On tensor approximation of Green iterations for Kohn-Sham equations

By Boris N. Khoromskij. 2008

41 citations recorded by the source corpus.

In the present paper we discuss efficient rank-structured tensor approximation methods for 3D integral transforms representing the Green iterations for the Kohn-Sham equation. We analyse the local convergence of the Newton iteration to solve the Green’s function integral formulation of the Kohn-Sham model in electronic structure calculations. We prove the low-separation rank approximations for the arising discrete convolving kernels given by the Coulomb and Yukawa potentials 1/|x|, and e−λ|x|/|x|, respectively, with $$x \in {\mathbb{R}}^{d} $$. Complexity analysis of the nonlinear iteration with truncation to the fixed Kronecker tensor-product format is presented. Our method has linear scaling in the univariate problem size. Numerical illustrations demostrate uniform exponential convergence of tensor approximations in the orthogonal Tucker and canonical formats.

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): “…[152]; • computational tasks in electronic structure calculations, e.g., based on Hartree-Fock or DFT models [23,24,31,44,45,47,87,141–146,150,156–159]; • the solution of stochastic and parametric PDEs [68,69,76,77,161,165,171,189,202]; • approximation of Green’s functions of…”
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].”
Tensor-Structured Preconditioners and Approximate Inverse of Elliptic Operators in ℝd (2009, methodology): “In the present paper we prove the explicit estimate on the tensor rank of sinc-quadrature based preconditioners that ensures spectral equivalence.”
Tensors-structured Numerical Methods in Scientific Computing: Survey on Recent Advances (2012, methodology): “8, and applying the sinc-quadrature approximation in [49, 53, 4] (see Tables 3.”
An Adaptive Algebraic Multigrid Algorithm for Low-Rank Canonical Tensor Decomposition (2011, unknown): “A second class of problems is related to the decomposition of tensors arising from PDE discretization on high-dimensional regular lattices [27, 28, 17, 18].”
Multilevel Toeplitz Matrices Generated by Tensor-Structured Vectors and Convolution with Logarithmic Complexity (2013, methodology): “This makes the CP-structured convolution more favorable compared to the FFT-based convolution in the full format, for example, in quantum chemistry computations [21, 22, 23, 20, 24, 25] and other applications (see the paper [5] on linear filtering and references therein) and allows (up to efficiency”
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.”
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.”
QTT approximation of elliptic solution operators in higher dimensions (2011, methodology): “[20] and references therein) for solving the spectral problem,”
Computation of the Hartree-Fock Exchange by the Tensor-Structured Methods (2010, methodology): “multivariate functions and operators [10, 14, 13, 11] showed that methods of the multi-way analysis can be applied to the numerical solution of basic equations of mathematical physics having strong requirements on accuracy of the results.”
Numerical Solution of the Hartree - Fock Equation in Multilevel Tensor-Structured Format (2011, methodology): “This concept appears to be particularly attractive for simplified approaches, like the ab initio Hartree-Fock and density functional theory methods, reducing the dimensionality of the problem to d = 3, though the kernels of the nonlocal operators involved are functions of six spatial dimensions [14,”
Low-rank quadrature-based tensor approximation of the Galerkin projected Newton/Yukawa kernels (2012, methodology): “3) Analogous to the case of Newton kernel, we apply the gaussian integral to the Yukawa kernel in the form [8], e−λ‖x‖ ‖x‖ = 1 √ π ∫”