Inverse problems

According to J. B. Keller's Inverse problems, two problems are said to be "inverses" of each other if the formulation of one calls the other into question. This definition is somewhat arbitrary, and makes the two problems play a symmetrical role. A more operational definition is that an inverse problem consists in determining causes knowing effects. Thus, this problem is the inverse of the so-called direct problem, which consists in deducing the effects, the causes being known.

This second definition shows that we are more accustomed to studying "direct" problems. Indeed, since Newton, the notion of causality has been anchored in our scientific subconscious, and at a more prosaic level, we've learned to pose and then solve problems for which the causes are given, and we then look for the effects. This definition also shows that inverse problems are likely to pose particular difficulties. We'll see later that it is possible to give a mathematical content to the phrase "the same causes produce the same effects", in other words, that it is reasonable to demand that the direct problem be "well posed". On the other hand, it's easy to imagine - and we'll see many examples of this - that the same effects can come from different causes. This idea contains the seeds of the main difficulty in studying inverse problems: they can have several solutions, and additional information is needed to discriminate between them.

Predicting the future state of a physical system, knowing its current state, is a typical example of a direct problem. Various inverse problems can be envisaged: for example, reconstructing the past state of the system, knowing its current state (if this system is irreversible), or determining the parameters of the system, knowing (part of) its evolution. The latter problem is that of parameter identification, which will be our main concern in the second part of this article.

A practical difficulty in the study of inverse problems is that it often requires a good knowledge of the direct problem, resulting in the use of a wide variety of both physical and mathematical concepts. Success in solving an inverse problem generally depends on elements specific to that problem. There are, however, a few techniques that have a wide field of applicability, and this article is an introduction to the main ones: regularization of ill-posed problems, and the method of least squares, whether linear or non-linear.

The most important is the reformulation of an inverse problem as the minimization of an error functional between the real measurements and the "synthetic measurements" (i.e. the solution of the direct problem). It will be useful to distinguish between linear and non-linear problems. Non-linearity refers to the inverse...