Numerical solution of the Hartree-Fock equation by multilevel tensor-structured methods

By Venera Khoromskaia. 2011

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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];”
Quantized-TT-Cayley Transform for Computing the Dynamics and the Spectrum of High-Dimensional Hamiltonians (2011, methodology): “In fact, it allows implementing the basic linear and bilinear algebraic operations on tensors such as addition, scalar, Hadamard and convolution products with linear complexity in the univariate tensor size (see [22, 25, 19, 20, 5]).”
Quantized-TT-Cayley transform to compute dynamics and spectrum of high-dimensional Hamiltonians (2011, methodology): “Some tensor-structured numerical methods based on this approach were proposed and analyzed in [8, 21, 20, 26].”
QTT Representation of the Hartree and Exchange Operators in Electronic Structure Calculations (2011, methodology): “Owing to this concept, the grid-based tensor approximation of functions and operators in the Hartree-Fock equation led to efficient tensor numerical algorithms for electronic structure calculations: • low-rank representation of the Galerkin and collocation discretizations of the Newton and Yukawa co”
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]).”
DMRG Approach to Fast Linear Algebra in the TT-Format (2011, background): “Khoromskij [23, 24, 25, 26, 27] for the Tucker format, which also has nice stability properties, but cannot be used in high dimensions due to the exponential dependence on the number of dimensions.”
Møller-Plesset (MP2) energy correction using tensor factorization of the grid-based two-electron integrals (2014, methodology): “In tensor-structured numerical methods, calculation of the 3D convolution integrals is replaced by a sequence of 1D scalar and Hadamard products, and 1D convolution transforms [40,23].”
Approximation of the electron density of Aluminium clusters in tensor-product format (2012, methodology): “The numerics on the tensor-product convolution with the Newton potential in application to the electronic structure calculations is presented in [18, 19, 14].”
Multilevel Toeplitz Matrices Generated by Tensor-Structured Vectors and Convolution with Logarithmic Complexity (2013, methodology): “Alternatively, tensor methods (in the Tucker [44, 45], CP, TT or QTT format) may be applied directly to electronic density functions, potentials and other data and operators involved in computations (see [22, 23, 25, 24] for details and further references).”
Tensors-structured Numerical Methods in Scientific Computing: Survey on Recent Advances (2012, methodology): “The efficient multigrid accelerated canonical-to-Tucker rank reduction scheme, which scales linearly in the input parameters, became a powerful tool in 3D electronic structure calculations [48].”
Superfast Fourier Transform Using QTT Approximation (2012, methodology): “In this way, fast tensor-product convolution with the Newton kernel based on the canonical or/and Tucker tensor decompositions has been developed in [30, 33, 50] and applied to the fully discrete grid-based solution of the Hartree-Fock equation in [26, 35].”