Tensor decomposition in electronic structure calculations on 3D Cartesian grids

Khoromskij, Boris N.; Khoromskaia, Venera; Chinnamsetty, Sambasiva Rao; Flad, Heinz-Jürgen

doi:10.1016/j.jcp.2009.04.043

Cited by 47 publications

(55 citation statements)

References 33 publications

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“…3.3 (see also [2,19]). Low rank representation of G z (x) can be utilized in fast tensor convolution transform [20,21,23]. Construction of separable approximation to the nonoscillating fundamental solution G z for d > 3 can be based on similar sinc-methods.…”

Section: Discussionmentioning

confidence: 99%

“…This effect can be relaxed or completely avoided by a systematic application of low rank tensor-structured representations of the arising multivariate functions and related operators. Applications of tensor methods for representation of classical Green's kernels and elliptic resolvent [2-4, 10, 12, 18, 22], multidimensional convolution [6,19,20] and many other quantities arising in electronic structure calculations [16,21,23] have demonstrated surprising efficiency. In some applications the a priori fixed tensor subspace via sparse grids leads to efficient algorithms in higher dimensions [11,30].…”

Section: Introductionmentioning

confidence: 99%

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Tensor-Structured Preconditioners and Approximate Inverse of Elliptic Operators in ℝ d

Khoromskij

2009

Constr Approx

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In the present paper we analyze a class of tensor-structured preconditioners for the multidimensional second-order elliptic operators in R d , d ≥ 2. For equations in a bounded domain, the construction is based on the rank-R tensor-product approximation of the elliptic resolvent B R ≈ (L − λI ) −1 , where L is the sum of univariate elliptic operators. We prove the explicit estimate on the tensor rank R that ensures the spectral equivalence. For equations in an unbounded domain, one can utilize the tensor-structured approximation of Green's kernel for the shifted Laplacian in R d , which is well developed in the case of nonoscillatory potentials. For the oscillating kernels e −iκ x / x , x ∈ R d , κ ∈ R + , we give constructive proof of the rank-O(κ) separable approximation. This leads to the tensor representation for the discretized 3D Helmholtz kernel on an n × n × n grid that requires only O(κ | log ε| 2 n) reals for storage. Such representations can be applied to both the 3D volume and boundary calculations with sublinear cost O(n 2 ), even in the case κ = O(n).Numerical illustrations demonstrate the efficiency of low tensor-rank approximation for Green's kernels e −λ x / x , x ∈ R 3 , in the case of Newton (λ = 0), Yukawa (λ ∈ R + ), and Helmholtz (λ = iκ, κ ∈ R + ) potentials, as well as for the kernel functions 1/ x and 1/ x d−2 , x ∈ R d , in higher dimensions d > 3. We present numerical results on the iterative calculation of the minimal eigenvalue for the d-dimensional finite difference Laplacian by the power method with the rank truncation and based on the approximate inverse B R ≈ (− ) −1 , with 3 ≤ d ≤ 50.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Tensor-Structured Preconditioners and Approximate Inverse of Elliptic Operators in ℝ d

Khoromskij

2009

Constr Approx

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“…This scheme becomes attractive for the multiple calculation of the Yukawa potential when the exponents λ ≥ 0 vary during the computational process. It is also important to mention that the adaptive black-box scheme presented in this paper was successfully applied in electronic structure calculations [8,12,11].…”

Section: Introductionmentioning

confidence: 95%

“…Algorithm TGN was already successfully applied in numerical computations of various 3D convolution integrals [7] included in the Fock operator of the nonlinear Hartree-Fock equation in 3D, see [9,10,11,12]. In particular, this includes fast multiple computations of the Coulomb and exchange convolution integrals in the tensor-structured numerical methods for solving the ab initio Hartree-Fock equation on large n × n × n Cartesian grids, in the range n ≤ 10 4 , see [12].…”

Section: On the Rank Optimality And Conclusionmentioning

confidence: 99%

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

Bertoglio

Khoromskij

2012

Computer Physics Communications

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Tensor-product approximation provides a convenient tool for efficient numerical treatment of high dimensional problems that arise, in particular, in electronic structure calculations in R d . In this work we apply tensor approximation to the Galerkin representation of the Newton and Yukawa potentials for a set of tensor-product, piecewise polynomial basis functions. To construct tensor-structured representations, we make use of the well-known Gaussian transform of the potentials, and then approximate the resulting univariate integral in R by special sinc quadratures. The novelty of the approach lies on the heuristic optimisation of the quadrature parameters that allows to reduce dramatically the initial tensor-rank obtained by the standard sincquadratures. The numerical experiments show that this approach gives tensor-ranks close to the optimal in 3D computations on large spatial grids and with linear complexity in the univariate grid size. Particularly, this scheme becomes attractive for the multiple calculation of the Yukawa potential when the exponents in gaussian functions vary during the computational process.

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“…Tensor numerical methods are proved to be efficient for data-sparse representation of functions and operators in the Hartree-Fock and Kohn-Sham models in electronic structure calculations [7,39,59,57,23,76,47]. Fully tensorized numerical approach for solving the Hartree-Fock equation, discretized over N × N × N grid, by tensor truncated iteration of complexity O(N log N ), was recently presented in [58].…”

Section: On Tensor-structured Solution Of Multidimensional Equationsmentioning

confidence: 99%

Tensors-structured numerical methods in scientific computing: Survey on recent advances

Khoromskij¹

2012

Chemometrics and Intelligent Laboratory Systems

160

153

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In the present paper, we give a survey of the recent results and outline future prospects of the tensor-structured numerical methods in applications to multidimensional problems in scientific computing. The guiding principle of the tensor methods is an approximation of multivariate functions and operators relying on certain separation of variables. Along with the traditional canonical and Tucker models, we focus on the recent quantics-TT tensor approximation method that allows to represent N -d tensors with log-volume complexity, O(d log N ). We outline how these methods can be applied in the framework of tensor truncated iteration for the solution of the high-dimensional elliptic/parabolic equations and parametric PDEs. Numerical examples demonstrate that the tensor-structured methods have proved their value in application to various computational problems arising in quantum chemistry and in the multi-dimensional/parametric FEM/BEM modeling-the tool apparently works and gives the promise for future use in challenging high-dimensional applications.

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Tensor decomposition in electronic structure calculations on 3D Cartesian grids

Cited by 47 publications

References 33 publications

Tensor-Structured Preconditioners and Approximate Inverse of Elliptic Operators in ℝ d

Tensor-Structured Preconditioners and Approximate Inverse of Elliptic Operators in ℝ d

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

Tensors-structured numerical methods in scientific computing: Survey on recent advances

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