Inverse Obstacle scattering in two dimensions with multiple frequency   data and multiple angles of incidence
  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. 
Carlos Borges and Leslie Greengard
math.NA math-ph math.MP physics.comp-ph