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

发展的分析和计算程序，解决基本phenonema在广泛的尺度，导致了一个新的领域的出现多尺度建模。一个补充的方法，在许多应用中必不可少的，是使用随机扰动模型的细尺度组件。这些方法将由研究员在一个雄心勃勃的研究方案中开发，该方案将多尺度数值分析、数值方法开发和实施结合起来。这些技术将应用于地球科学和生物学中出现的问题，包括地下水污染（在强不均匀介质中浸润的多尺度分析）和微管与马达蛋白的动力学（建模为刚性随机系统）。基于宏观到微观策略的新的多尺度方法将大大降低计算复杂性，（传统的）直接方法，因为微尺度结构然后仅在采样域上被解析。数学框架将是一般的，足以适用于各种各样的问题，并允许建设的数值方法，而不是基于限制性的假设，如周期性或均匀性的微观结构。新的稳定化显式方法将被构造为刚性随机系统。由于其增强的稳定性特性，预计这种方法比标准的显式方法更有效。多时间尺度的随机问题也将被研究。

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.