On the Best Rank-1 and Rank-(R1,R2,. . .,RN) Approximation of Higher-Order Tensors
SIAM Journal on Matrix Analysis and Applications
Algorithms for Numerical Analysis in High Dimensions
SIAM Journal on Scientific Computing
Multilevel domain decomposition for electronic structure calculations
Journal of Computational Physics
The hyperbolic cross space approximation of electronic wavefunctions
Numerische Mathematik
Approximate iterations for structured matrices
Numerische Mathematik
On tensor approximation of Green iterations for Kohn-Sham equations
Computing and Visualization in Science
Tucker Dimensionality Reduction of Three-Dimensional Arrays in Linear Time
SIAM Journal on Matrix Analysis and Applications
Tensor decomposition in electronic structure calculations on 3D Cartesian grids
Journal of Computational Physics
Multigrid Accelerated Tensor Approximation of Function Related Multidimensional Arrays
SIAM Journal on Scientific Computing
Tensor Decompositions and Applications
SIAM Review
Journal of Computational and Applied Mathematics
Tensor-Structured Galerkin Approximation of Parametric and Stochastic Elliptic PDEs
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Approximation of the electron density of Aluminium clusters in tensor-product format
Journal of Computational Physics
Wedderburn Rank Reduction and Krylov Subspace Method for Tensor Approximation. Part 1: Tucker Case
SIAM Journal on Scientific Computing
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In this paper, we describe a novel method for a robust and accurate iterative solution of the self-consistent Hartree-Fock equation in $\mathbb{R}^3$ based on the idea of tensor-structured computation of the electron density and the nonlinear Hartree and (nonlocal) exchange operators at all steps of the iterative process. We apply the self-consistent field (SCF) iteration to the Galerkin discretization in a set of low separation rank basis functions that are solely specified by the respective values on a three-dimensional Cartesian grid. The approximation error is estimated by $O(h^3)$, where $h=O(n^{-1})$ is the mesh size of an $n\times n\times n$ tensor grid, while the numerical complexity to compute the Galerkin matrices scales linearly in $n\log n$. We propose the tensor-truncated version of the SCF iteration using the traditional direct inversion in the iterative subspace scheme enhanced by the multilevel acceleration with the grid-dependent termination criteria at each discretization level. This implies that the overall computational cost scales almost linearly in the univariate problem size $n$. Numerical illustrations are presented for the all electron case of H$_2$O and the pseudopotential case of CH$_4$ and CH$_3$OH molecules. The proposed scheme is not restricted to a priori given rank-1 basis sets, allowing analytically integrable convolution transform with the Newton kernel that opens further perspectives for promotion of the tensor-structured methods in computational quantum chemistry.