Gradient Domain Reconstruction for Monte Carlo PDE Solvers

Jiaqi Wu¹, Xuejun Hu¹, Shuang Zhao², Kun Xu¹

¹Tsinghua University ²University of Illinois Urbana-Champaign

ACM Transactions on Graphics (SIGGRAPH) 2026

SIGGRAPH 2026 Technical Papers Honorable Mention Award

PDF Code

Teaser comparison of heat-transfer reconstruction on a robot model — We simulate heat transfer within a robot by solving the Laplace equation. The robot’s lower body is immersed in water and its upper body is exposed to air, powered by an internal reactor. Under a simplified modeling assumption, we assign Dirichlet boundary conditions to the submerged lower-body surfaces to represent the water temperature, and Neumann boundary conditions to the exposed upper-body surfaces to model adiabatic heat transfer at the air interface. Internal sources account for heat generated by the reactor. We present temperature fields under two scenarios: a global slice solution, in which the solution is computed over a planar cross-section of the robot, and a local region-of-interest solution, restricted to a small area near the shoulder. All methods are constrained to a fixed 100-second time budget. Our method achieves lower error than other Monte Carlo PDE solvers in both scenarios.

Abstract

Grid-free Monte Carlo methods are capable of solving Poisson equations on highly complex domains. However, existing methods operate solely in the primal domain and can converge slowly due to high variance. Inspired by gradient-domain rendering, we introduce a gradient-domain framework for Poisson problems. Specifically, we devise a new Monte Carlo estimator that directly targets differences of the solution between spatially varying query locations. Further, we adopt state-of-the-art reconstruction techniques originated in gradient-domain rendering to allow efficient reconstruction of the solutions without incurring additional bias. We demonstrate the effectiveness of our technique by comparing solutions obtained using our method and several state-of-the-art baselines.

BibTeX

@article{wu2026gradient,
  author = {Wu, Jiaqi and Hu, Xuejun and Zhao, Shuang and Xu, Kun},
  title = {Gradient Domain Reconstruction for Monte Carlo PDE Solvers},
  journal = {ACM Transactions on Graphics},
  volume = {45},
  number = {4},
  articleno = {130},
  numpages = {11},
  year = {2026},
  month = {July},
  doi = {10.1145/3811295},
  publisher = {Association for Computing Machinery}
}