Off-Centered WoS-Type Solvers with Statistical Weighting

TL;DR

The paper tackles solving PDEs in graphics using stochastic WoS-type solvers and identifies correlation artifacts and bias when Green's functions are approximated. It introduces a statistically weighted off-centered estimator, forming the multi-domain estimator $\hat{u}(x)=\sum_{y\in H_x}\lambda_{x,y}\hat{I}_{x,y}$ with a similarity test based on $w^*$ and a bias-variance tuning parameter $\gamma$ to filter unreliable off-centered terms. The method extends to Dirichlet, Neumann, and mixed boundary conditions, supports gradient estimation, and applies to the screened Poisson equation. Experiments demonstrate consistent variance reduction and improved accuracy over vanilla WoS, mean value caching, and boundary value caching, while offering a tunable bias-variance trade-off via $\gamma$ and robust handling of correlation artifacts.

Abstract

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.

Figures (9)

• Figure 1: Illustration of multiple domain integration for the Walk on Spheres algorithm. For a certain evaluation point $x_0$, the solution can be estimated using integrals over different domains—i.e., maximal spheres centered at various locations. The sampling stage and reuse stage are illustrated by solid and dashed lines, respectively.
• Figure 2: Outlier values in off-centered estimators lead to remarkable artifacts in visual results. (a): off-centered estimator with the uniform weighting, extreme values affect all evaluation points that reuse them. Artifacts caused by outliers are marked by purple boxes (b): Our statistical weighting strategy can significantly reduce these artifacts.
• Figure 3: Left: the first step of the standard walk-on-star algorithm. Right: our method extends walk-on-star estimators by introducing an additional step in the random walk. Blue spheres represent added integration domains, and the green region shows the second step of the walk-on-star algorithm for a sample in one of these domains. The estimated value $\hat{u}(x_1)$ is used to estimate both center position $x_0$ and off-centered position $y$.
• Figure 4: Boundary geometry, division of different boundary conditions, and evaluation slices in our experimentation. For the experimentation of pure Dirichlet problems, we use all Dirichlet boundary conditions.
• Figure 5: Result and MSE of different weighting methods in the simple 2D domain. (a): For the simplest Laplace problem, our method has higher variance than the method by czekanski2024walking. (b): For the Poisson equations, the motivation of the Poisson bound weight no longer holds.