Felix Schindler: Error control and adaptivity for model reduction of parametric multi-scale problems

Abstract:

Many interesting problems, modeled by partial differential equations (PDEs), involve features on several length- or time-scales, the simulation of which would require grid-based approximation techniques such as Finite Element methods to fully resolve all relevant scales. The computation of such fully resolved approximations is often not feasible in practice, which gave rise to numerical multi-scale methods to lower the computational burden (usually by exploiting scale separation and by decoupling through localization).

On the other hand, many problems depend on a low-dimensional input (i.e., modeling the diffusion coefficient or boundary conditions in elliptic problems) and one is interested in computing a low-dimensional output (i.e., the mean temperature in a domain of interest). Such parametric problems, arising for instance in the context of optimization or uncertainty quantification, require a large number of approximations of PDEs, which gave rise to model reduction techniques, such as Reduced Basis (RB) methods, to quickly approximate an evaluation of a quantity of interest.

In this talk I will present a combination a numerical multi-scale and model reduction techniques (namely the localized Reduced Basis multi-scale method, LRBMS) for the efficient and accurate approximation of elliptic and parabolic parametric multi-scale problems, where one is interested in repeatedly solving a PDE, the data functions of which may depend on a low-dimensional input and at the same time inhibit multi-scale features.

In particular, I will discuss the approximation properties of the LRBMS in terms of a posteriori error control of the full approximation error (including the discretization as well as the model reduction error), introduce various aspects of adaptivity (going beyond the traditional offline/online splitting of RB methods) and present our software frameworks, which were used for the efficient discretization (dune-gdt, C++ based) and model reduction (pyMOR, Python based) of PDEs.