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Address: Dr. Venera Khoromskaia Max-Planck-Institute for Mathematics in the Sciences Inselstr., 22-26, D-04103, Leipzig, Germany Tel.: ++49 341-99 59 725 e-mail: vekh [at] mis.mpg.de

Reduced higher order singular value decomposition (RHOSVD) as part of the

canonical-to-Tucker (C2T) tensor transform (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 form of low-rank Cholesky factorization (2012)

(by "1D density fitting").

Superfast tensor method for summation of long-range electrostatic 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.

Range-separated tensor format for modeling the collective long-range electrostatic potentials of

multiparticle systems of general type. (2016-2017)

I had my University 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 tensor format which were not known before. It showed that for tensors resulting from the discretization of physically relevant multidimensional functions on the uniform grids, the error of the Tucker tensor approximation decays exponentially in the Tucker rank.

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'' (since RHOSVD does not need the construction of a full size tensor).

In 2006-2008 we with Boris Khoromskij have shown that when using the rank-structured tensor representation for functions and operators, calculation of the 3D convolution integrals with the Newton kernel is reduced to a sequence of 1D operations (1D Hadamard, 1D scalar products and 1D convolutions). [22-27].

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.

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. [22-26]. 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, TEI) and of the core Hamiltonian [16-19], and includes MP2 energy correction. We never compute the whole TEI but only a number of vectors for its Cholesky factorization, by using the grid-based "1D density fitting".

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. [9,13,15]. Computational cost is O(L) instead of O(L

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 03.03.2018 by V.Khoromskaia