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)...
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Published in | SIAM journal on scientific computing Vol. 34; no. 6; pp. A3016 - A3038 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
Philadelphia
Society for Industrial and Applied Mathematics
01.01.2012
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Subjects | |
Online Access | Get full text |
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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. |
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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 |