Towards Tensor-Train Solution of Vector Volume Integral Equation in 3D with $\log N $ Complexity

Tensor train decomposition is used first time to represent the matrix, excitation, and the solution vectors of the dense matrix equation resulting from MoM discretization of full-wave 3D Volume Integral Equation as a product of $O(\log(N))$ small matrices (tensors). Such QTT decomposition is shown to use $O(\log(N))$ CPU time and memory. To solve the matrix equation given the QTT representation of the SLAE matrix and vectors, we use iterative GMRES scheme performing fast evaluation of the matrix-vector products with only $O(\log(N))$ CPU time and memory use. At this stage the $O(\log(N))$ performance is limited, however, to the SLAEs with purely Toeplitz matrices corresponding to the scattering problems on homogeneous dielectric scatterers of cubic voxels.