A Framework for Solving Parabolic Partial Differential Equations on Discrete Domains

A Framework for Solving Parabolic Partial Differential Equations on Discrete Domains ACM Transactions on Graphics

LETICIA MATTOS DA SILVA¹, ODED STEIN², JUSTIN SOLOMON¹

¹Massachusetts Institute of Technology, USA; ²University of Southern California, USA

teaser — We show the time evolution of the Fokker-Planck equation under a certain flow on the discrete sphere with 81,920 triangle faces and 40,962 vertices, obtained using our framework. Our method can be used for any equation in a class of second-order parabolic PDE on triangle mesh surfaces.

Abstract

We introduce a framework for solving a class of parabolic partial differential equations on triangle mesh surfaces, including the Hamilton-Jacobi equation and the Fokker-Planck equation. PDE in this class often have nonlinear or stiff terms that cannot be resolved with standard methods on curved triangle meshes. To address this challenge, we leverage a splitting integrator combined with a convex optimization step to solve these PDE. Our machinery can be used to compute entropic approximation of optimal transport distances on geometric domains, overcoming the numerical limitations of the state-of-the-art method. In addition, we demonstrate the versatility of our method on a number of linear and nonlinear PDE that appear in diffusion and front propagation tasks in geometry processing.

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BibTeX

@article{10.1145/3666087,
          author = {Mattos Da Silva, Leticia and Stein, Oded and Solomon, Justin},
          title = {A Framework for Solving Parabolic Partial Differential Equations on Discrete Domains},
          year = {2024},
          issue_date = {October 2024},
          publisher = {Association for Computing Machinery},
          address = {New York, NY, USA},
          volume = {43},
          number = {5},
          issn = {0730-0301},
          url = {https://doi.org/10.1145/3666087},
          doi = {10.1145/3666087},
          journal = {ACM Trans. Graph.},
          month = jun,
          articleno = {156},
          numpages = {14}
      }

Acknowledgements

Leticia Mattos Da Silva acknowledges the generous support of the Schwarzman College of Computing Fellowship funded by Google Inc. and the MathWorks Fellowship. Oded Stein was supported by the Swiss National Science Foundation’s Early Postdoc.Mobility Fellowship. Justin Solomon acknowledges the generous support of Army Research Office grants W911NF2010168 and W911NF2110293, of Air Force Office of Scientific Research award FA9550-19-1-031, of National Science Foundation grant CHS-1955697, from the CSAIL Systems that Learn program, from the MIT–IBM Watson AI Laboratory, from the Toyota–CSAIL Joint Research Center, from a gift from Adobe Systems, and from a Google Research Scholar award.