Numerical Solution of the Hartree - Fock Equation in Multilevel Tensor-Structured Format

By Boris N. Khoromskij, Venera Khoromskaia, Heinz-Jürgen Flad. 2011

91 citations recorded by the source corpus.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Numerical Analysis

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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];”
Low-Rank Explicit QTT Representation of the Laplace Operator and Its Inverse (2012, methodology): “[25] B. N. Khoromskij and C. Schwab, Tensor-structured Galerkin approximation of parametric and stochastic elliptic PDEs, SIAM J. Sci. Comput., 33 (2011), pp. 364–385.”
Tensor-Train Decomposition (2011, background): “It is suitable for “small” dimensions, especially for the three-dimensional case [22, 30, 23].”
Fast and accurate 3D tensor calculation of the Fock operator in a general basis (2012, methodology): “It completes the concept of the numerical solution to the Hartree-Fock equation by the tensor-structured methods introduced in [1, 2, 3].”
Møller-Plesset (MP2) energy correction using tensor factorization of the grid-based two-electron integrals (2014, methodology): “A tensor-structured paradigm leads to ‘‘black-box’’ numerical treatment of the Hartree–Fock problem based on representation of the basis functions in a volume box, using an n×n×n 3D Cartesian grid positioned arbitrarily with respect to the atomic centers [22,23].”
Computation of the Hartree-Fock Exchange by the Tensor-Structured Methods (2010, methodology): “Tensor-structured computation of the Hartree-Fock exchange, along with the tensorbased algorithms for calculating the Coulomb matrix considered in [15] are the main building blocks in the recent grid-based numerical solution of the Hartree-Fock equation by the tensorstructured methods (see a 3D nonl”
Low-rank quadrature-based tensor approximation of the Galerkin projected Newton/Yukawa kernels (2012, methodology): “Application of the finite element and wavelet methods [13, 6] might be of the particular interest if the volume integral transforms can be evaluated efficiently [13, 6, 12].”
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].”
Quantized-TT-Cayley transform to compute dynamics and spectrum of high-dimensional Hamiltonians (2011, methodology): “Max-Planck-Institut für Mathematik in den Naturwissenschaften Leipzig Quantized-TT-Cayley transform to compute dynamics and spectrum of high-dimensional Hamiltonians (revised version: September 2011) by Ivan P. Gavrilyuk, and Boris N. Khoromskij Preprint no.”
Adaptive low rank wavelet methods and applications to two-electron Schrödinger equations (2012, background): “, in [101, 5, 96, 140], and for eigenvalue problems in [92, 6, 74, 102, 95, 140].”
Fast Solution of Parabolic Problems in the Tensor Train/Quantized Tensor Train Format with Initial Application to the Fokker-Planck Equation (2012, methodology): “, the multilevel solver for the Hartree–Fock equation in the Tucker format with the canonical representation of the core [27, 26].”
Quantics-TT Collocation Approximation of Parameter-Dependent and Stochastic Elliptic PDEs (2010, unknown): “AMS Subject Classification: 65F30, 65F50, 65N35, 65F10 Key words: elliptic equations, stochastic PDEs, the Karhunen-Loève expansion, separable approximation, quantics-TT tensors, preconditioners, tensor-truncated iteration.”
O(dlog N)-Quantics Approximation of N-d Tensors in High-Dimensional Numerical Modeling (2011, background): “AMS Subject Classification: 65F30, 65F50, 65N35, 65F10 Key words: High-dimensional problems, quantics folding of vectors, rank structured tensor approximation, matrix valued functions, FEM, material sciences, stochastic modeling.”
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”
Use of tensor formats in elliptic eigenvalue problems (2012, unknown): “The recent results on e! cient methods of separable tensor approximation of multivariate functions and their application in electronic structure calculations can be found in [12, 4, 6, 9, 16, 18, 19 ] and others.”