Math Blog #1. What is an Inverse Problem?

Inspired by a mini-symposium I visited at the 2013 SIAM Annual Meeting devoted to having a web presence as a research mathematician, I've decided to start doing periodic blogs with a mathematical bent. This one is the first.

My research area is inverse problems, which is broad in the sense that it overlaps with a variety of mathematical areas, from pure analysis, to applied mathematics, statistics, physics, engineering, and computer science, to name a few. But the international community of inverse problems researchers is not huge.

What are inverse problems? Consider for a moment the equation

z=Au,

where z is function representing an observed quantity (the collected image in astronomy or x-ray intensities in a CT scan); u is a function representing the unknown that we want to estimate (a higher resolution image in astronomy or the actual CT scan the doctor looks at); and A is the mathematical operator mapping u to z.

Inverse problems encompass a broad range of problems, but traditionally they are ill-posed, which means NOT well-posed. Well-posed problems were defined mathematically by Hadamard.

DEFINITION: The equation z=Au is well-posed if
1. for all z, there is a solution u such that z=Au;
2. the solution u is unique for all z;
3. the solution u depends continuously on z.

In my mind, it is really the lack of satisfying #3 that characterizes an inverse problem. Problems that don't satisfy #1 and/or #2 appear in many areas, for example, in basic linear models in stastics. A linear problem that fails #1 and/or #2, but not #3, can be made well-posed by computing the minimum norm, least squares solution, for example. But if #3 doesn't hold, the inverse mapping from z to u is unstable in finite dimensions and discontinuous in infinite dimensions. In such cases, a separate class of methods, known as regularization, must be used to make the problem well-posed.

NOTE: in practice, I have no problem with the broad use of  term 'inverse problem' to include problems ranging from small-scale, nonlinear parameter estimation, to problems that only fail #1 and/or #2, it's just that in my mind, it is the failure of #3 that makes inverse problems a unique discipline.