Walk-on-Spheres (WoS)

<figure> <p align="center"> <img src="./media/teaser_small.png", width="768px"> <br><em>Credit: <a href="https://www.cs.cmu.edu/~kmcrane/">Keenan Crane - CMU School of Computer Science</a></em> </p> </figure>

This is a Taichi Lang implementation of the Walk-on-Spheres (WoS) algorithm and its variants for solving PDEs without discretizing the problem domain using Monte-Carlo estimators. This project aims to implement recent methods proposed to solve various PDEs arising in geometry processing.

Get Started

Clone the repository and setup a Python environment with the required dependencies:

git clone https://github.com/DveloperY0115/wos.git
cd wos
conda env create -f environment_{linux|macos}.yml
conda activate wos
export PYTHONPATH=.

Check the installation by running simple examples that solve the Poisson's equation over a 2-sphere or a triangular mesh with a Dirichlet boundary condition:

python scripts/wos_solver/heat_sphere.py --out-dir {OUTPUT DIRECTORY} --use-gui
python scripts/wos_solver/heat_sphere.py --out-dir {OUTPUT DIRECTORY} --use-gui --eqn-type poisson
python scripts/wos_solver/heat_mesh.py --mesh-path data/spot_unit_cube.obj --out-dir {OUTPUT DIRECTORY} --use-gui

When executed, the scripts should display the following visualizations: the first panel shows the solution defined over a 2-sphere, while the second panel shows the impact of a heat source near the center of the sphere. The last panel shows the solution with a boundary condition defined over a triangular mesh. You may disable the GUI by removing the --use-gui flag and save the results to the specified output directory.

<table> <tr> <td> <figure> <img src="./media/sphere_heat_boundary.gif", width="252px"> </figure> </td> <td> <figure> <img src="./media/sphere_heat_source.gif", width="252px"> </figure> </td> <td> <figure> <img src="./media/mesh_heat_boundary.gif", width="252px"> </figure> </td> </tr> </table>

TODOs

• [ ] Implement the solver for spatially varying coefficient introduced in Sawhney et al., 2022, ACM ToG 2022
• [ ] Fix numerically unstable math functions (e.g., NaNs in Harmonic Green Functions and Yukawa Potentials, etc)
• [ ] Fix bugs in the Bounding Volume Hierarchy (BVH) on GPUs when certain meshes are used (e.g., CUDA, Metal, etc)
• [ ] Design a data structure for triangular meshes encapsulating:
  • Geometry with acceleration structures (e.g., BVH)
  • Boundary conditions
  • (Optional) Textures and materials for visualization

Readings

This project is greatly inspired by the following papers:

• Monte Carlo Geometry Processing: A Grid-Free Approach to PDE-Based Methods on Volumetric Domains, ACM ToG 2020
• Grid-Free Monte Carlo for PDEs with Spatially Varying Coefficients, ACM ToG 2022
• Boundary Value Caching for Walk on Spheres, ACM ToG 2023
• Walk on Stars: A Grid-Free Monte Carlo Method for PDEs with Neumann Boundary Conditions, ACM ToG 2023
• Walkin’ Robin: Walk on Stars with Robin Boundary Conditions, ACM ToG 2024