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A Feynman–Kac path-integral implementation for Poisson’s equation using an h-conditioned Green’s function

Author

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  • Hwang, Chi-Ok
  • Mascagni, Michael
  • Given, James A.

Abstract

This study presents a Feynman–Kac path-integral implementation for solving the Dirichlet problem for Poisson’s equation. The algorithm is a modified “walk on spheres” (WOS) that includes the Feynman–Kac path-integral contribution for the source term. In our approach, we use an h-conditioned Green’s function instead of simulating Brownian trajectories in detail to implement this path-integral computation. The h-conditioned Green’s function allows us to represent the integral of the right-hand-side function from the Poisson equation along Brownian paths as a volume integral with respect to a residence time density function: the h-conditioned Green’s function. The h-conditioned Green’s function allows us to solve the Poisson equation by simulating Brownian trajectories involving only large jumps, which is consistent with both WOS and our Green’s function first-passage (GFFP) method [J. Comput. Phys. 174 (2001) 946]. As verification of the method, we tabulate the h-conditioned Green’s function for Brownian motion starting at the center of the unit circle and making first-passage on the boundary of the circle, find an analytic expression fitting the h-conditioned Green’s function, and provide results from a numerical experiment on a two-dimensional Poisson problem.

Suggested Citation

  • Hwang, Chi-Ok & Mascagni, Michael & Given, James A., 2003. "A Feynman–Kac path-integral implementation for Poisson’s equation using an h-conditioned Green’s function," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 62(3), pages 347-355.
    • Handle: RePEc:eee:matcom:v:62:y:2003:i:3:p:347-355
    DOI: 10.1016/S0378-4754(02)00224-0
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    Citations

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    Cited by:

    1. Cameron Martin & Hongyuan Zhang & Julia Costacurta & Mihai Nica & Adam R Stinchcombe, 2022. "Solving Elliptic Equations with Brownian Motion: Bias Reduction and Temporal Difference Learning," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1603-1626, September.
    2. Maire, Sylvain & Nguyen, Giang, 2016. "Stochastic finite differences for elliptic diffusion equations in stratified domains," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 121(C), pages 146-165.
    3. Maire Sylvain & Tanré Etienne, 2013. "Monte Carlo approximations of the Neumann problem," Monte Carlo Methods and Applications, De Gruyter, vol. 19(3), pages 201-236, October.
    4. Zhang, Bolong & Yu, Wenjian & Mascagni, Michael, 2019. "Revisiting Kac’s method: A Monte Carlo algorithm for solving the Telegrapher’s equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 156(C), pages 178-193.
    5. Xuxin Yang & Antti Rasila & Tommi Sottinen, 2017. "Walk On Spheres Algorithm for Helmholtz and Yukawa Equations via Duffin Correspondence," Methodology and Computing in Applied Probability, Springer, vol. 19(2), pages 589-602, June.

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