Solving partial differential equations in participating media

We consider the problem of solving partial differential equations (PDEs) in domains with complex microparticle geometry that is impractical, or intractable, to model explicitly. Drawing inspiration from volume rendering, we propose tackling this problem by treating the domain as a participating medium that models microparticle geometry stochastically, through aggregate statistical properties (e.g., particle density). We first introduce the problem setting of PDE simulation in participating media. We then specialize to exponential media and describe the properties that make them an attractive model of microparticle geometry for PDE simulation problems. We use these properties to develop two new algorithms, volumetric walk on spheres and volumetric walk on stars, that generalize previous Monte Carlo algorithms to enable efficient and discretization-free simulation of linear elliptic PDEs (e.g., Laplace) in participating media. We demonstrate experimentally that our algorithms can solve Laplace boundary value problems with complex microparticle geometry more accurately and more efficiently than previous approaches, such as ensemble averaging and homogenization.

@article{miller2025_2506.08237,
  title={ Solving partial differential equations in participating media },
  author={ Bailey Miller and Rohan Sawhney and Keenan Crane and Ioannis Gkioulekas },
  journal={arXiv preprint arXiv:2506.08237},
  year={ 2025 }
}