### 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.

Original language | English |
---|---|

Pages (from-to) | 23-38 |

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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Numerical Analysis
- Computational Mathematics
- Applied Mathematics

### Cite this

**A Tensor Decomposition Algorithm for Large ODEs with Conservation Laws.** / Dolgov, Sergey V.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - A Tensor Decomposition Algorithm for Large ODEs with Conservation Laws

AU - Dolgov, Sergey V.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - Alternating Iteration

KW - Conservation Laws

KW - Differential Equations

KW - DMRG

KW - High-Dimensional Problems

KW - Tensor Train Format

UR - http://www.scopus.com/inward/record.url?scp=85053496467&partnerID=8YFLogxK

U2 - 10.1515/cmam-2018-0023

DO - 10.1515/cmam-2018-0023

M3 - Article

VL - 19

SP - 23

EP - 38

JO - Computational Methods in Applied Mathematics

JF - Computational Methods in Applied Mathematics

SN - 1609-4840

IS - 1

ER -