Statistical and Computational Inverse Problems (Applied Mathematical Sciences) (v. 160) [Hardcover]

Jari Kaipio (Author), E. Somersalo (Author)

Book Description

ISBN-10: 0387220739 | ISBN-13: 978-0387220734 | Publication Date: December 1, 2004 | Edition: 1

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.

Editorial Reviews

Review

From the reviews: "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.

Product Details

• Hardcover: 360 pages
• Publisher: Springer; 1 edition (December 1, 2004)
• Language: English
• ISBN-10: 0387220739
• ISBN-13: 978-0387220734
• Product Dimensions: 9.3 x 6.3 x 0.8 inches
• Shipping Weight: 1.4 pounds (View shipping rates and policies)
• Average Customer Review: 3.0 out of 5 stars  See all reviews (3 customer reviews)
• Amazon Best Sellers Rank: #226,388 in Books (See Top 100 in Books)

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Most Helpful Customer Reviews

6 of 6 people found the following review helpful:

4.0 out of 5 stars Nice, short introduction for those who speak math, January 18, 2009

By 

Ian Langmore (New York, NY, USA) - See all my reviews
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This review is from: Statistical and Computational Inverse Problems (Applied Mathematical Sciences) (v. 160) (Hardcover)

This book is aimed at mathematicians who want an introduction to Bayesian inverse problems. This special application of Bayesian inference deals with problems of the sort: Y = F(X,E), where the measurement 'Y' depends on the unknown 'X' and the noise 'E' through 'F'. What differentiates this from typical Bayesian inference is that F is a computationally intensive solution operator (most often to a PDE). Moreover, for fixed E=e, the map F(X,e) to X is ill-posed (does not exist, not unique, or doesn't depend continuously on X.

I used this text for a one semester course taught to applied mathematicians, biomedical engineers, and statisticians. Not surprisingly then, the mathematicians thought it was great, the engineers thought it was too mathematical, and the statisticians were in for a surprise. I thought the book was well-written and I would recommend it to any mathematician. I would like future editions to include much more theoretical content.

Pros: The writing style is for the most part clear. It provides an introduction to the subject without too much fuss--in fact, this was my introduction. The organization was excellent, each page motivating the next.

Cons: Chapters 3 and 4 are the only ones with much statistical inversion theory. The book lacked any material on diagnosing convergence of MCMC methods, or even much on basic Monte Carlo integration (importance sampling for example). A teaching issue is that the second chapter (providing the introduction to ill-posedness) is done in infinite dimensions using results from functional analysis. This complicated the material for engineers. Unnecessarily so since latter chapters were done in R^n. A similar complaint is that the appendix introduction to probability is done from the abstract probability-space standpoint, whereas the body content is done using densities (and distribution functions in the MCMC section). The last 80 pages of the book seem to be lifted almost directly from papers and could have simply been referenced.

2 of 3 people found the following review helpful:

4.0 out of 5 stars extended use of Bayesians, October 9, 2006

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W Boudville (Terra, Sol 3) - See all my reviews
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This review is from: Statistical and Computational Inverse Problems (Applied Mathematical Sciences) (v. 160) (Hardcover)

The book is an extended application of Bayesian analysis. Complicated by the presence of noise, which is an unfortunate reality in all practical cases. Thus the book also decribes various models of noise. Where typically, but not always, the noise is assumed to be additive to the signal in question. Long standing Monte Carlo simulation ideas are applied in a Markov manner. Some of this should be familiar to readers in fields like materials science and image processing.

Numerous applications are given in the text. Notably for XRay and optical wavelength tomography. But more generally, to finding the source in a system modelled Maxwell's Equations, where the source is reflecting or radiating.

0 of 1 people found the following review helpful:

1.0 out of 5 stars Poor printing quality., October 29, 2011

By 

Pedro Vilanova - See all my reviews

Amazon Verified Purchase

This review is from: Statistical and Computational Inverse Problems (Applied Mathematical Sciences) (v. 160) (Hardcover)

This review is about the material quality of the printing in the copy I received. This is not about the content.

I have access to a real copy of this edition in the local library. It is the usual high quality hardcover: it has a matte cover with texture, beautifully bound; the paper inside is high-quality, very soft and slightly off-white; and the printing of the text is very sharp. The version I received from Amazon claimed to be exactly the same, but was very different:
- The hardcover was shiny, did not have texture, and had a natural tendency to bend strongly outwards, it even cannot stay opened if I leave it alone, it will close.
- The paper inside is whiter, horribly white, like standard printing A4 paper;
- The text printing looks like a cheap photocopy of the original. It don't even match a home laser printer. Some formulas are difficult to read. Moreover, some pages are not even centered.

It looks and feels like a cheap knock-off photocopy done in a garage. When I pay a lot of money for a hardcover edition I want the real thing, not a cheap knock-off. Authors should avoid their work being degraded with this cheap printing.