Low-rank quadrature-based tensor approximation of the Galerkin projected Newton/Yukawa kernels

Bertoglio, C; Khoromskij, Boris N.

doi:10.1016/j.cpc.2011.12.016

Cited by 30 publications

(64 citation statements)

References 20 publications

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“…In this section, we recall the grid-based method for the low-rank canonical representation of a spherically symmetric kernel function p( x ), x for x ∈ R 3 . The single reference potential, like 1/ x , can be represented on a fine 3D n × n × n Cartesian grid in the form of a low-rank canonical tensor [26,8]. Further, we confine ourselves to the case d = 3.…”

Section: 2mentioning

confidence: 99%

Range-Separated Tensor Format for Many-Particle Modeling

Benner

Khoromskaia

Khoromskij

2018

SIAM J. Sci. Comput.

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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.

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Section: 2mentioning

confidence: 99%

Range-Separated Tensor Format for Many-Particle Modeling

Benner

Khoromskaia

Khoromskij

2018

SIAM J. Sci. Comput.

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“…where P N ∈ R n×n×n is the rank-R N canonical tensor approximating the Newton kernel in (2.8) [15,6]. Then, the entries of the 4-th order tensor B = [b µνκλ ] N b µνκλ=1 can be evaluated by bilinear tensor operations as a sequence of (n log n) convolutions, and 1D Hadamard and scalar products [22,25] b…”

Section: Grid-based Two-electron Integralsmentioning

confidence: 99%

“…Here, we use the quantized version of the Laplace operator introduced in [19]. For ∆ 3 in (3.4), the rank-2 tensor train representation [38] is introduced in [19] as 6) where the sign ⊗ b denotes the matrix product of block core matrices, with blocks being multiplied by means of the tensor product. Suppose that n = 2 L , then the quantized representation of ∆ 1 , takes the form [19] …”

Section: D Laplace Operator In O(log N) Complexitymentioning

confidence: 99%

“…The tensor approximation of the Newton potential in a canonical tensor format with the rank-R N introduced in [32,6], is based on the application of the sinc approximation and the Laplace transform [15], and piecewise constant discretization of the resulting representation on the equidistant tensor grid ω 3,n . Due to the tensor approximation P c,ν of the Newton kernel, we represent the nuclear potential for the molecule as a sum of M tensors placed in the coordinates of nuclei,…”

Section: D Nuclear Potential Operator In 1d Complexitymentioning

confidence: 99%

“…These algorithms and approximations, as well as the tensor-product convolution developed in [29], enable tractable numerical treatment of the multivariate functions and operators [23]. The important ingredient of the multidimensional tensor product convolution is the canonical tensor approximation to the Newton kernel introduced in [14,15,6,9]. Thus, it became possible to avoid the cubic scaling of computational work in the numerical treatment of functions and operators in 3D problems.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Black-Box Hartree–Fock Solver by Tensor Numerical Methods

Khoromskaia

2014

Computational Methods in Applied Mathematics

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The Hartree-Fock eigenvalue problem governed by the 3D integro-differential operator is the basic model in ab initio electronic structure calculations. Several years ago the idea to solve the Hartree-Fock equation by fully 3D grid based numerical approach seemed to be a fantasy, and the tensor-structured methods did not exist. In fact, these methods evolved during the work on this challenging problem. In this paper, our recent results on the topic are outlined and the black-box Hartee-Fock solver by tensor numerical methods is presented. The approach is based on the rank-structured calculation of the core hamiltonian and of the two-electron integrals tensor using the problem adapted basis functions discretized on n × n × n 3D Cartesian grids. The arising 3D convolution transforms with the Newton kernel are replaced by a combination of 1D convolutions and 1D Hadamard and scalar products. The approach allows huge spatial grids, with n 3 ≃ 10 15 , yielding high resolution at low cost. The two-electron integrals are computed via multiple factorizations. The Laplacian Galerkin matrix can be computed "on-the-fly" using the quantized tensor approximation of O(log n) complexity. The performance of the black-box solver in Matlab implementation is compatible with the benchmark packages based on the analytical (pre)evaluation of the multidimensional convolution integrals. We present ab initio Hartree-Fock calculations of the ground state energy for the amino acid molecules, and of the "energy bands" for the model examples of extended (quasi-periodic) systems.

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A dual‐space multilevel kernel‐splitting framework for discrete and continuous convolution

Jiang,

Greengard

2024

Comm Pure Appl Math

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We introduce a new class of multilevel, adaptive, dual‐space methods for computing fast convolutional transformations. These methods can be applied to a broad class of kernels, from the Green's functions for classical partial differential equations (PDEs) to power functions and radial basis functions such as those used in statistics and machine learning. The DMK (dual‐space multilevel kernel‐splitting) framework uses a hierarchy of grids, computing a smoothed interaction at the coarsest level, followed by a sequence of corrections at finer and finer scales until the problem is entirely local, at which point direct summation is applied. Unlike earlier multilevel summation schemes, DMK exploits the fact that the interaction at each scale is diagonalized by a short Fourier transform, permitting the use of separation of variables, but without relying on the FFT. This requires careful attention to the discretization of the Fourier transform at each spatial scale. Like multilevel summation, we make use of a recursive (telescoping) decomposition of the original kernel into the sum of a smooth far‐field kernel, a sequence of difference kernels, and a residual kernel, which plays a role only in leaf boxes in the adaptive tree. At all higher levels in the grid hierarchy, the interaction kernels are designed to be smooth in both physical and Fourier space, admitting efficient Fourier spectral approximations. The DMK framework substantially simplifies the algorithmic structure of the fast multipole method (FMM) and unifies the FMM, Ewald summation, and multilevel summation, achieving speeds comparable to the FFT in work per gridpoint, even in a fully adaptive context. For continuous source distributions, the evaluation of local interactions is further accelerated by approximating the kernel at the finest level as a sum of Gaussians (SOG) with a highly localized remainder. The Gaussian convolutions are calculated using tensor product transforms, and the remainder term is calculated using asymptotic methods. We illustrate the performance of DMK for both continuous and discrete sources with extensive numerical examples in two and three dimensions.

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Low-rank quadrature-based tensor approximation of the Galerkin projected Newton/Yukawa kernels

Cited by 30 publications

References 20 publications

Range-Separated Tensor Format for Many-Particle Modeling

Range-Separated Tensor Format for Many-Particle Modeling

Black-Box Hartree–Fock Solver by Tensor Numerical Methods

A dual‐space multilevel kernel‐splitting framework for discrete and continuous convolution

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