带统计加权的偏心WoS类求解器
Off-Centered WoS-Type Solvers with Statistical Weighting
AI总结：
针对偏心WoS类蒙特卡洛PDE求解器存在的相关伪影与偏差问题，提出带统计加权的偏心估计量，通过局部相似性滤波和加权策略平衡偏差与方差，在多种PDE任务上优于现有求解器，还可扩展至梯度场估计与混合边界问题。
摘要：
随机偏微分方程（PDE）求解器已成为传统基于离散化的PDE求解方法的有力替代方案，尤其在几何处理与图形学领域应用广泛。尽管偏心估计量提升了WoS类蒙特卡洛求解器的样本复用效率，但在近似格林函数时会引入相关伪影和偏差。本文提出一种带统计加权的偏心WoS类估计量，利用局部相似性滤波对相邻评估点的样本进行选择性组合。该方法通过一种有理论依据的加权策略抑制不可靠估计量，实现偏差与方差的平衡。我们在多种PDE（包括屏蔽泊松方程及各类边界条件）上验证了方法的有效性，相比现有求解器（如基础Walk on Spheres、均值缓存、边值缓存）取得了一致的性能提升。该方法还可自然扩展至梯度场估计与混合边界问题。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
