## Abstract

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.

Original language | English |
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Pages (from-to) | A3016-A3038 |

Number of pages | 23 |

Journal | SIAM Journal on Scientific Computing |

Volume | 34 |

Issue number | 6 |

DOIs | |

Publication status | Published - 31 Dec 2012 |

## Keywords

- Density matrix renormalization group
- Dumbbell model
- Fokker-Planck equation
- Higher dimensions
- Parabolic problems
- QTT-format
- Tensor methods

## ASJC Scopus subject areas

- Computational Mathematics
- Applied Mathematics