Gradient Domain Reconstruction for Monte Carlo PDE Solvers

Grid-free Monte Carlo methods are capable of solving Poisson equations on highly complex domains. However, existing methods operate solely in the primal domain and can converge slowly due to high variance. Inspired by gradient-domain rendering, we introduce a gradient-domain framework for Poisson problems. Specifically, we devise a new Monte Carlo estimator that directly targets differences of the solution between spatially varying query locations. Further, we adopt state-of-the-art reconstruction techniques originated in gradient-domain rendering to allow efficient reconstruction of the solutions without incurring additional bias. We demonstrate the effectiveness of our technique by comparing solutions obtained using our method and several state-of-the-art baselines.