Application of multilevel structured matrices for the solution of direct and inverse electromagnetic problems

We consider the problem of electromagnetic wave scattering in the heterogeneous 3D half-space bounded by a perfectly conducting plane. Using a local heterogeneity model, we reduce this problem to a volume integral equation. Applying the Galerkin discretization on uniform Cartesian grids with special basis functions, we obtain a linear system with a three-level block matrix structured as TTT+THT. Taking into account this special structure of the matrix, we propose a parallel algorithm for the solution of the problem under consideration. The employment of this algorithm makes it possible to perform a numerical simulation of measurements with an accuracy sufficient for the solution of the inverse problem, i.e., for the study of heterogeneity structure. The results of solving the inverse problem with the use of Born approximation show a high accuracy of the method proposed. The work is partially supported by the Russian Foundation for Basic Research (04-07-90336, 05-01-00721) according to the programme of high-priority fundamental research of the Department of Mathematical Sciences of RAS “Computational and Information Technologies for the Solution of Large-Scale Problems”.