8902122 Sacks The subject of inverse scattering includes a number of interesting and important scientific problems, arising in the areas of geophysics, quantum mechanics, nondestructive testing, medical imaging and elsewhere. Mathematical formulation of such problems often require the determination of one or more coefficient functions in a partial differential equation (or system of equations), given as data certain limited knowledge about special solutions to the equation. One finds two substantial sources of difficulty associated with most problems of this type, in theory and in application, namely nonlinearity and ill-posedness. There is a need for much more mathematical analysis. The objective of this project is the investigation of the mathematical structure of certain model problems arising in several areas of inverse scattering. As a general rule the approach to a particular nonlinear problem will be via the study of linearized approximations. Such linearized inverse problems are also of independent interest. Specific questions under investigation include the uniqueness of the solution, the continuous dependence of the solution on the given data, the extent of validity of linearization techniques, and how the answers to the previous questions can be used in the design of numerical solution methods.
