Heriot-Watt University

In various fields of application, one must solve very large scale boundary value problems using parallel solvers and supercomputers. The domain decomposition approach partitions the large computational domain into smaller computational subdomains. In order to speed up the convergence, we have several ``optimized'' algorithm that use Robin transmission conditions across the artificial interfaces (FETI-2LM). It is known that this approach alone is not sufficient: as the number of subdomains increases, the number of iterations required for convergence also increases and hence the parallel speedup is lost. A known solution for classical Schwarz methods as well as FETI algorithms is to incorporate a ``coarse grid correction'', which is able to transmit low-frequency information more quickly across the whole domain. Such algorithms are known to ``scale weakly'' to large supercomputers. A coarse grid correction is also necessary for FETI-2LM methods. In this talk, we will introduce and analyze coarse grid correction algorithms for FETI-2LM domain decomposition methods.

We present numerical schemes for nonlinear stochastic differential equations whose solution evolves on a smooth finite dimensional manifold. Given a Lie group action that generates transport along the manifold, we pull back the stochastic flow on the manifold to the Lie group via the action and subsequently to the corresponding Lie algebra.

We construct an approximation to the stochastic flow in the Lie algebra via closed operations and then push back to the manifold, thus ensuring our approximation lies in the manifold. We call such schemes stochastic Munthe-Kaas methods after their deterministic counterparts. We also present stochastic Lie group integration schemes based on Castell--Gaines methods. They become stochastic Lie group integrator schemes if we use Munthe-Kaas methods as the underlying ordinary differential integrator. Lastly, we demonstrate our methods by presenting some numerical examples