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Computational methods for inverse problems. (English) Zbl 1008.65103
Frontiers in Applied Mathematics. 23. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). xvi, 183 p. (2002).
This book introduces to techniques for solving inverse problems formulated as convolution integral equations which are then discretized. The first chapters are standard. They introduce to the topic, present analytical tools (regularization theory, optimization methods, the method of Tikhonov regularization), numerical ones (steepest descent method, gradient methods, Newton’s method, line search) and statistical ones (various usual estimation methods).
The statistical and optimization methods are brought together (an originality of this monograph) for a problem of image reconstruction in dimension two. A next chapter comes back to general methods in identification and some more elaborate nonlinear optimization algorithms are quickly presented. A way of selecting a regularization parameter in a statistical context is the topic of chapter seven.
In chapter eight the optimization cost contains a term involving the total variation of a function. This helps preserving the blocky structure of an image (in an image reconstruction problem). Positivity constraints are included in the last chapter.

MSC:
65R32 Numerical methods for inverse problems for integral equations
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
45Q05 Inverse problems for integral equations
65R20 Numerical methods for integral equations
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
Software:
Matlab
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