Error Control Based Model Reduction for Multiscale Problems

In this contribution we review a posteriori based discretization methods for variational multiscale problems and suggest a suitable conceptual approach for an efficient numerical treatment of parametrized variational multiscale problems where the parameters are either chosen from a low dimensional parameter space or consists of parameter functions from some compact low dimensional manifold that is embedded in some high dimensional or even infinite dimensional function space. The approach is based on combinations of ideas from established numerical multiscale methods and efficient model reduction approaches such as the reduced basis method.