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

0541908 - MÚ 2022 RIV GB eng J - Journal Article
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
R&D Projects: GA ČR(CZ) GA20-01074S
Institutional support: RVO:67985840
Keywords : a posteriori error bounds * high-dimensional reaction–diffusion problems
OECD category: Pure mathematics
Impact factor: 2.713, year: 2021
Method of publishing: Limited access
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.
Permanent Link: http://hdl.handle.net/11104/0319401