Knowles The investigator studies parameter identification inverse problems. Typically, in such a problem one uses certain given information about the solution of a partial differential equation to compute one or more of the coefficients in the equation. Such problems are invariably ill-posed (sometimes badly so) and, to make matters worse, in processing the data in most 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 consequence, in many areas, it is well documented that algorithms for parameter identification that are sufficiently reliable to engender widespread practical use are not yet available. In this project the investigator develops a new approach to a broad class of parameter identification problems. At the core of the approach 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 key observation. Recent experiments have shown that the same numerical stability that is associated with the Dirichlet principle approach to finding solutions is also present in the constrained minimization, and furthermore the method appears to be quite robust in the presence of noise in the data. The class of parameter identification problems to which the approach is applicable is, roughly speaking, those that involve elliptic equations that may be solved by means of a Dirichlet principle; as is well known, this is a large class, having considerable practical significance. The bulk of the work in the project involves applying these ideas to producing working algorithms for the solution of the aquifer transmissivity problem (a crucial step in the modeling of under ground water systems) and the problem of imaging inside the human body with electrical impedance tomography. Inverse problems dealing with parameter identification in the presence of data error are of fundamental practical importance in a number of areas, including, but by no means limited to, medical and industrial imaging, high energy physics, geophysics, and hydrology. Such problems are mathematically ill-posed in the sense that small variations in the data (caused by measurement error for example) can cause uncontrollably large errors to appear in the calculated parameters. In this proposal a new optimization approach to a broad class of parameter identification problems is presented that already shows significant promise for handling the moderately ill-posed problems in this class, and there are indications that such an approach may help solve the most egregiously ill-posed examples. The project concentrates on two test cases, the aquifer transmissivity problem (a necessary and crucial step in monitoring the flow of contaminants in underground water systems, for example) and the problem of (noninvasive and nondestructive) imaging inside the human body using electrical impedance tomography. If suitably effective algorithms can be developed with these ideas in the test cases, similar algorithms should be possible for a wide range of related applications in the areas outlined above, and elsewhere.
