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Guaranteed a posteriori error bounds for low-rank tensor approximate solutions

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    0541908 - MÚ 2022 RIV GB eng J - Článek v odborném periodiku
    Dolgov, S. - Vejchodský, Tomáš
    Guaranteed a posteriori error bounds for low-rank tensor approximate solutions.
    IMA Journal of Numerical Analysis. Roč. 41, č. 2 (2021), s. 1240-1266. ISSN 0272-4979. E-ISSN 1464-3642
    Grant CEP: GA ČR(CZ) GA20-01074S
    Institucionální podpora: RVO:67985840
    Klíčová slova: a posteriori error bounds * high-dimensional reaction–diffusion problems
    Kód oboru RIV: BA - Obecná matematika
    Obor OECD: Pure mathematics
    Impakt faktor: 2.601, rok: 2020
    https://doi.org/10.1093/imanum/draa010

    We propose a guaranteed and fully computable upper bound on the energy norm of the error in low-rank tensor train (TT) approximate solutions of (possibly) high-dimensional reaction–diffusion problems. The error bound is obtained from Euler–Lagrange equations for a complementary flux reconstruction problem, which are solved in the low-rank TT representation using the block alternating linear scheme. This bound is guaranteed to be above the energy norm of the total error, including the discretization error, the tensor approximation error and the error in the solver of linear algebraic equations, although quadrature errors, in general, can pollute its evaluation. Numerical examples with the Poisson equation and the Schrödinger equation with the Henon–Heiles potential in up to 40 dimensions are presented to illustrate the efficiency of this approach.
    Trvalý link: http://hdl.handle.net/11104/0319401
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