Localized model reduction for parameterized problems

TL;DR

This paper surveys localized model order reduction techniques for parameterized PDEs, focusing on constructing local reduced spaces, coupling methods, error estimation, adaptivity, and practical implementation, with applications to multiscale problems.

Contribution

It provides a comprehensive overview of localized model reduction methods, including construction, coupling, error analysis, adaptivity, and implementation, with numerical examples.

Findings

Optimal local approximation spaces can be constructed and approximated by random sampling.

Various coupling strategies for local spaces are analyzed and compared.

Localized error estimates guide adaptive basis enrichment.

Abstract

In this contribution we present a survey of concepts in localized model order reduction methods for parameterized partial differential equations. The key concept of localized model order reduction is to construct local reduced spaces that have only support on part of the domain and compute a global approximation by a suitable coupling of the local spaces. In detail, we show how optimal local approximation spaces can be constructed and approximated by random sampling. An overview of possible conforming and non-conforming couplings of the local spaces is provided and corresponding localized a posteriori error estimates are derived. We introduce concepts of local basis enrichment, which includes a discussion of adaptivity. Implementational aspects of localized model reduction methods are addressed. Finally, we illustrate the presented concepts for multiscale, linear elasticity and fluid-flow problems, providing several numerical experiments. This work has been accepted as a chapter in P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W.H.A. Schilders, L.M. Sileira. Handbook on Model Order Reduction. Walter De Gruyter GmbH, Berlin, 2019+.