  Walk on Spheres algorithms leverage properties of Brownian Motion to create
Monte Carlo estimates of solutions to a class of elliptic partial differential
equations. Until recently, estimates were constructed pointwise and did not
utilize the relationship between solutions at nearby points within a domain. We
propose a new caching strategy which leverages the continuity of paths of
Brownian Motion. In the case of Laplace's equation with Dirichlet boundary
conditions, our novel algorithm has improved asymptotic runtime compared to
previous approaches. This is achieved by passing information from a cache of
constant size. We also provide bounds on the performance of our algorithm and
demonstrate its performance on an example problem.
