Range-Separated Tensor Format for Many-Particle Modeling

Abstract

Abstract. We introduce and analyze the new range-separated (RS) canonical/Tucker tensor format which aims for numerical modeling of the 3D long-range interaction potentials in multiparticle systems. The main idea of the RS tensor format is the independent grid-based low-rank representation of the localized and global parts in the target tensor, which allows the efficient numerical approximation of N -particle interaction potentials. The single-particle reference potential, described by the radial basis function p( x ), x ∈ R d , say p( x ) = 1/ x for d = 3, is split into a sum of localized and long-range low-rank canonical tensors represented on a fine 3D n×n×n Cartesian grid. The smoothed long-range contribution to the total potential sum is represented on the 3D grid in O(n) storage via the low-rank canonical/Tucker tensor. We prove that the Tucker rank parameters depend only logarithmically on the number of particles N and the grid size n. Agglomeration of the short-range part in the sum is reduced to an independent treatment of N localized terms with almost disjoint effective supports, calculated in O(N ) operations. Thus, the cumulated sum of shortrange clusters is parametrized by a single low-rank canonical reference tensor with local support, accomplished by a list of particle coordinates and their charges. The RS canonical/Tucker tensor representations defined on the fine n × n × n Cartesian grid reduce the cost of multilinear algebraic operations on the 3D potential sums, arising in modeling of the multidimensional data by radial basis functions. For instance, computation of the electrostatic potential of a large biomolecule and the interaction energy of a many-particle system, 3D integration and convolution transforms as well as low-parametric fitting of multidimensional scattered data all are reduced to 1D calculations.

Regularization of Poisson--Boltzmann Type Equations with Singular Source Terms Using the Range-Separated Tensor Format

“…However, all these techniques do not efficiently separate the long-and short-range components in the total potential sum, hence there is a need to apply so-called interface (or jump) conditions at the interface between the molecular region and the solvent region in order to reduce discontinuities. In what follows, we describe the RS tensor format developed in [6], applied as the main tool in our paper to modify the PBE to increase the accuracy of its numerical approximation. A more detailed description of the new solution decomposition technique is provided in Sections 4.1 and 4.2.…”

“…Moreover, in general, the radial function p( x ) has a singularity or a cusp at the origin,x = 0, making its accurate grid representation problematic. An efficient numerical scheme for the grid-based calculation of P (x) in multiparticle systems can be constructed by using the RS tensor format [6].…”

“…Here, we present a new approach for the regularization of the PBE by using the rangeseparated (RS) canonical tensor format introduced and analyzed in [6]. The RS tensor format relies on the independent grid-based low-rank tensor representation of the long-and shortrange parts in the total sum of single-particle electrostatic potentials discretized on a fine 3D n×n×n Cartesian grid Ω n in the computational box Ω ⊂ R 3 .…”

“…Section 2 provides a short overview of existing solution decomposition schemes for the PBE problem. Section 3 describes the principles of the rank-structured tensor approximation to the long-range electrostatic potential and sketches the RS tensor decomposition techniques [6] for the free space electrostatic potential of many particle systems. The main Section 4 explains how the application of the RS tensor format leads to the new regularization scheme for solving the PBE.…”

Range-separated tensor decomposition of the discretized Dirac delta and elliptic operator inverse

“…where rank(A 0 ) ≤ R 0 and diam(suppU 0 ) ≤ 2γ in the index size.The storage size for the RS-canonical tensor A in (2.17) is estimated by ([3], Lemma 3.9), stor(A) ≤ dRn + (d + 1)N + dR 0 γ.…”