Workshop 4: December 12-16, 2011
Numerical Analysis of Multiscale Problems & Stochastic Modelling
Organizers
Massimo Fornasier, Johann Radon Institute, Austria
Ivan G. Graham, University of Bath, UK
Markus Melenk, Vienna University of Technology, Austria
Robert Scheichl, University of Bath, UK
Jörg Willems, Johann Radon Institute, Austria
Synopsis and Main Topics
The solution of complex multiscale and multiphysics problem requires, amongst other things, effective computational
tools, uncertainty quantification, and, ultimately, (adaptive) modelling error control. This workshop will focus on
the numerical analysis of multiscale problems with an emphasis on those problem classes arising in the context of
the preceding workshops of the special semester.
In the last decade, several novel non-standard methods have been proposed to solve multiscale problems. Examples
include the multiscale FEM (MS-FEM), the heterogeneous multiscale method (HMM), various upscaling methods, and
"knowledge-based" techniques such as the generalised FEM, XFEM, quasi-continuum ("bridging the scales") methods,
and, more recently, mimetic finite difference methods. Typically, these methods are formulated as coarsescale methods
that incorporate fine-scale information in the overall scheme. This can done in various ways. For example, it is
possible to use special shape functions (either derived analytically or computed numerically); alternatively, one
can aim at obtaining effective material properties by a suitable sampling strategy on the fine scale. The analysis
and the application of these methods in areas related to the special semester (e.g., flow and wave propagation in
porous and heterogeneous media) will be of particular interest.
These multiscale methods are naturally related to uncertainty quantification and stochastic modelling. A second
emphasis of the workshop will therefore be on numerical methods for PDEs with stochastic input data. Several methods
currently compete in this area. We mention in particular, Monte Carlo and Quasi-Monte Carlo methods on the one hand,
and stochastic Galerkin/collocation methods on the other hand. The latter techniques are based on transforming the stochastic
problem into a deterministic one, which, however, is high dimensional. Here, multilevel approximation techniques
such as sparse grids or hyperbolic cross approximation have to be brought to bear. Of great interest are also
questions of adaptivity and further compression since even with sparse multilevel techniques, current computing power
puts stringent limitations on the number of dimensions that can effectively be handled.
Although complex, realistic applications naturally call for the concurrent use of numerical methods for multiscale
problems and the appropriate numerical treatment of stochasticity, this is rarely done in practice. The aim of the
workshop is to encourage cross-fertilisation between the (numerical) analysis community and the practitioners in
fields directly related to applications in energy and the environment, so that recent mathematical progress in e.g.
multiscale approximation, fast solvers and the treatment of uncertainty can be brought to bear on practical problems
of a wide interest. The workshop will be mostly mathematical but key speakers from the practitioners' communities
which appeared in the previous workshops will be invited to present cutting edge problems.