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Overview of Inverse Problems

1.1. Direct and inverse problems

According to Keller [KEL 76], two problems are said to be the inverse of one another if the formulation of one of them involves the other. This definition includes a degree of arbitrariness and confers a symmetric role to both problems under consideration. A more operational definition is that an inverse problem consists of determining the causes knowing the effects. Thus, this problem is the inverse of what is called a "direct problem", consisting of the deduction of the effects, the causes being known.

This second definition shows that it is more usual to study direct problems. As a matter of fact, since Newton, the notion of causality is rooted in our scientific subconscious, and at a more prosaic level, we have learned to pose, and then solve, problems for which the causes are given, where the objective is to find the effects. This definition also shows that inverse problems may give rise to particular difficulties. We will see further that it is possible to attribute a mathematical content to the sentence "the same causes produce the same effects"; in other words, it is reasonable to require that the direct problem is well-posed. On the other hand, it is easy to imagine, and we will see numerous examples, that the same effects may originate from different causes. At the origin, this idea contains the main difficulty of the study of inverse problems: they can have several solutions and it is important to have additional information in order to discriminate between them.

The prediction of the future state of a physical system, knowing its current state, is the typical example of a direct problem. We can consider various inverse problems: for example to reconstitute the past state of the system knowing its current state (if this system is irreversible), or the determination of parameters of the system, knowing (part of) its evolution. This latter problem is that of the identification of parameters, which will be our main concern in the following.

A practical challenge of the study of inverse problems is that it often requires a good knowledge of the direct problem, which is reflected in the use of a large variety of both physical and mathematical concepts. The success in solving an inverse problem is based, in general, on elements specific to this problem. However, some techniques present an extended application domain and this book is an introduction to the principal techniques: the regularization of ill-posed problems and the least squares method.

The most important technique is the reformulation of an inverse problem in the form of the minimization of an error functional between the actual measurements and the synthetic measurements (that is the solution to the direct problem). It will be convenient to distinguish between linear and nonlinear problems. It should be noted here that the nonlinearity in question refers to the inverse problem, and that the direct problem itself may or may not be linear.

In the case of linear problems, resorting to linear algebra and to functional analysis allows accurate results as well as efficient algorithms to be obtained. The fundamental tool here is the singular value decomposition of the operator, or of the matrix, being considered. We will study the regularization method in detail, which consists of slightly "modifying" the problem under study by another that has "better" properties. This will be specified in Chapters 4 and 5.

Nonlinear problems are more difficult and there exist less overall results. We will study the application of optimization algorithms to problems obtained by the reformulation referred to above. A crucial technical ingredient (from the numerical point of view) is the calculation of the gradient of the functional to be minimized. We will study the adjoint state method in Chapter 7. It allows this calculation at a cost that is a (small) multiple of that of solving the direct problem.

As it can be seen, the content of this book primarily aims to present numerical methods to address inverse problems. This does not mean that theoretical questions do not exist, or are devoid of interest. The deliberate choice of not addressing them is dictated by the practical orientation of the course, by the author's taste and knowledge, but also by the high mathematical level that these issues require.

1.2. Well-posed and ill-posed problems

In his famous book, Hadamard [HAD 23] introduced, as early as 1932, the notion of a well-posed problem. It concerns a problem for which:

1. - a solution exists;
2. - the solution is unique;
3. - the solution depends continuously on the data.

Of course, these concepts must be clarified by the choice of space (and of topologies) to which the data and the solution belong.

In the same book, Hadamard suggested (and it was a widespread opinion until recently) that only a well-posed problem could properly model a physical phenomenon. After all, these three conditions seem very natural. In fact, we shall see that inverse problems often do not satisfy either of these conditions, or even the three all together. Upon reflection, this is not so surprising:

1. - a physical model being established, the experimental data available are generally noisy and there is no guarantee that such data originate from this model, even for another set of parameters;
2. - if a solution exists, it is perfectly conceivable (and we will see examples of this) that different parameters may result in the same observations.

The absence of one or any other of the three Hadamard's conditions does not have the same importance with respect to being able to solve (in a sense that remains to be defined) the associated problem:

1. - the fact that the solution of an inverse problem may not exist is not a serious difficulty. It is usually possible to restore the existence by relaxing the concept of solution (a classic procedure in mathematics);
2. - the non-uniqueness is a more serious problem. If a problem has several solutions, there should be a means of distinguishing between them. This requires additional information (we speak of a priori information);
3. - the lack of continuity is probably the most problematic, in particular in view of an approximate or a numerical solution. Lack of continuity means that it is not possible (regardless of the numerical method) to approach a satisfactory solution of the inverse problem, since the data available will be noisy, therefore close to the actual data, but different from the actual data.

A problem that is not well-posed within the meaning of the definition above is said to be ill-posed. We now give an example that, although very simple, illustrates the difficulties that may be found in more general situations.

EXAMPLE 1.1.- Differentiation and integration are two problems that are the inverse of each other. It would seem more natural to consider differentiation as the direct problem and integration as the inverse problem. In fact, integration has good mathematical properties that lead to consider it as the direct problem. In addition, differentiation is the prototypical ill-posed problem, as we shall see in the following.

Consider the Hilbert space , and the integral operator A defined by

It is easy to directly see that , or theorem 3.1, can be applied (see example 3.1). This operator is injective; however, its image is the vector subspace

where is the Sobolev space. In effect, the equation

is equivalent to

The image of A is not closed in (of course, it is closed in . As a result, the inverse of A is not continuous on as shown in the following example.

Consider a function and let . Let

then

Simple calculations show that

whereas

Thus, the difference between f´ and may be arbitrarily large, even though the difference between f and f´ is arbitrarily small. The derivation operator (the inverse of A) is thus not continuous, at least with this choice of norms.

The instability of the inverse is typical of ill-posed problems. A small perturbation over the data (here f) can have an arbitrarily large influence on the result (here f´).

A second class of inverse problems is the estimation of parameters in differential equations. We are going to discuss a very simple example of this situation.

EXAMPLE 1.2.- Considering the elliptic problem in one dimension:

This equation, or other similar although more complex, arises in several examples in the following chapter. In this example, we choose and the solution , which gives

The direct problem consists of calculating u, given a and f. For the inverse problem, we shall consider that f is known, and we will try to recover the coefficient a from a measurement of u. For this example, voluntarily simplified, we shall assume that u is measured over the whole interval ] - 1, 1[, which is obviously unrealistic. We shall see that even in this optimistic situation, we are likely to face difficulties.

By integrating equation [1.2], and by dividing by...