. The advantage of a discretized point of view for problems involving functions is that the mathematics is easier. The disadvantage is that some simplifications arising in a general approach can be hidden when using a discrete formulation. (Discretizing the forward problem and setting a discrete inverse problem is not always equivalent to setting a general inverse problem and discretizing for the practical computations.)
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.
Table of Contents
- The General Discrete Inverse Problem
- Monte Carlo Methods
- The Least-Squares Criterion
- Least-Absolute-Values Criterion and Minimax Criterion
- Functional Inverse Problems
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Number of pages: 358