Within the larger subject of inverse problems, the most well studied inverse problem involves the solution of a linear operator equation \(\mathrm{T}x=y\), where the data \(y\) and the unknown \(x\) live in Hilbert spaces and \(\mathrm{T}\) is a linear operator. These problems can be ill-posed in the sense that a solution fails to exist, or that the solution is non-unique, or that the solution is unstable with respect to small perturbations in \(y\). In practice, we typically settle for least squares solutions which minimize the norm of \(\mathrm{T}x-y\). If the least squares solution is not unique, then we select the least squares solution \(x\) that has minimum norm.

When \(\mathrm{T}\) is a matrix and \(x\) and \(y\) are vectors, the pseudoinverse of \(\mathrm{T}\) can be used to obtain the minimum norm least squares solution and that solution is more or less, depending on the norm of the pseudoinverse of \(\mathrm{T}\), stable with respect to small perturbations in \(y\). However, in infinite dimensional spaces, if the range of \(\mathrm{T}\) is not closed, then the pseudoinverse of \(\mathrm{T}\) is an unbounded operator, and the solution to the inverse problem is unstable with respect to perturbations in \(y\).

Tikhonov regularization is a technique that can be used to stabilize the solution of the inverse problem. Gockenbach’s book gives a focused presentation of the basic theory of ill-posed linear inverse problems on Hilbert spaces, Tikhonov regularization, compact operators and the singular value expansion, and regularization with seminorms. Readers should have some background in analysis and Hilbert spaces, but otherwise, the book is largely self-contained.

The strength of this book is in the very clear exposition and examples. It is the most easily accessible introduction to the analysis of linear inverse problems. However, there is little on applications and no real discussion of numerical methods. Furthermore, there are no exercises. Thus additional support would be needed to use this book as the text for a course.

Brian Borchers is a professor of Mathematics at the New Mexico Institute of Mining and Technology. His interests are in optimization and applications of optimization in parameter estimation and inverse problems.

## Comments

## What I Learned

I was completely ignorant about inverse problems when I started reading this book. Two things delighted me:

1) The first chapter has a delightful example of an inverse problem, one that is easy enough to show an undergraduate audience (which I did!).

2) Tikhonov regularization turns out to depend on approximating a discontinuous function by continuous ones, which can only be done when the convergence is pointwise rather than uniform. Given the "pointwise bad, uniform good" message one gets in real analysis, I thought this was striking.

I'm the editor of the Carus Monographs series, so I'm biased. I think it's a very good book, one that might motivate its readers to tackle more advanced books on the subject.