# Walk on Stars: A Grid-Free Monte Carlo Method for PDEs with Neumann Boundary Conditions

arXiv: https://arxiv.org/abs/2302.11815 · HTML: https://mlsanity.com/paper/2302.11815/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Feb 2023.

## Built on by (most cited first, 8 of 8)

- [Solving Poisson Equations using Neural Walk-on-Spheres](https://mlsanity.com/paper/2406.03494/) (2024)
- [Robust Biharmonic Skinning Using Geometric Fields](https://mlsanity.com/paper/2406.00238/) (2024)
- [Neural Control Variates with Automatic Integration](https://mlsanity.com/paper/2409.15394/) (2024)
- [Operator Learning Using Weak Supervision from Walk-on-Spheres](https://mlsanity.com/paper/2603.01193/) (2026)
- [Walk-on-Interfaces: A Monte Carlo Estimator for an Elliptic Interface Problem with Nonhomogeneous Flux Jump Conditions and a Neumann Boundary Condition](https://mlsanity.com/paper/2508.16767/) (2025)
- [HeatMat: Simulation of City Material Impact on Urban Heat Island Effect](https://mlsanity.com/paper/2601.22796/) (2026)
- [MC2: Monte Carlo Correction for Fast Elliptic PDE Solving](https://mlsanity.com/paper/2605.09288/) (2026)
- [Neural Harmonic Measure Operator](https://mlsanity.com/paper/2609.35752/) (2026)

## Abstract (arXiv)

Grid-free Monte Carlo methods based on the walk on spheres (WoS) algorithm solve fundamental partial differential equations (PDEs) like the Poisson equation without discretizing the problem domain or approximating functions in a finite basis. Such methods hence avoid aliasing in the solution, and evade the many challenges of mesh generation. Yet for problems with complex geometry, practical grid-free methods have been largely limited to basic Dirichlet boundary conditions. We introduce the walk on stars (WoSt) algorithm, which solves linear elliptic PDEs with arbitrary mixed Neumann and Dirichlet boundary conditions. The key insight is that one can efficiently simulate reflecting Brownian motion (which models Neumann conditions) by replacing the balls used by WoS with star-shaped domains. We identify such domains via the closest point on the visibility silhouette, by simply augmenting a standard bounding volume hierarchy with normal information. Overall, WoSt is an easy modification of WoS, and retains the many attractive features of grid-free Monte Carlo methods such as progressive and view-dependent evaluation, trivial parallelization, and sublinear scaling to increasing geometric detail.