BRANCHING DIFFUSION REPRESENTATION FOR NONLINEAR CAUCHY PROBLEMS AND MONTE CARLO APPROXIMATION

• 成果类型: Article
• 署名作者: Henry-Labordere, Pierre; Touzi, Nizar
• 署名单位: Universite Paris Cite; Institut Polytechnique de Paris; ENSTA Paris
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/20-AAP1649
• 发表日期: 2021
• 页码: 2350-2375
• 关键词: numerical-solution; equation; simulation; algorithm; pdes

摘要

We provide probabilistic representations of the solution of some semi-linear hyperbolic and high-order PDEs based on branching diffusions. These representations pave the way for an approximation of the solution by the standard Monte Carlo method, whose error estimate is controlled by the standard central limit theorem, thus partly bypassing the curse of dimensionality. We illustrate the numerical implications in the context of some popular PDEs in physics such as nonlinear Klein-Gordon equation, a simplified scalar version of the Yang-Mills equation, a fourth-order nonlinear beam equation and the Gross-Pitaevskii PDE as an example of nonlinear Schrodinger equations.