Tensor decomposition in electronic structure calculations on 3D Cartesian grids

By Boris N. Khoromskij, Venera Khoromskaia, Sambasiva Rao Chinnamsetty, Heinz-Jürgen Flad. 2009

52 citations recorded by the source corpus.

Subfields: Atomic and Molecular Physics, and Optics, Computational Mathematics, Computational Theory and Mathematics

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “, based on Hartree-Fock or DFT models [23, 25, 32, 45, 46, 48, 88, 142–147, 151, 157–160];”
Tensor-Structured Preconditioners and Approximate Inverse of Elliptic Operators in ℝd (2009, methodology): “Low rank representation of Gz(x) can be utilised in fast tensor convolution transform [21, 22 , 24].”
Concepts of Data-Sparse Tensor-Product Approximation in Many-Particle Modelling (2008, methodology): “We want to discuss two alternative approaches based on canonical (4.1) and Tucker (4.2) tensor-product decompositions which have been presented in [14] and [ 15 ], respectively.”
Numerical Solution of the Hartree - Fock Equation in Multilevel Tensor-Structured Format (2011, methodology): “The core of our method is the tensor-structured computation of the electron density and the Galerkin matrices of the nonlinear Hartree and (nonlocal) Hartree-Fock exchange operators at all steps of iterations on nonlinearity, based on the systematic use of the ranktruncated linear tensor operations ”
Tensors-structured Numerical Methods in Scientific Computing: Survey on Recent Advances (2012, methodology): “5 On tensor-structured solution of multidimensional equations 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].”
Adaptive low rank wavelet methods and applications to two-electron Schrödinger equations (2012, unknown): “Separable expansions for potential terms generated by exponential sum approximations have been used for a variety of related purposes, for instance, in [109, 25, 48] as part of an integration scheme for higher-dimensional integrals of potentials with wavelets, in [12, 79, 60] for the approximation o”
Computation of the Hartree-Fock Exchange by the Tensor-Structured Methods (2010, methodology): “The tensor-structured methods developed in [14, 10, 13] have been successfully applyed for the highly accurate grid-based numerical computation of the Hartree potential and the Coulomb matrix in the Hartree-Fock equation [15, 17].”
Wedderburn Rank Reduction and Krylov Subspace Method for Tensor Approximation. Part 1: Tucker Case (2010, methodology): “This was done in [4] using Tucker-ALS algorithm [21, 7], in [19] by Tucker-ALS with initial guess obtained from the coarser grids, in [10] by Cross3D algorithm [24], in [25] by individual cross approximation of canonical factors and in [31] by cross approximation of Gram matrices of unfoldings.”
QTT Representation of the Hartree and Exchange Operators in Electronic Structure Calculations (2011, background): “Moreover, there are interesting results on the tensor-product approximation of the 3D Newton and Yukawa potentials in the framework of the waveletand polynomial-based multiresolution schemes [16, 4], as well as using the grid-based approximations [13, 8, 25, 29].”
The tensor product approximation to single-electron systems (2009, methodology): “Recently, tensor product approximation [5, 6, 14, 11] has been applied in electronic structure calculations for efficient computation of Coulomb integrals.”
Constructive Representation of Functions in Low-Rank Tensor Formats (2012, background): “Low-parametric approximation of multivariate functions is very important in high-dimensional applications (ﬁnancial mathematics [33, 36], quantum chemistry [18, 19, 22], multiparametric and stochastic problems [3], and numerical analysis and sci-entiﬁc computing [10, 20, 23]).”
F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig Multigrid Accelerated Tensor Approximation of Function Related Multi-dimensional Arrays Multigrid Accelerated Tensor Approximation of Function Related Multidimensional Arrays * (unknown): “…approximation methods have opened challenging perspectives for a feasible numerical solution of multidimensional problems arising in large scale electronic and molecular structure calculations [1] [2] [7] [8] [10] [12] [15] [4] [18] [28], in stochastic PDEs [24], and in financial mathematics.”
Superfast Fourier Transform Using QTT Approximation (2012, background): “We refer also to [8, 12, 27, 29, 34, 43, 47, 51] for the detailed discussions on recent progress in this ﬁeld.”
Mathematik in den Naturwissenschaften Leipzig Tensor-Structured Galerkin Approximation of parametric and stochastic Elliptic PDEs (2010, background): “Tensor-structured techniques in higher dimensions [7, 13, 15, 23, 26] have already demonstrated their solid potential in electronic structure calculations [3, 12, 23, 24, 25], in PDEs with random input [34, 37, 35, 4], and in machine learning [1].”
Canonical Tensor Products as a Generalization of Gaussian-type Orbitals (2010, methodology): “For the approximation of orbitals, orbital products and electron densities, the canonical and Tucker formats seem to be most appropriate [36,42,43].”
O(dlog N)-Quantics Approximation of N-d Tensors in High-Dimensional Numerical Modeling (2011, methodology): “3 Toward asymptotically optimal numerical schemes Tensor-structured numerical methods of the linear scaling O(dN), are proved to be efficient for representation of functions and operators in the Hartree-Fock and Kohn-Sham models in electronic structure calculations [3, 18, 29, 27, 12, 38, 22].”