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Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

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A Tensor Decomposition Algorithm for Large ODEs with Conservation Laws

Sergey V. Dolgov
Published Online: 2018-09-11 | DOI: https://doi.org/10.1515/cmam-2018-0023

Abstract

We propose an algorithm for solution of high-dimensional evolutionary equations (ODEs and discretized time-dependent PDEs) in the Tensor Train (TT) decomposition, assuming that the solution and the right-hand side of the ODE admit such a decomposition with a low storage. A linear ODE, discretized via one-step or Chebyshev differentiation schemes, turns into a large linear system. The tensor decomposition allows to solve this system for several time points simultaneously using an extension of the Alternating Least Squares algorithm. This method computes a reduced TT model of the solution, but in contrast to traditional offline-online reduction schemes, solving the original large problem is never required. Instead, the method solves a sequence of reduced Galerkin problems, which can be set up efficiently due to the TT decomposition of the right-hand side. The reduced system allows a fast estimation of the time discretization error, and hence adaptation of the time steps. Besides, conservation laws can be preserved exactly in the reduced model by expanding the approximation subspace with the generating vectors of the linear invariants and correction of the Euclidean norm. In numerical experiments with the transport and the chemical master equations, we demonstrate that the new method is faster than traditional time stepping and stochastic simulation algorithms, whereas the invariants are preserved up to the machine precision irrespectively of the TT approximation accuracy.

Keywords: High-Dimensional Problems; Tensor Train Format; DMRG; Alternating Iteration; Differential Equations; Conservation Laws

MSC 2010: 65F10; 65L05; 65L04; 65M22; 65M70; 65D15; 15A23; 15A69

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About the article

Received: 2017-10-27

Revised: 2018-01-10

Accepted: 2018-05-02

Published Online: 2018-09-11


Funding Source: Engineering and Physical Sciences Research Council

Award identifier / Grant number: EP/M019004/1

The author acknowledges funding from the EPSRC fellowship EP/M019004/1.


Citation Information: Computational Methods in Applied Mathematics, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2018-0023.

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