Double Greedy Algorithms: Reduced Basis Methods for Transport Dominated Problems

TL;DR

This paper introduces reduced basis methods with provable rate-optimality for transport dominated problems, utilizing tight surrogates and well-conditioned formulations, demonstrated through numerical experiments.

Contribution

It develops a new reduced basis framework with rigorous performance guarantees for transport problems, improving computational efficiency and understanding of solution smoothness.

Findings

Achieves rate-optimal performance relative to Kolmogorov n-widths.

Constructs computationally feasible tight surrogates.

Provides numerical validation for convection-diffusion and transport equations.

Abstract

The central objective of this paper is to develop reduced basis methods for parameter dependent transport dominated problems that are rigorously proven to exhibit rate-optimal performance when compared with the Kolmogorov $n$-widths of the solution sets. The central ingredient is the construction of computationally feasible "tight" surrogates which in turn are based on deriving a suitable well-conditioned variational formulation for the parameter dependent problem. The theoretical results are illustrated by numerical experiments for convection-diffusion and pure transport equations. In particular, the latter example sheds some light on the smoothness of the dependence of the solutions on the parameters.