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

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)...

Full description

Saved in:
Bibliographic Details
Published inSIAM journal on scientific computing Vol. 34; no. 6; pp. A3016 - A3038
Main Authors Dolgov, S V, Khoromskij, B N, Oseledets, I V
Format Journal Article
LanguageEnglish
Published Philadelphia Society for Industrial and Applied Mathematics 01.01.2012
Subjects
Algorithms
Approximation
Linear systems
Liquids
Mathematical analysis
Mathematical models
Methods
Solvers
Tensors
Time integration
Trains
Online AccessGet full text

Cover

More Information
Summary: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.
Bibliography:ObjectType-Article-2
SourceType-Scholarly Journals-1
ObjectType-Feature-1
content type line 23
ObjectType-Article-1
ObjectType-Feature-2
ISSN:1064-8275
1095-7197
DOI:10.1137/120864210