Multiscale methods for nonlinear problems in strongly heterogeneous media and stiff stochastic systems

点击翻译按钮获取中文摘要

The development of analytical and computational procedures which resolve essential phenonema over a wide range of scales has led to the emergence of a new field of multiscale modelling. A complementary approach, essential in many applications, is to model fine-scale components using stochastic perturbations. These methods will be developed by the fellow in an ambitious programme of research incorporating multiscale numerical analysis, numerical method development and implementation. The techniques will be applied to problems arising in the geosciences and biology, including the pollution of groundwater (multiscale analysis of infiltration in strongly heterogeneous media) and the dynamics of microtubules with motor proteins (modelled as stiff stochastic systems).New multiscale methods based on a macro-to-micro strategy will considerably reduce the computational complexity of (traditional) direct approaches, since the microscale structures are then resolved only on sampling domains. The mathematical framework will be general enough to be applied to a wide variety of problems and allows for the construction of numerical methods which are not based on restrictive assumptions such as periodicity or homogeneity of the microstructures. New stabilised explicit methods will be constructed for stiff stochastic systems. Such methods are expected to be more efficient than standard explicit methods due to their enhanced stability properties. Stochastic problems with multiple time scales will also be studied.