Efficient probabilistic Poisson solver for complex geometries based on sparse markov matrices

Abstract

This article introduces an efficient stochastic method to approximate the solution to Poisson’s Equation (PE). The method starts by computing sparse Transition Probability Matrices (TPMs), assembled by averaging short continuous random walks based on the transient diffusion equation with a source. The TPMs can be used within a false-transient approach to approximate the solution to PE in the steady-state limit. The method’s efficiency and memory usage are comparable to the Finite-Volume method. However, the ease of implementation, stability, and versatility makes the method attractive in complex scenarios. Moreover, given that the code can be easily implemented in parallel, it displays superior efficiency and practical importance in scenarios where a large number of nodes is mandatory.

Keywords:

• Meshless methods
• Monte Carlo
• Poisson equation
• variance reduction

Acknowledgments

The author Manuel Ramirez, acknowledges the financial support from CONAHCyT through program Estancias Posdoctorales por México-Modalidad Académica grant 3020924.

Disclosure statement

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Moreover, this manuscript has not been published elsewhere and that it has not been submitted simultaneously for publication elsewhere.