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Low-rank tensor methods for partial differential equations

Part of: Partial differential equations, boundary value problems Numerical analysis in abstract spaces Approximations and expansions Computer aspects of numerical algorithms Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  11 May 2023

Markus Bachmayr Open the ORCID record for Markus Bachmayr [Opens in a new window]
Markus Bachmayr*
Affiliation:
Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany E-mail: bachmayr@igpm.rwth-aachen.de
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Abstract

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Low-rank tensor representations can provide highly compressed approximations of functions. These concepts, which essentially amount to generalizations of classical techniques of separation of variables, have proved to be particularly fruitful for functions of many variables. We focus here on problems where the target function is given only implicitly as the solution of a partial differential equation. A first natural question is under which conditions we should expect such solutions to be efficiently approximated in low-rank form. Due to the highly nonlinear nature of the resulting low-rank approximations, a crucial second question is at what expense such approximations can be computed in practice. This article surveys basic construction principles of numerical methods based on low-rank representations as well as the analysis of their convergence and computational complexity.

MSC classification

Primary: 41A46: Approximation by arbitrary nonlinear expressions; widths and entropy 41A63: Multidimensional problems (should also be assigned at least one other classification number in this section)
Secondary: 65J10: Equations with linear operators (do not use ) 65M12: Stability and convergence of numerical methods 65N12: Stability and convergence of numerical methods 65N25: Eigenvalue problems 65Y20: Complexity and performance of numerical algorithms

Information

Type
Research Article
Information
Acta Numerica , Volume 32 , May 2023 , pp. 1 - 121
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2023. Published by Cambridge University Press

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