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Range-separated tensor formats for numerical modeling of many-particle interaction potentials
Peter Benner, Venera Khoromskaia and Boris N. Khoromskij
We introduce and analyze the new range-separated (RS) canonical and Tucker tensor formats and apply them to numerical modeling of the arbitrarily distributed 3D long-range interaction potentials in multi-particle systems.
The main idea of the hybrid RS tensor formats is the independent low-rank representation of the localized and global parts in the target tensor which reduces the costs of related multi-linear algebra and enhances numerical treatment of many-particle interactions.
The reference interaction potential is precomputed in the form of a single low-rank canonical tensor on a 3D $n\times n\times n$ Cartesian grid. The smooth long-range contribution to the total potential sum is represented in $O(n)$ storage via the compressed canonical/Tucker tensor, which is obtained by the multigrid canonical-to-Tucker decomposition of this sum. We prove that the resultant canonical/Tucker rank depends only logarithmically on the number of particles $N$ and the grid-size $n$.
The respective rank bounds provide theoretical justification of the new approach to summation of arbitrarily distributed interaction potentials. Agglomeration of the short range sum is reduced to the independent treatment of $N$ localized terms with almost disjoint effective supports, calculated using $O(N)$ operations. Thus the cumulated set of short range clusters is parametrized by a single low-rank canonical reference tensor with a local support, complemented by a list of particle coordinates and their charges, thus providing the irreducible storage costs.
The RS-canonical or RS-Tucker tensor representations simplify algebraic operations on the 3D potential sums arising in multi-dimensional data modeling by radial basis functions, in 3D integration and convolution, computation of gradients, forces and the interaction energy of a system etc., by reducing all of them to 1D calculations. In particular, we introduce the new regularized formulation for the Poisson-Boltzmann equation that may be useful for efficient computation of the electrostatic potential of proteins.