Black-box solver for one dimensional multiscale modelling using the QTT format

Ivan V. Oseledets, Maxim V. Rakhuba, Andrei V. Chertkov

    Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

    1 Citation (Scopus)

    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 250 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 250 grid points with no problems with conditioning, while total computational time is around several seconds.

    Original languageEnglish
    Title of host publicationECCOMAS Congress 2016 - Proceedings of the 7th European Congress on Computational Methods in Applied Sciences and Engineering
    EditorsM. Papadrakakis, V. Plevris, G. Stefanou, V. Papadopoulos
    PublisherNational Technical University of Athens
    Pages7938-7947
    Number of pages10
    ISBN (Electronic)9786188284401
    DOIs
    Publication statusPublished - 2016
    Event7th European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS Congress 2016 - Crete, Greece
    Duration: 5 Jun 201610 Jun 2016

    Publication series

    NameECCOMAS Congress 2016 - Proceedings of the 7th European Congress on Computational Methods in Applied Sciences and Engineering
    Volume4

    Conference

    Conference7th European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS Congress 2016
    Country/TerritoryGreece
    CityCrete
    Period5/06/1610/06/16

    Keywords

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

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