416

RWTH Publication No: 463811 2015 IGPM416.pdf 
TITLE 
How to Best Sample a Solution Manifold? 
AUTHORS 
Wolfgang Dahmen 
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 high lights 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
solution manifold at judiceously chosen parameter values usually determined in a greedy fashion.
The corresponding space growth concepts are closely related to so called 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 Kolmogorov nwidths rates. Such algorithms can be interpreted as
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 wellconditioned 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 nwidth 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 wellconditioned 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
quantity of interest is only of a functional of the parameter dependent solutions.

KEYWORDS 
tight surrogates, stable variational formulations, saddle point problems, double greedy schemes, greedy stabilization, rateoptimality, transport equations, convectiondiffusion equations

DOI 
10.1007/9783319197494_11 
PUBLICATION 
Sampling Theory, a Renaissance, Applied and Numerical Harmonic Analysis, pp 403–435, 2015 