Sparsity-Aware Precorrected Tensor Train Algorithm for Fast Solution of 2-D Scattering Problems and Current Flow Modeling on Unstructured Meshes

Acceleration of the method of moments (MoM) solution of the volume integral equation (VIE) on unstructured meshes is performed using a precorrected tensor train (P-TT) algorithm. The elements of the MoM’s unstructured mesh are projected onto a regular Cartesian grid. This enables representation of the MoM matrix as the Toeplitz matrix of point-to-point interactions pre- and post-multiplied by sparse matrices projecting MoM’s basis and testing functions on the Cartesian grid. The Toeplitz matrix is subsequently cast into the form of a multidimensional tensor. The latter is decomposed into the product of smaller dimensional matrices also known as tensor train (TT). TT allows to store Toeplitz matrix in
$O(\log N)$
memory for VIE with the Laplace kernel and in
$O(N \log N)$
memory for VIE with the Helmholtz kernel. Unlike the FFT-based fast algorithms, the P-TT method enables further memory reduction due to the sparsity of sources on the Cartesian grid. This sparsity pattern can be accounted for in construction of TT cores. It further reduces memory use as well as CPU time required for the multiplication of the Toeplitz matrix with a vector. It is shown that upon sufficient sparsity in representation of the sources on the grid, the P-TT algorithm can outperform both in CPU time and memory its FFT-based counterpart known as the precorrected FFT algorithm.