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

分析和计算程序的发展解决了大范围尺度上的基本现象，导致了一个新的多尺度模拟领域的出现。一种在许多应用中必不可少的补充方法是使用随机扰动来对精细组件进行建模。这些方法将由该研究员在一项雄心勃勃的研究计划中开发，该计划包括多尺度数值分析、数值方法的开发和实施。这些技术将被应用于地学和生物学中出现的问题，包括地下水污染(强非均匀介质中渗透的多尺度分析)和含有马达蛋白的微管动力学(被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(传统的)直接方法的计算复杂性，因为微尺度结构只在采样域上求解。数学框架将足够普遍，可以应用于各种各样的问题，并允许建立数值方法，而不是基于限制性假设，如微观结构的周期性或同质性。对于刚性随机系统，将构造新的稳定化显式方法。这种方法由于其增强的稳定性，预计将比标准显式方法更有效。还将研究具有多个时间尺度的随机问题。

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.