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Computing the principal eigenvalue of the Laplace operator by a stochastic method

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  • Lejay, Antoine
  • Maire, Sylvain

Abstract

We describe a Monte Carlo method for the numerical computation of the principal eigenvalue of the Laplace operator in a bounded domain with Dirichlet conditions. It is based on the estimation of the speed of absorption of the Brownian motion by the boundary of the domain. Various tools of statistical estimation and different simulation schemes are developed to optimize the method. Numerical examples are studied to check the accuracy and the robustness of our approach.

Suggested Citation

  • Lejay, Antoine & Maire, Sylvain, 2007. "Computing the principal eigenvalue of the Laplace operator by a stochastic method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 73(6), pages 351-363.
    • Handle: RePEc:eee:matcom:v:73:y:2007:i:6:p:351-363
    DOI: 10.1016/j.matcom.2006.06.011
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    References listed on IDEAS

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    1. Gobet, Emmanuel, 2000. "Weak approximation of killed diffusion using Euler schemes," Stochastic Processes and their Applications, Elsevier, vol. 87(2), pages 167-197, June.
    2. Madalina Deaconu & Antoine Lejay, 2006. "A Random Walk on Rectangles Algorithm," Methodology and Computing in Applied Probability, Springer, vol. 8(1), pages 135-151, March.
    3. Gobet, Emmanuel & Menozzi, Stéphane, 2004. "Exact approximation rate of killed hypoelliptic diffusions using the discrete Euler scheme," Stochastic Processes and their Applications, Elsevier, vol. 112(2), pages 201-223, August.
    4. Campillo Fabien & Lejay† Antoine, 2002. "A Monte Carlo method without grid for a fractured porous domain model," Monte Carlo Methods and Applications, De Gruyter, vol. 8(2), pages 129-148, December.
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    Cited by:

    1. Deaconu, M. & Herrmann, S. & Maire, S., 2017. "The walk on moving spheres: A new tool for simulating Brownian motion’s exit time from a domain," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 135(C), pages 28-38.
    2. Lejay, Antoine, 2018. "A Monte Carlo estimation of the mean residence time in cells surrounded by thin layers," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 143(C), pages 65-77.
    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.

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