A Hierarchically Semiseparable Preconditioner for the Numerical Solution of 3D Electromagnetic Scattering Problems

Abstract

We consider the numerical solution of linear systems arising from the discretization of the Electric Field Integral Equation (EFIE) in scattering problems for arbitrarily shaped targets. For some geometries the associated matrix can be poorly conditioned making the use of a preconditioner mandatory to obtain convergence. In this thesis the linear system resulting from the discretized electromagnetic scattering problem is solved by means of a preconditioned GMRES in the context of the multilevel fast multipole method (MLFMM) . For this purpose an approximate hierarchically semiseparable (HSS) representation of the near-field matrix, the part of the matrix capturing interactions among nearby groups in the MLFMM, is constructed and used as preconditioner for the GMRES iterations. As experience shows, the efficiency of an ILU preconditioning for such systems essentially depends on a sufficient fill-in, which apparently sacrifices the sparsity of the near-field matrix and thus leads, for electrically large structures, to storage problems. In the light of this experience we propose to approximate the near-field matrix by means of a HSS approximant so as to build a structured preconditioner to the impedance matrix. In this manner we impose a low-rank structure on the arising fill-in, thus alleviating the storage requirements of the preconditioner. To this end the group distribution at the finest level of the MLFMM is exploited so as to obtain the HSS approximant to the near-field matrix. As a result, the storage problems stated in a previous work by the author are mitigated substantially. The numerical results presented in the thesis show that this kind of algebraic preconditioning can substantially reduce the number of iterations in the solution of the resulting system of equations.