Venera Khoromskaia, Dr. rer. nat.

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  Tel.:  ++49 341-99 59 763  
  e-mail: vekh[at]

  Dr. Venera Khoromskaia
  Max-Planck-Institute for Mathematics in the Sciences 
  Inselstr., 22-26, D-04103, 
  Leipzig, Germany 
Main results:

Canonical-to-Tucker (C2T) tensor transform for tensor rank compression (2006-2007).

Calculation of multi-dimensional (convolution) integrals in 1D complexity (2006-2010)
(starting tensor numerical methods in electronic structure calculations).

Tensor-based solver(s) for the Hartree-Fock equation (2009-2013).

Calculation of the 4D electron repulsion integrals tensor in a low-rank factorized form (2012)
(by "1D density fitting").

Superfast tensor method for summation of long-range interaction potentials on finite 3D lattices (2013-2015)
(for lattices with defects and non-rectangular geometries).

Tensor-based Hartree-Fock solver for the finite lattices (at work).

Bethe-Salpeter excitation energies using low-rank tensor factorizations.

I had my Diploma in Physics in 1974, and next 21 years I have been working at the Joint Institute for Nuclear Research (JINR) in Dubna near Moscow, which at that time had close collaboration with CERN in Geneva. I worked on processing of large scale data from experiments on nuclear particles accelerators in high energy physics.

In 2006 I started working at the Max-Planck Institute for Mathematics in the Sciences on Tucker tensor decomposition for function related tensors (together with B. Khoromskij). This application revealed a number of properties of the Tucker format which were not known before. It showed that tensors resulting from the discretization of physically relevant multidimensional functions on the uniform grids can be represented by low-rank approximation, which only logarithmically depends on the size of the grid.
On the basis of this knowledge we developed the canonical-to-Tucker decomposition (C2T, 2007) and the reduced higher order singular value decomposition (RHOSVD), which promoted the tensor formats ``avoiding the curse of dimensionality''.

In 2006-2008 we with Boris Khoromskij have first used rank-structured tensor representation of 3D functions for calculation of the 3D convolution integrals with the Newton kernel. We showed that it is reduced to a sequence of 1D operations (1D Hadamard, 1D scalar products and 1D convolutions). [20-17].

In this way, in 2007 the development of the tensor-structured numerical methods was started by solving some problems of 3D grid based ``ab initio`` electronic structure calculations. One of the advantages of this approach for the Hartree-Fock equation is that there is no obligation to usage of the Gaussian-type basis functions, since the analytical integration is completely avoided.
We have developed two completely 3D grid-based Hartree-Fock solvers by tensor numerical methods (in Matlab) :

1. The multilevel Hartree-Fock solver (No. 1, in 2009) employs the direct 3D grid-based computation of the Hartree and nonlocal exchange operators in 1D complexity, ''on the fly'', on a sequence of dyadically refined 3D Cartesian grids. [16-19]. Its performance in time is slower compared with the standard packages (due to Matlab loops for the exchange operator) but it was a proof of concept for tensor numerical methods.

2. The black-box Hartree-Fock solver (No. 2, in 2013) uses 3D grid-based factorized tensor calculation of the electron repulsion integrals (two-electron integrals) and of the core Hamiltonian [10-13], and includes MP2 energy correction.
Performance of this solver in time and accuracy is close to standard quantum chemical packages based on analytical evaluation of the electron repulsion integrals.
Time of one iteration step for small amino acid molecules is about several seconds (Matlab).

Together with Boris Khoromskij we invented a new method for summation of long-range electrostatic potentials on finite lattices using assembling of canonical/Tucker vectors of the tensor representation of a single potential. [4,5,7,8]. Computational cost of the method is O(L) instead of O(L3 ) by usual Ewald-type summation methods.

For example, a 3D lattice potential sum for a million of atoms on a 3D lattice in a box (128 x 128 x 64 H atoms) is computed in 10 seconds in Matlab (no parallelization, etc).
The method is valid for lattices with multiple defects, as well as for shaped lattices (L-shape, O-shape, hexagonal lattices etc.). These algorithms can be useful in computer modeling of large finite atomic/molecular clusters (nano-structures).

last modified 03.02.2017 by V.Khoromskaia