NDLI: Fast Solution of Parabolic Problems in the Tensor Train/Quantized Tensor Train Format with Initial Application to the Fokker--Planck Equation

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Oseledets, I. V. Khoromskij, B. N. Dolgov, S. V.
Copyright Year	2012
Abstract	In this paper we propose two schemes of using the so-called quantized tensor train (QTT)-approximation for the solution of multidimensional parabolic problems. First, we present a simple one-step implicit time integration scheme using a solver in the QTT-format of the alternating linear scheme (ALS) type. As the second approach, we use the global space-time formulation, resulting in a large block linear system, encapsulating all time steps, and solve it at once in the QTT-format. We prove the QTT-rank estimate for certain classes of multivariate potentials and respective solutions in $(x,t)$ variables. The log-linear complexity of storage and the solution time is observed in both spatial and time grid sizes. The method is applied to the Fokker--Planck equation arising from the beads-springs models of polymeric liquids.
Starting Page	A3016
Ending Page	A3038
Page Count	23
File Format	PDF
ISSN	10648275
DOI	10.1137/120864210
e-ISSN	10957197
Issue Number	6
Volume Number	34
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2012-12-12
Access Restriction	Subscribed
Subject Keyword	Multilinear algebra, tensor products Fokker--Planck equation Initial-boundary value problems for second-order parabolic equations parabolic problems density matrix renormalization group tensor methods Sparse matrices Algorithms for functional approximation dumbbell model Numerical approximation and evaluation higher dimensions Iterative methods for linear systems Fokker-Planck equations Polymers QTT-format
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

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