SIGGRAPH 2025
Tianyu Huang, School of Software and BNRist, Tsinghua University
Jingwang Ling, School of Software and BNRist, Tsinghua University
Shuang Zhao, University of California Irvine
Feng Xu*, School of Software and BNRist, Tsinghua University
* Corresponding author.
Abstract
Walk on stars (WoSt) has shown its power in being applied to Monte Carlo methods for solving partial differential equations, but the sampling techniques in WoSt are not satisfactory, leading to high variance. We propose a guiding-based importance sampling method to reduce the variance of WoSt. Drawing inspiration from path guiding in rendering, we approximate the directional distribution of the recursive term of WoSt using online-learned parametric mixture distributions, decoded by a lightweight neural field. This adaptive approach enables importance sampling the recursive term, which lacks shape information before computation. We introduce a reflection technique to represent guiding distributions at Neumann boundaries and incorporate multiple importance sampling with learnable selection probabilities to further reduce variance. We also present a practical GPU implementation of our method. Experiments show that our method effectively reduces variance compared to the original WoSt, given the same time or the same sample budget.
Citation
TBANote
A concurrent master’s thesis “An Efficient Fourier Caching Algorithm for Walk on Spheres” by Zihong Zhou at Dartmouth College has explored a completely different guiding approach using FDM solution for 2D problems. We think this is also a meaningful direction (but it has not yet been published), so we note it here.
Acknowledgements
We thank anonymous reviewers for their valuable feedback. The Bob model is provided by Keenan Crane; the Dragon and Bunny models are from the Stanford Computer Graphics Laboratory; the Gear model is courtesy of Hu et al. [2018]; and the Spot model is from Crane et al. [2013]. Fille and Ladybug are from Orzan et al. [2008].
This work was supported by the National Key R&D Program of China (2023YFC3305600), the Zhejiang Provincial Natural Science Foundation (LDT23F02024F02), and the NSFC (No.61822111, 62021002). This work was also supported by THUIBCS, Tsinghua University, and BLBCI, Beijing Municipal Education Commission.
Tianyu Huang was supported by Undergraduate Disruptive Innovation Talent Cultivation Program of Tsinghua University. He would like to thank Honghao Dong for valuable discussions, as well as Toshiya Hachisuka, Ryusuke Sugimoto, Li Yi, and EECG for their support.