On the Relationship between Nonuniqueness of Electromagnetic Scattering Integral Equations and Krylov Subspace Methods

摘要

Some relevant operator equations used to describe natural phenomena can degenerate. This is actually the case of the electric (EFIE) and magnetic (MFIE) field integral equations in scattering theory, both failing to provide a unique solution (at least in a standard sense) in correspondence of some set of frequencies (known as resonances), for which they lack injectivity.rnOne of the most common expedient to the problem deals with the concept of a generalized solution (say a pseudo-solution), whose computation is numerically related to the Moore-Penrose pseudo-inverse of some large and generally dense square matrix A ∈ C~(n,n) resulting by the discretization of the integral model via the Method of Moments (MoM) or different projective schemes and its consequent approximation (in the sense of the uniform operator limit) by means of a linear system of algebraic equations such as Ax = b. Ultimately this involves the very time-consuming task of computing the singular value decomposition (SVD) of A, in order to filter the noisy effects introduced on the unknown current by the almost-singularity of A when EFIE or MFIE are applied in the neighborhood of a resonance.rnIn this paper, we try to give an answer and provide theoretical motivations and numerical evidence to prove it should be the right one, relying on the inherent capability of some Krylov subspace methods to extract the minimum norm solution to the linear system Ax = b to the extent that b lies in the range of A.