标题：Numerical methods for a class of nonlocal diffusion problems with the use of backward SDEs
作者：Zhang, Guannan; Zhao, Weidong; Webster, Clayton; Gunzburger, Max
作者机构：[Zhang, Guannan; Webster, Clayton] Oak Ridge Natl Lab, Comp Sci & Math Div, Oak Ridge, TN 37831 USA.; [Zhao, Weidong] Shandong Univ, Sch Math, Jinan 更多
会议名称：Conference on Advances in Scientific Computing and Applied Mathematics
会议日期：OCT 09-12, 2015
来源：COMPUTERS & MATHEMATICS WITH APPLICATIONS
关键词：Backward stochastic differential equation with jumps; Nonlocal diffusion; equations; Superdiffusion; Compound Poisson process theta-scheme;; Adaptive approximation
摘要：We propose a novel numerical approach for nonlocal diffusion equations Du et al. (2012) with integrable kernels, based on the relationship between the backward Kolmogorov equation and backward stochastic differential equations (BSDEs) driven by Levy jumps processes. The nonlocal diffusion problem under consideration is converted to a BSDE, for which numerical schemes are developed. As a stochastic approach, the proposed method completely avoids the challenge of iteratively solving non-sparse linear systems, arising from the nature of nonlocality. This allows for embarrassingly parallel implementation and also enables adaptive approximation techniques to be incorporated in a straightforward fashion. Moreover, our method recovers the convergence rates of classic deterministic approaches (e.g. finite element or collocation methods), due to the use of high-order temporal and spatial discretization schemes. In addition, our approach can handle a broad class of problems with general inhomogeneous forcing terms as long as they are globally Lipschitz continuous. Rigorous error analysis of the new method is provided as several numerical examples that illustrate the effectiveness and efficiency of the proposed approach. (C) 2015 Elsevier Ltd. All rights reserved.