Tensor Train Acceleration of Method of Moments Solution of Volume Integral Equation on Structured and Unstructured Meshes

Abstract

Method of Moments (MoM) discretization of the Integral Equations (IEs) of electromagnetics results in dense matrix equation. Such matrix equations require prohibitively large computational resources when the number of basis functions used in discretization reaches hundreds of thousands and higher. Tensor Train (TT) decomposition of the MoM dense matrix equations has been recently proposed [1] to drastically reduce both the memory use for matrix storage and the CPU time required for its multiplication with a vector. Toeplitz matrix resulting from MoM discretization of Volume Integral Equation (VIE) can be represented as a multi-dimensional matrix and stored as a product of smaller dimensional matrices (tensors). Such product of smaller dimensional matrices, also known as the tensor train (TT), can reduce the matrix storage from O(N2) to O(logN) at low frequencies and O(NlogN) at high frequencies. The product of the matrix in the TT form with a vector can be computed in O(NlogN) operations. In order to accelerate MoM solution of practical scattering problems we recently developed Conjugate-Gradient-Tensor-Train (CG-TT) [2] and Precorrected-Tensor-Train (P-TT) [3] algorithms.