IDEAS home Printed from https://ideas.repec.org/a/eee/matcom/v62y2003i3p347-355.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378475402002240
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/S0378-4754(02)00224-0?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:matcom:v:62:y:2003:i:3:p:347-355. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/mathematics-and-computers-in-simulation/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.