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Abstract
We propose an algorithm for solution of highdimensional evolutionary equations (ODEs and discretized timedependent PDEs) in the Tensor Train (TT) decomposition, assuming that the solution and the righthand side of the ODE admit such a decomposition with a low storage. A linear ODE, discretized via onestep 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 offlineonline 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 righthand 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.
Original language  English 

Pages (fromto)  2338 
Number of pages  16 
Journal  Computational Methods in Applied Mathematics 
Volume  19 
Issue number  1 
Early online date  11 Sep 2018 
DOIs  
Publication status  Published  1 Jan 2019 
Keywords
 Alternating Iteration
 Conservation Laws
 Differential Equations
 DMRG
 HighDimensional Problems
 Tensor Train Format
ASJC Scopus subject areas
 Numerical Analysis
 Computational Mathematics
 Applied Mathematics
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Projects
 1 Finished

Tensor product numerical methods for highdimensional problems in probability and quantum calculations
1/01/16 → 31/12/18
Project: Research council
Profiles

Sergey Dolgov
Person: Research & Teaching