This project centers on parameter identification inverse problems. Typically, in such a problem one is given certain information about the solution of a partial differential equation and one is hopeful of using this information to compute one or more of the coefficients in the equation. Such problems are typically mathematically ill-posed and, to make matters worse, in processing the data in many practical applications, one must contend not only with measurement error but sometimes also with the fact that the readings may only be taken at a rather sparse collection of measurement points. In this proposal we outline a continuation of work on a new and very promising approach to a broad class of parameter identification problems. At the core of the method is the Dirichlet principle for elliptic boundary value problems, that the solution may be obtained by the minimization of a certain energy functional; that the same energy functional can also be used to compute coefficients in the elliptic equation, if an appropriate constrained minimization is employed, is the significant observation that drives much of the work in this proposal. Extensive numerical experiments have shown that the same numerical stability that is associated with the Dirichlet principle approach to finding solutions is also present (assuming the appropriate stabilization is employed) in the constrained minimization; in particular the method appears to be quite robust in the presence of noise in the data. The parameter identification problems amenable to this approach are, roughly speaking, those that involve elliptic equations that may besolved by means of a Dirichlet principle; as is well known this is a large class, having considerable practical significance. In addition, many parameter identification inverse problems involving parabolic (diffusion) and hyperbolic (wave propagation) equations may be restated to a form covered by this theory, and thus the eventual impact is expected to be even greater.
The first part of the project involves producing working algorithms to identify, from easily obtainable measurements, the various parameters needed to quantify flow in porous media. This is a crucial step in the modeling of underground water systems and a necessary first step when one wishes for example to manage water resources efficiently, or to predict the effects on an aquifer caused by environmental factors such as flooding or industrial pollution. The second part of the project is concerned with the problem of imaging inside the human body with electrical impedance tomography. Here, one attempts, by means of low voltage electrical measurements taken at the surface of a subject, to form an image of the interior. Such imaging has the advantage of being both non-invasive and non-destructive, and inexpensive; current disadvantages include poor image quality, and it is hoped that improvements can be accomplished by methods related to those used above. Related practical applications of these ideas include non-destructive evaluations involving the determination of gas pores, impurities, and cracks in cast metals; such problems are of interest in a variety of manufacturing situations, including the inspection of airplane parts, and also in the manufacture and testing of nuclear reactor containment vessels.
