Numerical approximation of nonlinear stochastic ordinary and partial differential equations
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：Arnulf. Jentzen（香港中文大学深圳分校/德国明斯特大学）
：2023-02-03 16:00
：腾讯会议ID：920-657-936（无密码）
报告人：Arnulf. Jentzen（香港中文大学深圳分校/德国明斯特大学）
时 间：2月3日下午16:00
地 点：腾讯会议ID：920-657-936（无密码）
内容摘要:
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].
个人简介：
Arnulf. Jentzen , 国际著名随机微分方程数值解及机器学习领域专家，香港中文大学(深圳分校)及德国明斯特大学教授，2020年获得欧洲数学学会Felix Klein Prize，2022年获得Joseph F. Traub Prize, 目前为数学期刊Annals of Applied probability, SIAM J. Numer. Anal. SIAM J. Sci. Computing, SIAM JUQ, J. Machine Learning等期刊编委。
联系人：黄灿
