Reduced Basis Methods: Success, Limitations and Future Challenges

TL;DR

This paper analyzes the success, limitations, and future challenges of reduced basis methods in parametric model order reduction, focusing on theoretical convergence properties and recent nonlinear approximation approaches.

Contribution

It provides a comprehensive analysis of the theoretical aspects of reduced basis methods, including convergence and failure cases, and discusses recent nonlinear techniques to address limitations.

Findings

Reduced basis methods enable high-fidelity real-time simulations.

Convergence properties are well-understood, but limitations exist.

Recent approaches using nonlinear approximation aim to overcome current challenges.

Abstract

Parametric model order reduction using reduced basis methods can be an effective tool for obtaining quickly solvable reduced order models of parametrized partial differential equation problems. With speedups that can reach several orders of magnitude, reduced basis methods enable high fidelity real-time simulations of complex systems and dramatically reduce the computational costs in many-query applications. In this contribution we analyze the methodology, mainly focussing on the theoretical aspects of the approach. In particular we discuss what is known about the convergence properties of these methods: when they succeed and when they are bound to fail. Moreover, we highlight some recent approaches employing nonlinear approximation techniques which aim to overcome the current limitations of reduced basis methods.