Tucker tensor method for fast grid-based summation of long-range potentials on 3D lattices with defects
Venera Khoromskaia and Boris N. Khoromskij
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Submission date: 21. Aug. 2014 (revised version: October 2014)
MSC-Numbers: 65F30, 65F50, 65N35, 65F10
Keywords and phrases: Lattice sums, Tucker tensor decomposition, long-range potentials
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We introduce the Tucker tensor method for the grid-based assembled summation of long-range interaction potentials over large 3D lattices in a box. This method is a generalization of our previous approach on the low-rank canonical tensor summation of electrostatic potentials on a rectangular 3D lattice. In the new technique we first approximate (with a guaranteed precision) the single kernel function represented on large N × N × N 3D grid in a bounding box by a low-rank reference Tucker tensor. Then each 3D singular kernel function involved in the summation is approximated on the same grid by the shift of the reference Tucker tensor. Directional vectors of the Tucker tensor representing a full lattice sum are assembled by the 1D summation of the corresponding Tucker vectors for shifted potentials, while the core tensor remains unchanged. The Tucker ranks of the resultant tensor sum on the 3D rectangular L × L × L lattice are proven to be the same as for the single kernel function. The required storage scales linearly in the 1D grid-size, O(N), while the numerical cost is estimated by O(NL). With the slight modifications our approach applies in the presence of defects, such as vacancies, impurities and non-rectangular geometries of a set of active lattice points, as well as for the case of hexagonal lattices. For potential sums with defects the Tucker rank of the resultant tensor may increase, so that we apply the 𝜀-rank truncation procedure based on the generalized reduced HOSVD approximation combined with the ALS iteration. We prove the error bounds and stability for the HOSVD Tucker approximation to a sum of canonical/Tucker tensors. Numerical tests confirm the efficiency of the presented tensor summation method. In particular, we show that a sum of millions of Newton kernels on a 3D lattice with defects/impurities can be computed in about a minute in Matlab implementation. The approach is beneficial for functional calculus with the lattice potential sum represented on large 3D grids in the Tucker/canonical formats. Indeed, the interpolation, scalar product with a function, integration or differentiation can be performed easily in tensor arithmetics with 1D complexity.