A Level - Set Approach for Inverse Problems Involving

An approach for solving inverse problems involving obstacles is proposed. The approach uses a level-set method which has been shown to be eeective in treating problems of moving boundaries, particularly those that involve topological changes in the geometry. We develop two computational methods based on this idea. One method results in a nonlinear time-dependent partial diierential equation for the level-set function whose evolution minimizes the residual in the data t. The second method is an optimization that generates a sequence of level-set functions that reduces the residual. The methods are illustrated in two applications: a deconvolution problem and a diiraction screen reconstruction problem. 1. Inverse problems involving obstacles There is a host of inverse problems wherein the desired unknown is a region in IR 2 or IR 3. The region is possibly multiply connected or consisting of several subregions. A classical example is the inverse scattering problem for an obstacle (see Colton and Kress 3]). Other examples include a problem in mine detection (Friedman 5]), reconstruction of a diiraction screen (Sondhi 11]; Magnanini and Papi 8]). A common goal in these problems is to determine the set of an unknown characteristic function given remotely measured data. Abstractly, they can be posed as: Find D in the equation A(u) = g; (1a) where u(x) = u int for x 2 D u ext for x = 2 D :