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An exit probability approach to solve high dimensional Dirichlet problems

by F. B. Buchmann and W. P. Petersen

(Report number 2004-09)

Abstract
We present an approach to solve high dimensional Dirichlet problems in bounded domains which is based on a variant of the Feynman-Kac formula that connects solutions to elliptic partial differential equations and functional integration. We integrate the resulting system of stochastic differential equations numerically with the Euler scheme. To correct for possible intermediate excursions of the simulated paths and to find good approximations for first exit times, we extend to higher dimensions a strategy which we have introduced in [1] for stopping problems in $1d$. In addition, for Dirichlet's problem, good approximations of the first exit points are needed to evaluate the boundary condition. To this end, we sample a random point on the tangent hyperplane of the boundary of the domain once a first exit was estimated. To detect excursions and adequately compute first exit times, we apply the same local half space approximation. We therefore assume that the domain is smooth enough such that these approximations are possible. Numerical experiments in dimensions up to 128 show that resulting approximations are of very high quality. In particular, we observe that first order convergence behavior of the Euler scheme can be maintained.

Keywords:

@Techreport{BP04_336,
  author = {F. B. Buchmann and W. P. Petersen},
  title = {An exit probability approach to solve high dimensional Dirichlet problems},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2004-09},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2004/2004-09.pdf },
  year = {2004}
}

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