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Reduced basis methods for time-dependent problems

Published online by Cambridge University Press:  09 June 2022

Jan S. Hesthaven Open the ORCID record for Jan S. Hesthaven [Opens in a new window] ,
Cecilia Pagliantini  and
Gianluigi Rozza
Jan S. Hesthaven
Affiliation:
Ecole Polytechnique Federale de Lausanne (EPFL), CH-1015Lausanne, Switzerland E-mail: jan.hesthaven@epfl.ch
Cecilia Pagliantini
Affiliation:
Eindhoven University of Technology, 5600MBEindhoven, Netherlands E-mail: c.pagliantini@tue.nl
Gianluigi Rozza
Affiliation:
SISSA – International School for Advanced Studies, 34136Trieste, Italy E-mail: gianluigi.rozza@sissa.it
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Abstract

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Numerical simulation of parametrized differential equations is of crucial importance in the study of real-world phenomena in applied science and engineering. Computational methods for real-time and many-query simulation of such problems often require prohibitively high computational costs to achieve sufficiently accurate numerical solutions. During the last few decades, model order reduction has proved successful in providing low-complexity high-fidelity surrogate models that allow rapid and accurate simulations under parameter variation, thus enabling the numerical simulation of increasingly complex problems. However, many challenges remain to secure the robustness and efficiency needed for the numerical simulation of nonlinear time-dependent problems. The purpose of this article is to survey the state of the art of reduced basis methods for time-dependent problems and draw together recent advances in three main directions. First, we discuss structure-preserving reduced order models designed to retain key physical properties of the continuous problem. Second, we survey localized and adaptive methods based on nonlinear approximations of the solution space. Finally, we consider data-driven techniques based on non-intrusive reduced order models in which an approximation of the map between parameter space and coefficients of the reduced basis is learned. Within each class of methods, we describe different approaches and provide a comparative discussion that lends insights to advantages, disadvantages and potential open questions.

Type
Research Article
Information
Acta Numerica , Volume 31 , May 2022 , pp. 265 - 345
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), 2022. Published by Cambridge University Press

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