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

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 very dense line segments in the spectrum as being 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 on the operator level transforms the system to the form “identity plus compact.” On the matrix level this regularization reduces the length of the dense spectral segments roughly by a factor of four while preserving the ability to calculate the matrix-vector products using the FFT algorithm. 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.