The Linear Functional Strategy for Improperly Posed Problems

Abstract

In the construction of approximation procedures for the solution of inverse problems, some form of stabilization is needed if realistic and useful information about the properties of the solutions is to be generated. There are various ways in which such stabilization can be introduced. They include various forms of regularization, contrained optimization, linear programming inversion, and low dimensional parameterization in the characterization of the approximations. An alternative strategy is to simply limit the information determined and used for inference purposes about the solutions to bounded linear functionals defined on them, such as generalized moments.

Even if, when evaluating such a functional, the approximation to che solution is not very stable, the evaluation of a bounded functional will induce its own stabilization of the information being sought. This can be characterized in a number of independent ways. For example, by a statistical analysis of the effect of the functional on randomly perturbed data. A more cogent and useful characterization can be built around the fact that, by exploiting the mathematical structure of the problem under examination, the linear functionals defined on its solutions (the solution-functionals) can be transformed to linear functionals defined on the data (the data-functionals). The major advantage of this approach is that it gives an explicit characterization of the degree of improperly posedness of a solution functional in terms of the operations performed on the data by the data-functional.

The ill-posedness of the inverse problem has been transferred to the transformation which maps the solution-functionals into the data-functional s; and has thereby been circumvented, when the transformation can be solved exactly, as long as no attempt is made to reconstruct the solution from the computed estimates of these functionals derived from the available observational data. Even when the transformation must be inverted numerically, the situation is superior to that for the original inverse problem since the data is now known analytically, and therefore the full power of regularization can be exploited.

Thus, from the practical point of view of solving inverse problems which arise in applications, the aim of the linear functional strategy is to identify solution-functionals relevant to the problem context and, where possible, to simply work with them as the corresponding data-functionals for inference and decision making purposes.