## Tensor numerical methods of logarithmic complexity for multi-dimensional applications

Boris N. Khoromskij

Max-Planck-Institute for Mathematics in the Sciences, Leipzig, Allemagne

*bokh@mis.mpg.de*

Lundi 26 Novembre 2012, 11^{h}00

bibliothèque LCT, tour 12 - 13, 4e étage

Tensor numerical methods provide the efficient representation of multivariate functions and operators
on large *n*^{×d}
-grids, that allows the solution of *d*-dimensional PDEs with linear complexity scaling in
the dimension, *O*(*dn*). Modern methods of separable approximation combine the canonical, Tucker, as
well as the general matrix product state (MPS) formats in the framework of DMRG-MPS optimization
methods in FCI calculations in quantum chemistry.

The recent quantized tensor train (QTT) approximation [1] is proven to provide the logarithmic datacompression
for a wide class of functions and operators [1, 2]. It makes possible to solve highdimensional
steady-state and dynamical problems in quantized tensor spaces with the log-volume complexity
scaling in the full-grid size, *O*(*d* log *n*), instead of *O*(*n*^{d}).

We show how the approximation in quantized tensor spaces applies to hard problems arising in electronic
structure calculations, such as multi-dimensional convolution [3], many-electron integrals [4]
and FFT(d) [5]. The QTT method also applies to high-dimensional time-dependent models [6, 7], for
example, to molecular Schrödinger, Fokker-Planck and master equations. Numerical tests indicate the
logarithmic complexity of the QTT tensor methods with respect to both spacial and temporal grid-sizes.

*http://personal-homepages.mis.mpg.de/bokh*

[1] B. N. Khoromskij, Constr. Approx. 2011, 34, 257.

[2] B. N. Khoromskij, Lecture
Notes, Preprint 06-2011, University of Zuerich, Institute of Mathematics, 2011, pp. 1-238.

http://www.math.uzh.ch/fileadmin/math/preprints/06-11.pdf.

[3] B. N. Khoromskij
, J. Comp. Appl. Math. 2010, 234, 3122.

[4] V. Khoromskaia, B. N. Khoromskij, and R. Schneider,
Preprint 29/2012, MPI MIS, Leipzig, 2012 (SIAM J. Sci. Comput., submitted).

[5] S. V. Dolgov, B. N. Khoromskij, and D. Savostianov,
J. Fourier Anal. Appl., 2012, 18, 915.

[6] B. N. Khoromskij and I. Oseledets
, Preprint 68/2010, MPI MIS, Leipzig, 2010 (Math. Comp., submitted).

[7] S. V. Dolgov and B. N. Khoromskij
, Preprint 68/2012, MPI MIS,
Leipzig, 2012.