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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(\log (N))$</tex> small matrices (tensors). Such Quantized Tensor Train QTT decomposition is shown to use <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(\log (N))$</tex> CPU time and memory. To solve the matrix equation given the QTT representation of the system of linear algebraic equations (SLAE) matrix and vectors, we use iterative GMRES scheme performing fast evaluation of the matrix-vector products with only <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(\log (N))$</tex> CPU time and memory use. At this stage the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(\log (N))$</tex> performance is limited, however, to the SLAEs with purely Toeplitz matrices corresponding to the scattering problems on homogeneous dielectric scatterers of cubic voxels.