A hierarchically semiseparable preconditioner for the MFLMM-based solution of the EFIE

In this paper, the numerical solution of linear systems arising from the discretization of the electric field integral equation (EFIE) in scattering problems for arbitrarily shaped targets is considered. For some geometries the associated matrix can be poorly conditioned making the use of a preconditioner mandatory to obtain convergence. Here 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). The novelty of this work is the construction of an approximate hierarchically semiseparable (HSS) representation of the near-field matrix, the part of the matrix capturing interactions among nearby groups in the MLFMM. as preconditioner for the GMRES iterations. In the light of this experience this paper proposes 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 a low-rank structure on the arising fill-in is imposed, thus alleviating the storage requirements of the preconditioner. Unlike the bisection method, this paper now exploits the group distribution at the finest level of the MLFMM so as to obtain the HSS approximant to the near-field matrix. As a result, the storage problems are mitigated substantially. In the conference presentation it is shown that this kind of algebraic preconditioning can substantially reduce the number of iterations in the solution of the resulting system of equations