The complex bi-conjugate gradient solver applied to large electromagnetic scattering problems, computational costs, and cost scalings

Abstract

The complex biconjugate gradient iterative method is applied to an isoparametric boundary integral equation formulation for frequency-domain electromagnetic scattering problems. It is demonstrated to work well on large and geometrically complex examples, including a 20 wavelength slender dipole, the NASA almond, and a resonant cavity. On such problems, with asymmetric curvilinear irregular meshes and nontrivial geometries, the number of iterations required seems to increase rather more than linearly with body size, indicating an overall sixth power cost scaling. This scaling is essentially as for direct methods, but with costs still a small fraction of the direct approach. A method is proposed for the selection of a termination condition designed to avoid seeking the approximate answer too precisely; it typically permits a further halving of costs.

Lanczos-type variants of the COCR method for complex nonsymmetric linear systems

“…Inspired by the elegant achievement thanks to Pocock and Walker [1] mentioned before, the main attention of the present paper is to explore three Lanczos-type variants of the COCR method for the solution of complex nonsymmetric linear systems. The first two can be considered as mathematically equivalent but numerically improved popularizing versions of the BiCR and CRS methods for complex systems, respectively.…”

Surface integral equation formulations for solving electromagnetic scattering problems with iterative methods

“…In the case of metallic objects the fundamental integral equations are the electric field integral equation (EFIE), the magnetic field integral equation (MFIE) and the combined field integral equation (CFIE) [ Mautz and Harrington , 1978]. Iterative solution of these formulations have been studied [e.g., Wilton and Wheeler , 1991; Song and Chew , 1995; Pocock and Walker , 1997; Makarov and Vedantham , 2002].…”

Reconstruction of Two‐Dimensional Axisymmetric Media using the Variational Born Iteration Method

“…(3) To find a for the electric parameter, we use the bi-conjugate gradient(BCG) method [9] to solve Eq.(15). The solution a is then used to update the conductivity of the object region.…”