The Inverse Medium Problem

We now turn our attention to the problem of reconstructing the refractive index from a knowledge of the far field pattern of the scattered acoustic or electromagnetic wave. We shall call this problem the inverse medium problem. Of particular interest to us will be the use of a dual space method to determine the refractive index. This method has the numerical advantage of significantly reducing the number of unknowns in the nonlinear optimization step for determining the refractive index and, in addition, leads to a number of mathematical problems which are of interest in their own right. Our aim in this chapter is to develop the theory of the inverse medium problem to the point where an optimization scheme can be formulated for the solution such that under appropriate conditions the infimum of the cost functional is zero. However, since similar optimization schemes were analyzed in depth in Chapters 5 and 7, we shall not dwell on the specific optimization scheme itself, except in Section 10.6 where we present some numerical examples.