Analysis of a HMM time-discretization scheme for a system of Stochastic PDE's
Provides theoretical error bounds for multiscale numerical methods in infinite-dimensional SPDEs, advancing the mathematical foundation for HMM-type schemes.
The paper analyzes a time-discretization scheme for a system of stochastic PDEs with slow and fast components, proving strong and weak error bounds that generalize prior finite-dimensional results.
We consider the discretization in time of a system of parabolic stochastic partial differential equations with slow and fast components; the fast equation is driven by an additive space-time white noise. The numerical method is inspired by the Averaging Principle satisfied by this system, and fits to the framework of Heterogeneous Multiscale Methods.The slow and the fast components are approximated with two coupled numerical semi-implicit Euler schemes depending on two different timestep sizes. We derive bounds of the approximation error on the slow component in the strong sense - approximation of trajectories - and in the weak sense - approximation of the laws. The estimates generalize the results of \cite{E-L-V} in the case of infinite dimensional processes.