Tensor Numerical Methods for Multidimensional PDEs: Theoretical Analysis and Initial Applications
Boris N. Khoromskij
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Submission date: 07. Nov. 2014
published in: ESAIM / Proceedings, 48 (2015), p. 1-28
DOI number (of the published article): 10.1051/proc/201448001
MSC-Numbers: 35J15, 65F30, 65F50, 65N35, 65D30
Keywords and phrases: tensor numerical methods, low-rank approximation, high-dimensional PDEs, Electronic structure calculations, dynamical problems, stochastic PDEs, QTT tensor approximation, multilinear algebra
Link to arXiv:See the arXiv entry of this preprint.
We present a brief survey on the modern tensor numerical methods for multidimensional stationary and time-dependent partial diﬀerential equations (PDEs). The guiding principle of the tensor approach is the rank-structured separable approximation of multivariate functions and operators represented on a grid. Recently, the traditional Tucker, canonical, and matrix product states (tensor train) tensor models have been applied to the grid-based electronic structure calculations, to parametric PDEs, and to dynamical equations arising in scientiﬁc computing. The essential progress is based on the quantics tensor approximation method proved to be capable to represent (approximate) function related d-dimensional data arrays of size Nd with log-volume complexity, O(dlog N). Combined with the traditional numerical schemes, these novel tools establish a new promising approach for solving multidimensional integral and diﬀerential equations using low-parametric rank-structured tensor formats. As the main example, we describe the grid-based tensor numerical approach for solving the 3D nonlinear Hartree-Fock eigenvalue problem, that was the starting point for the developments of tensor-structured numerical methods for large-scale computations in solving real-life multidimensional problems. We also discuss a new method for the fast 3D lattice summation of electrostatic potentials by assembled low-rank tensor approximation capable to treat the potential sum over millions of atoms in few seconds. We address new results on tensor approximation of the dynamical Fokker-Planck and master equations in many dimensions up to d = 20. Numerical tests demonstrate the beneﬁts of the rank-structured tensor approximation on the aforementioned examples of multidimensional PDEs. In particular, the use of grid-based tensor representations in the reduced basis of atomics orbitals yields an accurate solution of the Hartree-Fock equation on large N × N × N grids with a grid size of up to N = 105.