Reducing runtime for Monte Carlo estimates to Partial Differential Equations

This paper discusses multiple methods available for reducing the run time for Monte Carlo estimates to Partial Differential Equations (PDEs). PDEs appear extensively in mathematical modeling across the sciences, but they are often very difficult to solve analytically. Monte Carlo approaches provide a robust approach for generating estimates for the solutions of PDEs by simulating many random walks that travel through the domain of the specific problem. These random walks can be performed using a discrete time Markov Chain transition matrix. Two specific properties of Markov Chains can then be used to reduce the computation time, namely raising the matrix to a higher power and the independence of the random walk. When combined with parallel processing, dramatic reductions in runtime for a simulation can be achieved. This paper uses the 2D Laplace equation on a square domain as an example problem and implements all three of the techniques to reduce the computation time by a factor of nearly 150. The techniques described can be applied to other PDEs and allow researchers to either obtain results faster than the baseline, or achieve a higher accuracy in a given amount of time.