  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.
