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Tensor numerical methods for multi-dimensional PDEs: Basic approximation theory and initial applications
first name:
Modern methods of rank-structured tensor decomposition allow an efficient data-sparse sepa-
rable approximation of multivariate functions and operators, providing linear complexity scaling
in the dimension [1, 2]. The successful applications of tensor methods include the challenging
high-dimensional problems arising in material sciences, bio-science, stochastic modeling, signal
processing, machine learning and data mining, quantum computing, many-body dynamics etc.
The recent quantized tensor train (QTT) technique is proved to provide the super-compressed
representation for a class of multidimensional data-arrays of size N d with the log-volume storage
complexity O(d log N ) [3]. The approach is based on data quantization to the 2×2×...×2-format in
the highest possible dimension D = d log N . Combined with the efficient multi-linear QTT-algebra,
this method opens the way to the profound deterministic numerical simulation of high-dimensional
PDEs getting rid of the “curse of dimensionality” and restrictions on the grid size N [4, 5, 6, 7].
However, in some cases, the approach is limited by the “curse of ranks” characterizing the essential
entanglement in the system of interest.
We address the basic theory on low QTT-rank approximation to a class of multivariate functions
and operators (matrices), focusing on the multivariate potentials, d-dimensional elliptic Green’s
function and the parabolic solution operator.
Numerical illustrations for stochastic PDEs, electronic structure calculations (the Hartree-Fock
equation), quantum molecular dynamics (the molecular Schrödinger equation) and stochastic modeling
for dynamics of liquids (the Fokker-Planck equation), are presented.

[1] B.N. Khoromskij. Tensors-structured Numerical Methods in Scientific Computing: Survey on Recent Advances.
Chemometr. Intell. Lab. Syst. (2011), DOI: 10.1016/j.chemolab.2011.09.001; Preprint 21/2010, MPI MiS,
Leipzig, 2010.
[2] B.N. Khoromskij. Introduction to Tensor Numerical Methods in Scientific Computing. 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. O(d log N )-Quantics Approximation of N -d Tensors in High-Dimensional Numerical Model-
ing. J. Constr. Approx. 2011, DOI: 10.1007/s00365-011-9131-1.
[4] B.N. Khoromskij, V. Khoromskaia, and H.-J. Flad. Numerical solution of the Hartree-Fock equation in multilevel
tensor-structured format. SIAM J. Sci. Comp., 33(1), 2011, 45-65.
[5] B.N. Khoromskij, and Ch. Schwab. Tensor-Structured Galerkin Approximation of Parametric and Stochastic
Elliptic PDEs. SIAM J. Sci. Comp., 33(1), 2011, 1-25.
[6] S.V. Dolgov, B.N. Khoromskij, and I. Oseledets. Fast solution of high-dimensional parabolic equations in the
TT/QTT tensor formats with application to the Fokker-Planck equation. MPI MiS, 2011 (in preparation).
[7] B.N. Khoromskij, and I. Oseledets. DMRG+QTT computation of the ground state for the molecular Schrödinger
operator. Preprint 68/2010, MPI MiS, Leipzig 2010 (Numer. Math., submitted).