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An Introduction to the Mathematical Theory of Inverse Problems

Andreas Kirsch
Publication Date: 
Number of Pages: 
Applied Mathematical Sciences 120
[Reviewed by
Collin Carbno
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Andreas Kirsch successfully wrote this book not only for mathematics students but also physics and engineering students. For understanding the basic results from functional analysis needed for following the book, Appendix A is invaluable. Without it, I suspect that I would have been googling for functional analysis definitions and concepts, whereas with Appendix A I could quickly cover a few pages of the basic ideas and be one back into the main stream of the book. Appendix A is also remarkable in that it provides an overview course in functional analysis in a mere thirty pages, compared to some 400+ pages I remember from a textbook for a course on this topic long ago.

Also of help to students is the fact that throughout the book (including the appendixes) the theorems and examples contain detailed, complete, explained, and easy to follow step-by-step derivations and proofs. The basic equations covered in the selected inverse problems (Inverse Eigenvalue, Electrical Impedance Tomography, Inverse scattering problems) should be familiar to most physics and engineering students. The inverse problem investigation, however, reveals a surprising repertoire of lesser known properties of those equations that most students will not have encountered, thus providing an interesting twist to the material. The considerable attention paid throughout the book to numerical examples, simulations, and numerical methods will help hold the interest of physics and engineering students. These numerical examples illuminate the material and connect it directly to applied mathematics concerns.

One of the main challenges in inverse problems is that they are often ill-posed; so that small errors in the “input” are magnified when one attempts to determine the inverse solution. To me, one of the big takeaways from the book was how useful functional analysis is in inverse problems, both from an analysis point of view and an applied mathematical point of view, especially with the regularization process.

At the end of each chapter, there are few exercises. I did random samples of those exercises and found they engaged me in the material while not being too difficult or too long. So, overall, I think that this book would work very nicely as a university textbook.

One might wonder whether the second edition is a minor update or major reworking of the material. While I have not seen the first edition, it is clear that book has been completely refreshed, with lots of new material, such as the factorization method. All sections of the books seemed sprinkled with the latest results, showing that there is a surprising amount of current research activity in this area. I recommend this book to anyone interested in inverse problems, and book’s index makes it a valuable reference volume for your book shelf.

Collin Carbno is a specialist in process improvement and methodology. He holds a Master’s of Science Degree in theoretical physics and completed course work for Ph.D. in theoretical physics (relativistic rotating stars) in 1979 at the University of Regina. He has been employed for over 30 years in various IT and process work at Saskatchewan Telecommunications and currently holds a Professional Physics Designation from the Canadian Association of Physicists, and the Information System Professional Designation from the Canadian Information Process Society.

Introduction and Basic Concepts
Regularization Theory for Equations of the First Kind
Regularization by Discretization
Inverse Eigenvalue Problems
An Inverse Problem in Electrical Impedance Tomography
An Inverse Scattering Problem