Speaker：Xiaoguang Han（The Chinese University of Hong Kong, Shenzhen/ University of Munster, Germany）

Time：2023-02-03, 16:00

Location：Tencent Meeting ID：920-657-936（No Password）

Abstract:

In talk we review some selected recent developments in the numerical approximation of nonlinear stochastic ordinary and partial differential equations of the evolutionary type. First, we recall the nowadays well-known fact that the most basic numerical method for approximately solving stochastic ordinary differential equations (SODEs), the Euler-Maruyama scheme, fails to converge strongly and numerically weakly in the case of SODEs with superlinearly growing nonlinearities such as polynomial nonlinearities [1]. Thereafter, we observe that this divergence phenomena also arises in the case of several stochastic partial differential equations (SPDEs) with polynomial nonlinearities such as in the case of the Allen-Cahn equation on the unit interval [2]. Then, we introduce variants of the recently introduced tamed numerical methods which provably overcome such divergence phenomena in the case of a large class of SODEs and SPDEs with superlinearly growing nonlinearities [3,4,7]. Thereafter, we observe that there exist certain SODEs with bounded and infinitely often differentiable coefficient functions to which basically all standard approximation methods converge without any rate of convergence [5,6]. This kind of slow convergence phenomena essentially reveals that there exist SODEs/SPDEs which can not be solved approximately by nearly any approximation method in polynomial time [6]. Finally, we comment on suitable hypotheses on the nonlinearities of the considered SODE/SPDE which are sufficient to ensure that such a slow convergence phenomena does not occur and, thus, that the considered SODE/SPDE can be solved approximately in polynomial time [7].