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abstract:
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Recent results on QTT-based approximations in electronic structure and dynamical calculations}
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\textbf{{\large Boris N. Khoromskij} } \\
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Max-Planck-Institute for Mathematics in the Sciences,\\
Inselstr.~22-26, D-04103 Leipzig, Germany;
{\tt $bokh$\char'100mis.mpg.de} \\
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Most powerful separable representations of high-dimensional tensors combine the canonical, Tucker, tensor train (TT), and the
quantized-TT (QTT) formats in the form QTT-canonical, and QTT-Tucker decompositions.
The QTT tensor decomposition \cite{KhQuant:09} was proven to provide the remarkable approximation properties
on the wide class of functions and operators,
providing a simple and efficient tool to solve steady-state and dynamical problems in quantized tensor spaces
with the log-volume complexity scaling in the size of full-grid discretizations.
We discuss the recent tensor numerical methods in the new QTT-Tucker and canonical-QTT formats
applied in electronic structure calculations (on the example of two-electron integrals)
and to the time-dependent problems (on the example of $d$-dimensional Fokker-Planck and master equations).
Our numerical tests indicate the logarithmic computational complexity
of the QTT-based tensor-structured methods.
This talk is based on the results of recent papers \cite{DoKhOsel:11} - \cite{DoKh_QTTT:12}.
\vspace{0.1cm} {\it http://personal-homepages.mis.mpg.de/bokh}
{\small
\begin{thebibliography}{99}
\bibitem{KhQuant:09} B.N. Khoromskij. \emph{ $O(d\log N)$-Quantics Approximation
of $N$-$d$ Tensors in High-Dimensional Numerical Modeling.}
J. Constr. Approx. v. 34(2), 257-289 (2011).
\bibitem {DoKhOsel:11} S.V. Dolgov, B.N. Khoromskij, and I. Oseledets.
\emph{Fast solution of multi-dimensional parabolic problems in the TT/QTT
formats with initial application to the Fokker-Planck equation.}
Preprint 80/2011, MPI MiS, Leipzig 2011 (SISC 2012, to appear).
\bibitem {KhorVBAndrae:12} V. Khoromskaia, D. Andrae, and B.N. Khoromskij.
\emph{Fast and Accurate 3D Tensor Calculation of the Fock Operator in a General Basis}.
Comp. Phys. Communications, 183 (2012) 2392-2404. DOI:10.1016/j.cpc.2012.06.007.
\bibitem {KhorVe4:11} V. Khoromskaia, B.N. Khoromskij, and R. Schneider.
\emph{Tensor-structured calculation of two-electron integrals in a general basis}.
Preprint MPI MiS, Leipzig 2012 (SISC, submitted).
\bibitem {OselKhSchn:12} I. Oseledets, B.N. Khoromskij, and R. Schneider.
\emph{Efficient time-stepping scheme for dynamics on TT-manifolds}.
Preprint 24/2012, MPI MIS, Leipzig 2012 (submitted).
\bibitem {DoKh_QTTT:12} S.V. Dolgov, and B.N. Khoromskij.
\emph{Two-level Tucker-TT-QTT format for optimized tensor calculus.}
Preprint 19/2012, MPI MiS, Leipzig 2012 (SIMAX, submitted).
\end{thebibliography}
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