Tensor-Train Accelerated Solution of 3D Vector Volume Integral Equation Solutions with logN Complexity

This study investigates the application of tensor train decomposition to model the matrix, excitation, and solution vectors of the dense matrix equation derived from the Method of Moments (MoM) discretization of the full-wave 3D Volume Integral Equation. These components are represented as a product of O(log(N)) small matrices (tensors). This quantized tensor train (QTT) decomposition demonstrates O(log(N)) efficiency in both CPU time and memory usage. To solve the matrix equation based on the QTT-represented system of linear algebraic equations (SLAE) with matrices and vectors, we implement an iterative GMRES scheme, which enables rapid matrix-vector product evaluations with O(log(N)) CPU time and memory requirements. Currently, this O(log(N)) performance applies only to SLAEs with purely Toeplitz matrices, particularly in scattering problems involving homogeneous dielectric scatterers made of cubic voxels.