High-accuracy solution of PDEs by iterative refinement of global stochastic solvers with applications to interior and exterior Dirichlet problems

Abstract

The statistical convergence barrier of Monte Carlo methods has long been regarded as the principal obstacle to high-accuracy stochastic solution of elliptic boundary value problems. In this paper, we show that this barrier can be overcome by a general residual-based iterative refinement framework applicable to stochastic solvers that permit evaluation of the residual. As a representative example, we develop the theory for the Global Random Walk on Spheres (RWS) method. Our main theoretical result is a proof of exponential convergence of the complete iterative algorithm, together with explicit error estimates and computational complexity bounds. The analysis shows that each additional order-of-magnitude reduction in the error requires only about a twofold increase in computational effort, whereas conventional Monte Carlo methods require approximately a hundredfold increase. Thus, the statistical convergence barrier of Monte Carlo no longer determines the computational complexity of high-accuracy stochastic computation. Numerical experiments for both interior and exterior Dirichlet problems confirm the theoretical predictions. Using global RWS as the underlying stochastic solver, the proposed framework achieves several orders of magnitude higher accuracy while substantially reducing the computational time compared with the standard RWS algorithm, making it particularly attractive for high-accuracy computations.