  We present an efficient method to solve the narrow capture and narrow escape
problems for the sphere. The narrow capture problem models the equilibrium
behavior of a Brownian particle in the exterior of a sphere whose surface is
reflective, except for a collection of small absorbing patches. The narrow
escape problem is the dual problem: it models the behavior of a Brownian
particle confined to the interior of a sphere whose surface is reflective,
except for a collection of small patches through which it can escape.
  Mathematically, these give rise to mixed Dirichlet/Neumann boundary value
problems of the Poisson equation. They are numerically challenging for two main
reasons: (1) the solutions are non-smooth at Dirichlet-Neumann interfaces, and
(2) they involve adaptive mesh refinement and the solution of large,
ill-conditioned linear systems when the number of small patches is large.
  By using the Neumann Green's functions for the sphere, we recast each
boundary value problem as a system of first-kind integral equations on the
collection of patches. A block-diagonal preconditioner together with a multiple
scattering formalism leads to a well-conditioned system of second-kind integral
equations and a very efficient approach to discretization. This system is
solved iteratively using GMRES. We develop a hierarchical, fast multipole
method-like algorithm to accelerate each matrix-vector product. Our method is
insensitive to the patch size, and the total cost scales with the number N of
patches as O(N log N), after a precomputation whose cost depends only on the
patch size and not on the number or arrangement of patches. We demonstrate the
method with several numerical examples, and are able to achieve highly accurate
solutions with 100,000 patches in one hour on a 60-core workstation.
