Tensor Train Accelerated Solution of Volume Integral Equation for 2-D Scattering Problems and Magneto-Quasi-Static Characterization of Multiconductor Transmission Lines

Abstract

The method of moments (MoM) discretization of volume integral equation (VIE) results in dense $N \times N$ matrix, $N$ being the number of MoM basis functions. Naive solution of such a system of linear algebraic equations (SLAE) is expensive when problems become large scale. Recently, tensor decomposition has been introduced for solving the SLAE by folding its matrix and its vectors into high-dimensional tensors. In this paper, we present detailed explanations for tensor train (TT) decomposition of the SLAE matrices and vectors resulting from MoM discretization of VIE for scalar 2-D scattering problems under TM-polarization and magneto-quasi-static characterization of multiconductor transmission lines. For Toeplitz matrices resulted from MoM discretization on structured meshes, the extraordinary performance of TT with scaling of $\log (N)$ in CPU time and memory is shown to directly solve SLAE with millions of unknowns within a few minutes and few megabytes of memory. Such $\log (N)$ performance is limited, however, to the SLAE with purely Toeplitz matrices corresponding to the scattering problems on homogeneous dielectric scatterers of the rectangular cross section. To overcome this limitation and solve the problems with arbitrarily shaped inhomogeneous objects, we propose an iterative conjugate gradient-TT (CG-TT) scheme for solving MoM discretized VIE, which utilizes TT decomposition for the fast evaluation of the matrix-vector products. The CG-TT shows CPU and memory scaling in the low- and high-frequency regimes with $O(N)$ and $O(r^{2}~N \log N)$ , respectively, $r$ being the highest rank in the TT carriages. Detailed analysis of memory and CPU time scaling with respect to a number of unknowns and time-harmonic frequency is presented.