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[colloquium]Tensor numerical methods for multi-dimensionaal PDEs
时间  Datetime
2017-04-13 14:00 — 15:00
地点  Venue
Large Conference Room, Math Building
报告人  Speaker
Boris N. Khoromskij
单位  Affiliation
Max-Planck-Institute for Mathematics in the Sciences
邀请人  Host
张镭
报告摘要  Abstract

The breaking through approach to low-parametric representation of multivariate functions andoperators is based on the principle of separation of variables which can be realized by usingapproximation in rank-structured tensor formats [2]. This allows the linear complexity scalingin dimension, hence breaking the ”curse of dimensionality”. The method of quantized tensortrain (QTT) approximation is proven to provide the logarithmic data-compression on a wideclass of discretized functions and operators [1].

We discuss how the tensor methods based on the canonical, Tucker, TT and QTT approxi-mation apply to calculation of electrostatic potential of many-particle systems, to the (post)Hartree-Fock eigenvalue problem for large 3D lattice-structured molecules (electrostatic po-tential on a lattice in electronic structure calculations) [3,4,5], to elliptic equations with highly-oscillating coefficients (homogenization theory) [6], and to parametric/stochastic elliptic PDEs.The other direction is related to tensor approache for parabolic equations in space-time d + 1formulation.

The efficiency of the tensor approach is illustrated by numerical examples.
[1] B.N. Khoromskij. O(d log N )-Quantics Approximation of N -d Tensors in High-Dimensional Numerical

Modeling. J. Constr. Approx. v. 34(2), 257-289 (2011); (Preprint 55/2009 MPI MiS, Leipzig 2009).
[2] Boris N. Khoromskij.
Tensor Numerical Methods for High-dimensional PDEs: Basic Theory and Initial

Applications. ESAIM: January 2015, Vol. 48, p. 1-28.
[3] V. Khoromskaia and B.N. Khoromskij.
Tensor numerical methods in quantum chemistry: from Hartree-Fock

to excitation energies. Phys. Chem. Chem. Phys., 17:31491 - 31509, 2015.

[4] P. Benner, V. Khoromskaia and B.N. Khoromskij. A reduced basis approach for calculation