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Grid-based lattice summation of electrostatic potentials by low-rank tensor approximation
Venera Khoromskaia and Boris N. Khoromskij
We introduce and study the grid-based rank-structured tensor method for fast and accurate calculation of the lattice sums of Coulomb interactions on large 3D periodic-structured compounds. The approach is based on the low-rank canonical tensor representation of the Newton kernels discretized in a computational box using fine $N\times N \times N$ 3D Cartesian grid.
This reduces the 3D summation to a sequence of tensor operations involving only 1D vector sums, where each $N$-vector represents the canonical component in the tensor approximation to the lattice-translated Newton kernel. In the case of a supercell consisting of $L\times L \times L$ unit cells in a box the numerical cost scales linearly in the grid-size, $n$ as $O(N L)$. For periodic boundary conditions, the storage demand remains proportional to the size of a unit cell, $N/L$, while the numerical cost reduces to $O(N)$, that outperforms the FFT-based Ewald summation approaches of the complexity $O(N^3 \log N)$. The complexity scaling in the grid parameter $n$ can be reduced even to the logarithmic scale $O(\log N)$ by the quantics tensor approximation method. We prove an upper bound of the quantics rank for the canonical vectors in the lattice sum. This opens the way to numerical simulations including large lattice sums in a supercell (i.e. as $L\to \infty$) and their multiple replicas in periodic setting. This approach is beneficial in applications which require further functional calculus with the lattice potential, say, scalar product with a function, integration or differentiation, which can be performed easily in tensor arithmetics on large 3D grids with 1D cost. Numerical tests illustrate the performance of the tensor summation method and confirm the estimated bounds on the quantics rank.