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The determination of the mean first passage time (MFPT) for a Brownianparticle in a bounded 2-D domain containing small absorbing traps is afundamental problem with biophysical applications. The average MFPT is theexpected capture time assuming a uniform distribution of starting points forthe random walk. We develop a hybrid asymptotic-numerical approach to predictoptimal configurations of $m$ small stationary circular absorbing traps thatminimize the average MFPT in near-disk and elliptical domains. For a generalclass of near-disk domains, we illustrate through several specific examples howsimple, but yet highly accurate, numerical methods can be used to implement theasymptotic theory. From the derivation of a new explicit formula for theNeumann Green's function and its regular part for the ellipse, a numericalapproach based on our asymptotic theory is used to investigate how the spatialdistribution of the optimal trap locations changes as the aspect ratio of anellipse of fixed area is varied. The results from the hybrid theory for theellipse are compared with full PDE numerical results computed from the closestpoint method \cite{IWWC2019}. For long and thin ellipses, it is shown that theoptimal trap pattern for $m=2,\ldots,5$ identical traps is collinear along thesemi-major axis of the ellipse. For such essentially 1-D patterns, athin-domain asymptotic analysis is formulated and implemented to accuratelypredict the optimal locations of collinear trap patterns and the correspondingoptimal average MFPT.

Optimization of the Mean First Passage Time in Near-Disk and Elliptical Domains in 2-D with Small Absorbing Traps