Fast Direct Solution of 2D Scalar Volume Integral Equation via Tensor Train Decomposition for Scatterers of Arbitrary Shape

Abstract

The method of moments (MoM) discretization of the volume integral equation (VIE) is an effective tool for calculating field responses in the presence of arbitrary scatterers. This method scales poorly with problem size, however, due to the inherent difficulty of inverting the relevant matrix equations. The application of tensor train (TT) decomposition techniques has recently shown promise in alleviating the complexity of such calculations, with dramatic efficiency boosts in both computational time and memory having been proven possible for very simple scatterers. For arbitrary scatterers, previous work has shown that a conjugate gradient- TT (CG- TT) procedure was still capable of producing favorable runtime and memory complexities, particularly in quasi-constant rank regimes. In this work, we consider a direct solution of TT decomposed MoM matrix equation approach that is largely insensitive to MoM matrix condition number, while maintaining the generality of CG- TT. Preliminary results indicate that this approach is capable of matching the efficiency of CG- TT in the quasi-constant rank regime for scatterers of arbitrary geometry.