Workshop 4: December 12-16, 2011

Numerical Analysis of Multiscale Problems & Stochastic Modelling


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.