The Convex Scattering Support in a Background Medium

We discuss inverse problems for the Helmholtz equation at fixed energy, specifically the inverse source problem and the inverse scattering problem from a medium or an obstacle. In [S. Kusiak and J. Sylvester, Comm. Pure Appl. Math., 56 (2003), pp. 1525–1548], we introduced the convex scattering support of a far field, a set which will be a subset of the convex hull of the support of any source or scattering inhomogeneity which can produce it. We extend these results and modify the methods to locate a source within a known inhomogeneous background medium, or a deviation from that medium, using observations of a single far field. We also describe some numerical examples that illustrate the robustness of the method.