Global Optimization of the Mean First Passage Time for Narrow Capture Problems in Elliptic Domains

TL;DR

This paper develops and applies global optimization methods to minimize the mean first passage time in elliptic domains with multiple traps, providing new asymptotic formulas and identifying optimal trap arrangements.

Contribution

It introduces a systematic approach to optimize trap configurations in elliptic domains, extending previous asymptotic formulas to multiple traps of arbitrary sizes.

Findings

Optimal trap arrangements vary with domain eccentricity.

Derived asymptotic formula for MFPT with arbitrary trap sizes.

Sample configurations include non-equal trap sizes.

Abstract

Narrow escape and narrow capture problems which describe the average times required to stop the motion of a randomly travelling particle within a domain have applications in various areas of science. While for general domains, it is known how the escape time decreases with the increase of the trap sizes, for some specific 2D and 3D domains, higher-order asymptotic formulas have been established, providing the dependence of the escape time on the sizes and locations of the traps. Such results allow the use of global optimization to seek trap arrangements that minimize average escape times. In a recent paper \cite{iyaniwura2021optimization}, an explicit size- and trap location-dependent expansion of the average mean first passage time (MFPT) in a 2D elliptic domain was derived. The goal of this work is to systematically seek global minima of MFPT for $1\leq N\leq 50$ traps in elliptic domains using global optimization techniques, and compare the corresponding putative optimal trap arrangements for different values of the domain eccentricity. Further, an asymptotic formula the for the average MFPT in elliptic domains with $N$ circular traps of arbitrary sizes is derived, and sample optimal configurations involving non-equal traps are computed.