Reduced Basis Methods: Success, Limitations and Future Challenges
For researchers in computational science and engineering, this paper provides a theoretical analysis of reduced basis methods, identifying their limitations and potential improvements.
This paper analyzes the theoretical convergence properties of reduced basis methods for parametric model order reduction, discussing when they succeed and fail, and highlights nonlinear approximation techniques to overcome current limitations.
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.