Simulation and Optimization of Mean First Passage Time Problems in 2-D using Numerical Embedded Methods and Perturbation Theory

TL;DR

This paper introduces new numerical and perturbation methods to compute and optimize the mean first passage time for Brownian particles in complex 2-D domains, addressing challenges in general geometries and trap configurations.

Contribution

It develops embedded numerical methods based on the Closest Point Method for MFPT simulations and introduces perturbation approaches for near-disk and thin domains, advancing the analysis of trap optimization.

Findings

Effective numerical methods for MFPT in complex domains.

Identification of optimal trap arrangements via optimization techniques.

Perturbation approaches for approximate solutions in specific geometries.

Abstract

We develop novel numerical methods and perturbation approaches to determine the mean first passage time (MFPT) for a Brownian particle to be captured by either small stationary or mobile traps inside a bounded 2-D confining domain. Of particular interest is to identify optimal arrangements of small absorbing traps that minimize the average MFPT. Although the MFPT, and the associated optimal trap arrangement problem, has been well-studied for disk-shaped domains, there are very few analytical or numerical results available for general star-shaped domains or for thin domains with large aspect ratio. Analytical progress is challenging owing to the need to determine the Neumann Green's function for the Laplacian, while the numerical challenge results from a lack of easy-to-use and fast numerical tools for first computing the MFPT and then optimizing over a class of trap configurations. In this direction, and for the stationary trap problem, we develop a simple embedded numerical method, based on the Closest Point Method (CPM), to perform MFPT simulations on elliptical and star-shaped domains. For periodic mobile trap problems, we develop a robust CPM method to compute the average MFPT. Optimal trap arrangements are identified numerically through either a refined discrete sampling approach or from a particle-swarm optimization procedure. To confirm some of the numerical findings, novel perturbation approaches are developed to approximate the average MFPT and identify optimal trap configurations for a class of near-disk confining domains or for an arbitrary thin domain of large aspect ratio.