Tensor Product Multiscale Many-Particle Spaces with Finite-Order Weights for the Electronic Schrödinger Equation

Griebel, Michael; Hamaekers, Jan

doi:10.1524/zpch.2010.6122

Cited by 21 publications

(18 citation statements)

References 30 publications

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“…In addition, sparse grid techniques were successfully applied for the solution of integral equations [25], for quadrature [14], for regression [12] and for time series prediction [5]. Moreover, the sparse grid method was supplemented with adaptive refinement schemes [5,14,22], was used for the construction of anisotropic sparse tensor product spaces [21,20] and was applied in the context of weighted mixed spaces [19,22]. Sparse grid based collocation schemes were for example discussed in [2,26,29,30,33,39].…”

Section: Michael Griebelmentioning

confidence: 99%

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Fast Discrete Fourier Transform on Generalized Sparse Grids

Griebel

Hamaekers

2014

Lecture Notes in Computational Science and Engineering

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In this paper, we present an algorithm for trigonometric interpolation of multivariate functions on generalized sparse grids and study its application for the approximation of functions in periodic Sobolev spaces of dominating mixed smoothness. In particular, we derive estimates for the error and the cost. We construct interpolants with a computational cost complexity which is substantially lower than for the standard full grid case. The associated generalized sparse grid interpolants have the same approximation order as the standard full grid interpolants, provided that certain additional regularity assumptions on the considered functions are fulfilled. Numerical results validate our theoretical findings.

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Section: Michael Griebelmentioning

confidence: 99%

“…H t,r mix (T n ), H t,r mix (T n ) and H Γ w , can be straightforward generalized to the case of many-particle spaces [17,18,19,27]. 6 For r ≥ 0 we could also use the weight…”

Section: Periodic Sobolev Spacesmentioning

confidence: 99%

Fast Discrete Fourier Transform on Generalized Sparse Grids

Griebel

Hamaekers

2014

Lecture Notes in Computational Science and Engineering

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“…Previous wavelet-based discretizations of the higher-dimensional Schrödinger equation have been based e.g. on globally supported Meyer wavelets [24], approximately orthogonal Gaussian frames [25,28] and semiorthogonal piecewise linear prewavelets [51].…”

Section: Suitable Wavelet Basesmentioning

confidence: 99%

“…Then there exist C s , C ψ,j0,s,τ > 0 independent of r, R, ε 0 and Λ such that 25) where η 1 = 1, and η s = 2 for s ∈ ( 3 2 , 2].…”

Section: Approximation Of Two-electron Operatorsmentioning

confidence: 99%

Hyperbolic wavelet discretization of the two-electron Schrödinger equation in an explicitly correlated formulation

Bachmayr

2012

ESAIM: M2AN

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Abstract. In the framework of an explicitly correlated formulation of the electronic Schrödinger equation known as the transcorrelated method, this work addresses some fundamental issues concerning the feasibility of eigenfunction approximation by hyperbolic wavelet bases. Focusing on the two-electron case, the integrability of mixed weak derivatives of eigenfunctions of the modified problem and the improvement compared to the standard formulation are discussed. Elements of a discretization of the eigenvalue problem based on orthogonal wavelets are described, and possible choices of tensor product bases are compared especially from an algorithmic point of view. The use of separable approximations of potential terms for applying operators efficiently is studied in detail, and estimates for the error due to this further approximation are given.Mathematics Subject Classification. 35B65, 35J10, 65T60, 81Q05.

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“…In particular, the wavelet multiresolution schemes [11], as well as the sparse grids approach in [49,17] have been proposed. The entirely wavelet-based method is successful only for small atomic systems with one or two electrons [4].…”

Section: Introductionmentioning

confidence: 99%

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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Tensor Product Multiscale Many-Particle Spaces with Finite-Order Weights for the Electronic Schrödinger Equation

Cited by 21 publications

References 30 publications

Fast Discrete Fourier Transform on Generalized Sparse Grids

Fast Discrete Fourier Transform on Generalized Sparse Grids

Hyperbolic wavelet discretization of the two-electron Schrödinger equation in an explicitly correlated formulation

Black-Box Hartree–Fock Solver by Tensor Numerical Methods

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