Model reduction for multiscale problems

In this contribution we present efficient numerical multiscale methods for flow in heterogeneous porous media, in particular also in situations where the resulting equations are to be solved repeatedly for varying parameters, as e.g. in the context of uncertainty quantification, time dependent scenarios or optimal control problems. We discuss a posteriori based discretization methods and suggest a suitable conceptual approach for an efficient numerical treatment of parameterized 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. Our general approach [14] covers a large class of numerical multi-scale schemes based on an additive splitting of function spaces into macroscopic and fine scale contributions combined with a tensor decomposition of function spaces in the context of multi query applications. In detail, let U, V denote suitable function spaces over a domain Ω ⊂ R and let us look at solutions u μ ∈ U of parameterized variational problems of the form