Preliminary list of abstracts

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Grid-based solution of the 3D Hartree-Fock equation with 1D complexity
Keywords: tensor-structured numerical methods; Hartree-Fock equation; 3D tensor rank reduction
Wed, 16:35--17:00
  • Khoromskaia, Venera (Scientific Computing Group, Max-Planck-Institute for Mathematics in the Sciences, Germany)

Numerical solution of the Hartree-Fock equation using grid-based multilevel tensor-structured methods is discussed. The accurate tensor-structured computation of the nonlinear Hartree and the (nonlocal) exchange operators in $\mathbb{R}^3$, discretized on a sequence of $n\times n\times n$ Cartesian grids is considered. The arising three- and six-dimensional integrations are replaced by the rank-structured Hadamard and scalar products and the corresponding 3D convolution implemented with $O(n\log n)$-complexity. The robust multigrid canonical-to-Tucker rank reduction algorithm with the controllable accuracy enables usage of fine Cartesian grids up to $n^3 \approx 10^{12}$. That yields high resolution of the involved computational entities and allows arbitrary location of atoms in a molecule as in the conventional mesh-free analytical-based solution of the Hartree-Fock equation. The new QTT format allows further to reduce the complexity of the algorithms at some steps of the "black box" solver to $O(\log n)$. Numerical results for all electron case of H$_2$O, C$_2$H$_6$, CH$_4$, and the pseudopotential case of CH$_3$OH, C$_2$H$_5$OH demonstrate efficiency of the tensor-structured methods using traditional and recent tensor formats in electronic structure calculations.