Stochastic PDE solvers have emerged as a powerful alternative to traditional discretization-based methods for solving partial differential equations (PDEs), especially in geometry processing and graphics. While off-centered estimators enhance sample reuse in WoS-type Monte Carlo solvers, they introduce correlation artifacts and bias when Green's functions are approximated. In this paper, we propose a statistically weighted off-centered WoS-type estimator that leverages local similarity filtering to selectively combine samples across neighboring evaluation points. Our method balances bias and variance through a principled weighting strategy that suppresses unreliable estimators. We demonstrate our approach's effectiveness on various PDEs,including screened Poisson equations and boundary conditions, achieving consistent improvements over existing solvers such as vanilla Walk on Spheres, mean value caching, and boundary value caching. Our method also naturally extends to gradient field estimation and mixed boundary problems.

翻译：随机偏微分方程求解器已成为解决偏微分方程的一种强大替代方案，尤其适用于几何处理和图形学领域，相较于传统的基于离散化的方法具有显著优势。尽管非中心化估计器提升了WoS型蒙特卡洛求解器中的样本复用效率，但在格林函数近似时引入了相关性伪影和偏差。本文提出了一种基于统计加权的非中心化WoS型估计器，该方法利用局部相似性滤波，选择性地整合相邻评估点的样本。通过一种抑制不可靠估计器的原理性加权策略，我们的方法实现了偏差与方差的平衡。我们在多种偏微分方程（包括屏蔽泊松方程及边界条件）上验证了该方法的有效性，相较于现有求解器（如原始球面行走法、均值缓存及边界值缓存）取得了稳定改进。此外，该方法可自然扩展至梯度场估计和混合边界问题。