Geometric Queries on Closed Implicit Surfaces for Walk on Stars

ACM SIGGRAPH Asia 2025 (technical communications short paper)

Tianyu Huang^†, Independent researcher and Tsinghua University

† denotes sole author.

Abstract

Walk on stars (WoSt) is currently one of the most advanced Monte Carlo solvers for PDEs. Unfortunately, the lack of reliable geometric query approaches has hindered its applicability to boundaries defined by implicit surfaces. This work proposes a geometric query framework over closed implicit surfaces for WoSt, under the scope of walkin’ Robin. Our key observation is that all WoSt queries can be formulated as constrained global optimization or constraint satisfaction problems. Based on our formulations, to solve the highly non-convex problems, we adopt a branch-and-bound approach based on interval analysis. To the best of our knowledge, our method is the first to study closest silhouette point queries and Robin radius bound queries on closed implicit surfaces. Our formulations and methods first enable mesh-free PDE solving via WoSt when boundaries are defined by closed implicit surfaces.

Note

A comprehensive version of this short paper is currently a work in progress. Solutions to many existing issues, e.g., too conservative bounds when query point is near the boundaries (and thus extremely long walk lengths), will be described in the full version.

Figures

Presentation Recording

TBA.

Acknowledgements

Tianyu Huang would like to thank Hao Pan, and Ryusuke Sugimoto for valuable discussions. Jingwang Ling helped with proofreading of this paper. This paper utilized resources from Feng Xu’s lab. Tianyu Huang also acknowledges the travel funding support from the Tsinghua University Spark Program.

Citation

@article{huang2025implicit,
  title = {Geometric Queries on Closed Implicit Surfaces for Walk on Stars},
  author = {Huang, Tianyu},
  journal = {ACM SIGGRAPH Asia 2025 Technical Communications (SA Technical Communications '25)},
  year = {2025},
  isbn = {979-8-4007-2136-6/2025/12},
  location = {Hong Kong, Hong Kong},
  doi = {10.1145/3757376.3771378},
  publisher = {Association for Computing Machinery},
  address = {New York, NY, USA}
}