Publications

We consider the problem of reconstructing the shape of an impenetrable sound-soft obstacle from scattering measurements. The input data is assumed to be the far-field pattern generated when a plane wave impinges on an unknown obstacle from one or more directions and at one or more frequencies. It is well known that this inverse scattering problem is both ill posed and nonlinear. It is common practice to overcome the ill posedness through the use of a penalty method or Tikhonov regularization. Here, we present a more physical regularization, based simply on restricting the unknown boundary to be band-limited in a suitable sense. To overcome the nonlinearity of the problem, we use a variant of Newton's method. When multiple frequency data is available, we supplement Newton's method with the recursive linearization approach due to Chen. During the course of solving the inverse problem, we need to compute the solution to a large number of forward scattering problems. For this, we use high-order accurate integral equation discretizations, coupled with fast direct solvers when the problem is sufficiently large.

We present a fast direct solver for the simulation of electromagnetic scattering from an arbitrarily-shaped, large, empty cavity embedded in an infinite perfectly conducting half space. The governing Maxwell equations are reformulated as a well-conditioned second kind integral equation and the resulting linear system is solved in nearly linear time using a hierarchical matrix factorization technique. We illustrate the performance of the scheme with several numerical examples for complex cavity shapes over a wide range of frequencies.

In this paper, we consider acoustic or electromagnetic scattering in two dimensions from an infinite three-layer medium with thousands of wavelength-size dielectric particles embedded in the middle layer. Such geometries are typical of microstructured composite materials, and the evaluation of the scattered field requires a suitable fast solver for either a single configuration or for a sequence of configurations as part of a design or optimization process. We have developed an algorithm for problems of this type by combining the Sommerfeld integral representation, high order integral equation discretization, the fast multipole method and classical multiple scattering theory. The efficiency of the solver is illustrated with several numerical experiments.