Tensor Train Accelerated Method of Moment Solution of Volume Integral Equations for Arbitrary Objects with Logarithmic Complexity

This paper presents a computational framework for solution of the full-wave scattering problems and problems of the magneto-quasistatics for objects of arbitrary shapes with polylogarithmic O(logp N) complexity in both CPU time and memory, N being the number of basis and test functions in Method of Moments (MoM) discretization of the pertinent volume integral equations (VIEs). The dramatic reduction in the computational operations and storage is enabled through tensor train (TT) decomposition of the matrices and vectors involved in MoM discretization of the VIE, accompanied by the specialized linear algebra operations performed on these tensorized data sets. Such TT decompositions are performed after MoM matrices and vectors are represented as multidimensional datasets according to recursively subdivided basis functions defined on a regular grid of square boxes of a pixelized object of interest. While such multidimensional representations stemming from the hierarchical partitioning of the geometry as well as pertinent matrices and vectors in principle sufficient for construction of the TT decomposition, the ranks in the cores of the train can be shown to grow as O(N) for most geometries other than canonical cases of an ideal square or a smooth Gaussian object. In this work we show that containment of TT ranks both in compressed object representation and pertinent MoM matrices to polylog O(logp N) scaling can be established through global Gaussian smoothing of the step-function like transitions in the material contrast function throughout the pixelated domain confining the object of interest. Numerical examples conducted on TT-accelerated MoM solutions of the full-wave and quasi-magnetostatic VIEs show that the proposed approach enables overall complexity reduction to polylogarithmic scaling with N for arbitrarily shaped objects and material distributions ranging from simple shapes to complex fractal geometries paving a way for TT accelerated MoM to be used for practically relevant applications.