A Greedy Look at Model Reduction

Treating complex design or optimization problems, especially under online constraints, is often practically feasible only when the underlying model is suitably reduced.

The so called Reduced Basis Method is an attractive variant of a model order reduction strategy for models based on parametric families of PDEs since it often allows one to rigorously monitor the accuracy of the reduced model. A key role is played by a greedy construction of reduced bases based on appropriate numerically feasible surrogates for the actual distance of the solution manifold from the reduced space.

While this concept has meanwhile been applied to a wide scope of problems the theoretical understanding concerning a certified accuracy is still essentially confined to the class of elliptic problems. This is reflected by the varying performance of these concepts, for instance, for transport dominated problems. We show that such a greedy space search is rate optimal, when compared with the Kolmogorov widths of the solution set, if the surrogates are in a certain sense tight. A key task is therefore to derive tight surrogates beyond the class of elliptic PDEs. We highlight the main underlying concepts centering on the convergence of greedy methods in Hilbert spaces and the derivation of well conditioned variational formulations for unsymmetric or singularly perturbed PDEs.