Optimization of Trap Locations for Narrow Capture Problems

The determination of the mean first passage time (MFPT) for a Brownian particle in a domain that contains a collection of small absorbing traps in its interior is a narrow capture problem with many biophysical applications. The average MFPT is the expected capture time assuming a uniform distribution of starting points for the random walk. We survey some results for determining the spatial locations of the traps that minimize the average MFPT for certain 2-D and 3-D domains. In the limit of small trap radii, the optimization of the average MFPT over the trap locations can be reduced to the problem of seeking global minima of a discrete energy for an interacting particle system that is encoded by the Neumann Green’s matrix. Open problems for the optimization of the average MFPT are discussed.