PI:David Ingerman DMS-9801710 This project is concerned with discrete and continuous inverse boundary problems. These are the problems of obtaining information about a surface or a graph, given some measurements on the boundary of the object, and the problems of characterizing possible boundary measurements. An example of such a problem would be the recovering of conductivity (or genus) of a surface given its Dirichlet-to-Neumann map (the map from boundary potential to current). Our results show that the graphs can provide an efficient way for computer modeling of continuous medium behavior. We are also interested in approximation of solutions of continuous inverse boundary problems by the discrete ones. The main subject of this project is understanding of the connection between continuous and discrete inverse boundary problems. These are problems of obtaining information about interior structure of a body from some measurements made on its surface. The problems take their origin in the oil exploration industry from attempts to non-invasively determine earth deep structure by inducing and measuring electromagnetic fields on the surface. We made a good progress in understanding of the discretized versions of the problem. Our goals in this project are modeling of continuous medium behavior using the discrete analogs and approximation of solutions of continuous inverse boundary problems by the discrete ones.
