Transverse Electric Scattering on Inhomogeneous Objects: Spectrum of Integral Operator and Preconditioning

Abstract

Abstract. The domain integral equation method with its FFT-based matrix-vector products is a viable alternative to local methods in free-space scattering problems. However, it often suffers from the extremely slow convergence of iterative methods, especially in the transverse electric (TE) case with large or negative permittivity. We identify the nontrivial essential spectrum of the pertaining integral operator as partly responsible for this behavior, and the main reason why a normally efficient deflating preconditioner does not work. We solve this problem by applying an explicit multiplicative regularizing operator, which transforms the system to the form 'identity plus compact', yet allows the resulting matrix-vector products to be carried out at the FFT speed. Such a regularized system is then further preconditioned by deflating an apparently stable set of eigenvalues with largest magnitudes, which results in a robust acceleration of the restarted GMRES under constraint memory conditions. Key words. Domain integral equation, singular integral operators, electromagnetism, TE scattering, spectrum of operators, essential spectrum, regularizer, deflation, preconditioner AMS subject classifications. 78A45, 65F08, 45E10, 47G10, 15A231. Introduction. There is a steady interest in the numerical simulation of the electromagnetic field in inhomogeneous media. The methods can be roughly divided into two categories -local and global -in accordance with the governing equations. The local methods based on the differential Maxwell's equations [9,10,14,15] are generally more popular due to the sparse nature of their matrices and the ease of programming. Alongside the local methods, global methods based on an equivalent integral equation formulation are also often employed, especially in free-space scattering. If an object is large and homogeneous or has a perfectly conducting boundary then the problem is usually reduced to a boundary integral equation with the fields (currents) at the interfaces being the fundamental unknowns [13]. If an object is continuously inhomogeneous or is a composite consisting of many small different parts, the most appropriate global method is the domain integral equation (DIE), in two dimensions, or the volume integral equation, in three dimensions [16,17,18,19,21,22,25,27,28]. Although the global methods produce dense matrices, they are generally more stable with respect to discretization than the local ones, and the convolution-type integral operators sometimes allow to compute matrix-vector products at the FFT speed. These properties make the DIE method a viable alternative to local methods for certain free-space scattering problems.The main difficulties with the DIE method are the non-normality of both the operator and the resulting system matrix, inherent to frequency-domain electromagnetic scattering, and the extremely slow convergence of the few iterative methods that can be applied with such matrices. Numerical experiments consistently show that the GMRES algorithm is the best choice [11],...

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