Monte Carlo PDE Solvers for Nonlinear Radiative Boundary Conditions

TL;DR

This paper introduces a Monte Carlo PDE solver framework for nonlinear radiative boundary conditions, combining a fixed-point iteration with a heteroscedastic regression denoising technique, validated on complex geometries.

Contribution

It presents a novel iterative approach for nonlinear boundary conditions in Monte Carlo PDE solving, improving accuracy and stability over existing linearization methods.

Findings

The method achieves higher accuracy than linearization strategies.

It remains stable and empirically convergent with proper relaxation.

Effective denoising reduces variance in boundary estimates.

Abstract

Monte Carlo PDE solvers have become increasingly popular for solving heat-related partial differential equations in geometry processing and computer graphics due to their robustness in handling complex geometries. While existing methods can handle Dirichlet, Neumann, and linear Robin boundary conditions, nonlinear boundary conditions arising from thermal radiation remain largely unexplored. In this paper, we introduce a Picard-style fixed-point iteration framework that enables Monte Carlo PDE solvers to handle nonlinear radiative boundary conditions. While strict theoretical convergence is not generally guaranteed, our method remains stable and empirically convergent with a properly chosen relaxation coefficient. Even with imprecise initial boundary estimates, it progressively approaches the correct solution. Compared to standard linearization strategies, the proposed approach achieves significantly higher accuracy. To further address the high variance inherent in Monte Carlo estimators, we propose a heteroscedastic regression-based denoising technique specifically designed for on-boundary solution estimates, filling a gap left by prior variance reduction methods that focus solely on interior points. We validate our approach through extensive evaluations on synthetic benchmarks and demonstrate its effectiveness on practical heat radiation simulations with complex geometries.