Statistical and Computational Inverse Problems (Applied Mathematical Sciences)

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9781441919649: Statistical and Computational Inverse Problems (Applied Mathematical Sciences)
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This book covers the statistical mechanics approach to computational solution of inverse problems, an innovative area of current research with very promising numerical results.

The techniques are applied to a number of real world applications such as limited angle tomography, image deblurring, electical impedance tomography, and biomagnetic inverse problems.

Contains detailed examples throughout and includes a chapter on case studies where such methods have been implemented in biomedical engineering.

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"The book is devoted to the development of the statistical approach to inverse problems ... . The content is written clearly and without citations in the main text. Every chapter has a section called ‘Notes and comments’ where the citations and further reading, as well as brief comments on more advanced topics, are provided. The book is aimed at postgraduate students ... . The book also will be of interest for many researchers and scientists working in the area of image processing." (Tzvetan Semerdjiev, Zentralblatt MATH, Vol. 1068, 2005)

"Inverse problems are usually ill-posed in the sense that a solution need not exist, need not be unique, and depends in a discontinuous way on the data ... . there have been two quite separate communities dealing with such problems, one basing their methods mainly on functional analysis, the other one on statistics. ... several attempts have been made to bridge the gap between these two groups. The book under review ... is a further, quite successful attempt in this direction." (Heinz W. Engel, SIAM Review, Vol. 48 (1), 2006)

From the Back Cover:

The book develops the statistical approach to inverse problems with an emphasis on modeling and computations. The framework is the Bayesian paradigm, where all variables are modeled as random variables, the randomness reflecting the degree of belief of their values, and the solution of the inverse problem is expressed in terms of probability densities. The book discusses in detail the construction of prior models, the measurement noise modeling and Bayesian estimation. Markov Chain Monte Carlo-methods as well as optimization methods are employed to explore the probability distributions. The results and techniques are clarified with classroom examples that are often non-trivial but easy to follow. Besides the simple examples, the book contains previously unpublished research material, where the statistical approach is developed further to treat such problems as discretization errors, and statistical model reduction. Furthermore, the techniques are then applied to a number of real world applications such as limited angle tomography, image deblurring, electrical impedance tomography and biomagnetic inverse problems. The book is intended to researchers and advanced students in applied mathematics, computational physics and engineering. The first part of the book can be used as a text book on advanced inverse problems courses.

The authors Jari Kaipio and Erkki Somersalo are Professors in the Applied Physics Department of the University of Kuopio, Finland and the Mathematics Department at the Helsinki University of Technology, Finland, respectively.

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