  The determination of the mean first passage time (MFPT) for a Brownian
particle in a bounded 2-D domain containing small absorbing traps is a
fundamental problem with biophysical applications. The average MFPT is the
expected capture time assuming a uniform distribution of starting points for
the random walk. We develop a hybrid asymptotic-numerical approach to predict
optimal configurations of $m$ small stationary circular absorbing traps that
minimize the average MFPT in near-disk and elliptical domains. For a general
class of near-disk domains, we illustrate through several specific examples how
simple, but yet highly accurate, numerical methods can be used to implement the
asymptotic theory. From the derivation of a new explicit formula for the
Neumann Green's function and its regular part for the ellipse, a numerical
approach based on our asymptotic theory is used to investigate how the spatial
distribution of the optimal trap locations changes as the aspect ratio of an
ellipse of fixed area is varied. The results from the hybrid theory for the
ellipse are compared with full PDE numerical results computed from the closest
point method \cite{IWWC2019}. For long and thin ellipses, it is shown that the
optimal trap pattern for $m=2,\ldots,5$ identical traps is collinear along the
semi-major axis of the ellipse. For such essentially 1-D patterns, a
thin-domain asymptotic analysis is formulated and implemented to accurately
predict the optimal locations of collinear trap patterns and the corresponding
optimal average MFPT.
