Off-Centered WoS-Type Solvers with Statistical Weighting

TL;DR

This paper introduces a statistically weighted off-centered WoS-type estimator that improves stochastic PDE solving by reducing bias and variance through local similarity filtering and principled weighting, enhancing performance on various PDEs.

Contribution

It presents a novel weighting strategy for off-centered WoS estimators that balances bias and variance, improving accuracy and efficiency in stochastic PDE solvers.

Findings

Achieves consistent improvements over existing solvers.

Extends naturally to gradient and mixed boundary problems.

Effectively balances bias and variance in PDE solutions.

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.