## Abstract

In the present work we propose an efficient black-box solver for one-dimensional multiple scaled diffusion equation. For this problem it has been recently shown [1] that the solution can be represented in a certain low parametric representation, namely the quantized tensor train (QTT) format [2]. The key idea of the QTT format is to make the real space data multidimensional by introducing virtual dimensionalities. The next step is to apply the tensor train (TT) representation [3] to multidimensional data, which leads us to the logarithmic complexity. Hence very fine grids that describe the finest scale can be used. Since the solution of second order multi-scale problems can be represented in the QTT format, simple and efficient solvers can be developed using the existing software for the approximate solution of linear systems in the TT-format. However, if equations are discretized using standard finite element/difference methods, it is not possible to get to very fine meshes, say with 2^{50} grid points due to the condition number. On the other hand, the theory guarantees the existence of a good QTT-FEM approximant of the continuous problem. Thus, another discretization should be used to compute it numerically. Our idea is to rewrite the initial formulation in a certain form without derivatives. After that we get an explicit formula, which consists of the inversion of a diagonal matrix and the multiplication by a dense matrix. The latter can be multiplied with logarithmic complexity in the QTT format due to a special structure. The numerical experiment show that this formula gives accurate results and can be used for 2^{50} grid points with no problems with conditioning, while total computational time is around several seconds.

Original language | English |
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Title of host publication | ECCOMAS Congress 2016 - Proceedings of the 7th European Congress on Computational Methods in Applied Sciences and Engineering |

Editors | M. Papadrakakis, V. Plevris, G. Stefanou, V. Papadopoulos |

Publisher | National Technical University of Athens |

Pages | 7938-7947 |

Number of pages | 10 |

ISBN (Electronic) | 9786188284401 |

DOIs | |

Publication status | Published - 2016 |

Event | 7th European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS Congress 2016 - Crete, Greece Duration: 5 Jun 2016 → 10 Jun 2016 |

### Publication series

Name | ECCOMAS Congress 2016 - Proceedings of the 7th European Congress on Computational Methods in Applied Sciences and Engineering |
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Volume | 4 |

### Conference

Conference | 7th European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS Congress 2016 |
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Country/Territory | Greece |

City | Crete |

Period | 5/06/16 → 10/06/16 |

## Keywords

- Blackbox solver
- Multiscale modelling
- QTT-format
- Stable PDE discretization
- TT-format