How to best sample a solution manifold?

Abstract

Model reduction attempts to guarantee a desired "model quality", e.g. given in terms of accuracy requirements, with as small a model size as possible. This article highlights some recent developments concerning this issue for the so called Reduced Basis Method (RBM) for models based on parameter dependent families of PDEs. In this context the key task is to sample the {\em solution manifold} at judiciously chosen parameter values usually determined in a {\em greedy fashion}. The corresponding {\em space growth} concepts are closely related to so called {\em weak greedy} algorithms in Hilbert and Banach spaces which can be shown to give rise to convergence rates comparable to the best possible rates, namely the {\em Kolmogorov $n$-widths} rates. Such algorithms can be interpreted as {\em adaptive sampling} strategies for approximating compact sets in Hilbert spaces. We briefly discuss the results most relevant for the present RBM context. The applicability of the results for weak greedy algorithms has however been confined so far essentially to well-conditioned coercive problems. A critical issue is therefore an extension of these concepts to a wider range of problem classes for which the conventional methods do not work well. A second main topic of this article is therefore to outline recent developments of RBMs that do realize $n$-width rates for a much wider class of variational problems covering indefinite or singularly perturbed unsymmetric problems. A key element in this context is the design of {\em well-conditioned variational formulations} and their numerical treatment via saddle point formulations. We conclude with some remarks concerning the relevance of uniformly approximating the whole solution manifold also when the {\em quantity of interest} is only of a {\em functional} of the parameter dependent solutions.