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RANGE-SEPARATED TENSOR FORMAT FOR MANY-PARTICLE MODELING
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DOI：10.1137/16M1098930.png)
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Abstract
En 中文
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(parallel to x parallel to), x is an element of R-d, say P(parallel to x parallel to) = 1/parallel to x parallel to for d = 3, is split into a sum of localized and long-range low-rank canonical tensors represented on a fine 3D nxnxn 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 short-range 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 nxnxn 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.
Keywords:
low-rank tensors
summation of electrostatic potentials
long-range many-particle interactions
radial basis functions
canonical and Tucker tensor formats
Ewald summation
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