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

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