Lanczos Biconjugate A-Orthonormalization Methods for Surface Integral Equations in Electromagnetism

Abstract

Abstract| We present a novel class of Krylov projection methods computed from the Lanczosbiconjugate A-Orthonormalization algorithm for the solution of dense complex non-Hermitianlinear systems arising from the Method of Moments discretization of electromagnetic scatter-ing problems expressed in an integral formulation. Their competitiveness with other popularKrylov solvers, especially when memory is a concern, is illustrated on a set of model problemsrepresentative of realistic radar-cross section calculations. The results presented in this studywill contribute to assess the potential of iterative Krylov methods for solving electromagneticscattering problems from large structures and to enrich the database of this technology.1. INTRODUCTION In the last decades, the development of boundary element methods (BEM) for solving electromag-netic (EM) scattering problems in two and three dimensions is receiving a vigorous impulse. BEMsolvers are computationally attractive because they require a simple description of the surface of thetarget by means of triangular facets simplifying considerably the mesh generation especially in thecase of moving objects. The Method of Moments (MoM) discretization of integral equations leads tothe solution of large dense linear systems of equations that are often ill-conditioned. For solving suchsystems, iterative Krylov algorithms can be a valid alternative to dense direct methods, providedthey are combined with the Multilevel Fast Multipole Algorithm (MLFMA) [6,11] to carry outapproximate matrix by vector products (M-V), and with suitable preconditioners. Many integralformulations for surface scattering and hybrid surface/volume discretizations yield non-Hermitianlinear systems that cannot be solved using the Conjugate Gradient (CG) algorithm. The General-ized Minimal Residual method (GMRES) [9] and its variant Flexible GMRES (FGMRES) [8] arebroadly used in application codes due to their proved robustness and smooth convergence, see e.g.,experiments reported in [1,6] for solving very large boundary element equations involving milliondiscretization points. On the other hand, iterative methods based on Lanczos biconjugation, suchas the BiConjugate Gradient Stabilized method (BiCGSTAB) [11] and the Quasi-Minimal Residualmethod (QMR) [2] can be cheaper in terms of memory demands but they may require many moreiterations to converge especially on realistic geometries [1,7]. In this paper we present a novel classof Krylov projection methods based on the recently developed Lanczos A-orthonormalization pro-cedure for solving dense complex non-Hermitian linear systems arising from the MoM discretizationof EM scattering problems expressed in an integral formulation. The flrst solver is named Biconju-gate A-Orthogonal Residual (BiCOR). Two variants of BiCOR are also considered, which do notrequire multiplication by the Hermitian of