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Guaranteed a posteriori error bounds for low-rank tensor approximate solutions
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SYSNO ASEP 0541908 Document Type J - Journal Article R&D Document Type Journal Article Subsidiary J Článek ve WOS Title Guaranteed a posteriori error bounds for low-rank tensor approximate solutions Author(s) Dolgov, S. (GB)
Vejchodský, Tomáš (MU-W) RID, SAI, ORCIDSource Title IMA Journal of Numerical Analysis. - : Oxford University Press - ISSN 0272-4979
Roč. 41, č. 2 (2021), s. 1240-1266Number of pages 27 s. Language eng - English Country GB - United Kingdom Keywords a posteriori error bounds ; high-dimensional reaction–diffusion problems Subject RIV BA - General Mathematics OECD category Pure mathematics R&D Projects GA20-01074S GA ČR - Czech Science Foundation (CSF) Method of publishing Limited access Institutional support MU-W - RVO:67985840 UT WOS 000651815700014 EID SCOPUS 85116905135 DOI 10.1093/imanum/draa010 Annotation 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. Workplace Mathematical Institute Contact Jarmila Štruncová, struncova@math.cas.cz, library@math.cas.cz, Tel.: 222 090 757 Year of Publishing 2022 Electronic address https://doi.org/10.1093/imanum/draa010
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