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A Framework for Solving Parabolic Partial Differential Equations on Discrete Domains

Leticia Mattos Da Silva, Oded Stein, Justin Solomon

TL;DR

This work develops a framework for solving a broad class of second-order parabolic PDE on triangle mesh surfaces, including Hamilton–Jacobi and Fokker–Planck equations, by coupling Strang splitting with a convex optimization substep to handle nonlinear terms. The method enables stable, implicit-like integration without large nonlinear solves, and introduces a per-vertex spatial discretization compatible with curved meshes. It shows concrete benefits in entropy-regularized optimal transport, Wasserstein barycenters, measure interpolation, and graphically relevant PDEs like the G-equation, while delivering robust behavior across meshes and time steps. The approach advances geometry processing by providing a practical, stable solver for nonlinear/parabolic PDE on discrete geometries with broad applications in diffusion, front propagation, and stochastic-density evolution.

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.

A Framework for Solving Parabolic Partial Differential Equations on Discrete Domains

TL;DR

This work develops a framework for solving a broad class of second-order parabolic PDE on triangle mesh surfaces, including Hamilton–Jacobi and Fokker–Planck equations, by coupling Strang splitting with a convex optimization substep to handle nonlinear terms. The method enables stable, implicit-like integration without large nonlinear solves, and introduces a per-vertex spatial discretization compatible with curved meshes. It shows concrete benefits in entropy-regularized optimal transport, Wasserstein barycenters, measure interpolation, and graphically relevant PDEs like the G-equation, while delivering robust behavior across meshes and time steps. The approach advances geometry processing by providing a practical, stable solver for nonlinear/parabolic PDE on discrete geometries with broad applications in diffusion, front propagation, and stochastic-density evolution.

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.
Paper Structure (37 sections, 8 theorems, 25 equations, 16 figures, 2 tables)

This paper contains 37 sections, 8 theorems, 25 equations, 16 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. PDE-driven geometry processing
  4. Second-order Parabolic PDE
  5. Heat diffusion and entropy-regularized optimal transport
  6. $G$-equation
  7. Fokker--Planck equation
  8. Convex Optimization for Regularized Geodesics
  9. Mathematical Preliminaries
  10. General Preliminaries
  11. Example PDE
  12. Nonlinear diffusion
  13. G-equation
  14. Fokker-Planck equation
  15. Viscosity Solutions
  16. ...and 22 more sections

Key Result

proposition 1

Suppose $v(x,t)$ is such that $v(x,t)>0$ for all $(x,t)\in\mathcal{M}\times[0,\infty)$ and $v$ satisfies the heat equation: Define $u(x,t):=\log v(x,t)$. Then, $u(x,t)$ satisfies (eq:logheat). Moreover, if $v$ satisfies Dirichlet or Neumann boundary conditions, then $u$ satisfies the same boundary conditions up to applying $\log$. In sum, $u$ satisfies the boundary conditions satisfied by $v$ up

Figures (16)

  • Figure 1: Wasserstein barycenters (green) between two distributions on the unit line (grey) with $1{,}000$ elements for varying pairs of weights. Our method obtains these barycenters for a tiny amount of entropy, i.e., $\gamma=10^{-7}$, whereas the method in solomon2015convolutional fails.
  • Figure 2: Level-sets of the $G$-equation under cellular flow obtained via our method with varying values of viscosity: $\varepsilon=0, 10^{-4},10^{-3},$ and $10^{-2}$ (from top to bottom, respectively). See Figure \ref{['fig:types-flow']} for an illustration of cellular flow.
  • Figure 3: Three typical examples of vector fields $\Phi$ used when evolving second-order parabolic PDE that involve terms with vector fields.
  • Figure 4: Time evolution of the Fokker-Planck equation \ref{['eq:fokker-planck']} on a $100\times100$ triangle grid, obtained using our method, under constant drift (top), shear flow (middle), and no drift (bottom). See Figure \ref{['fig:types-flow']} for an illustration of constant drift and shear flow.
  • Figure 5: A two dimensional illustration of a viscosity supersolution $\varphi$.
  • ...and 11 more figures

Theorems & Definitions (11)

  • proposition 1
  • Remark
  • definition 1: Viscosity subsolutions, and resp., supersolutions
  • definition 2: Viscosity solution
  • theorem 1: Continuously differentiable viscosity solution
  • theorem 2: Comparison principle
  • theorem 3
  • corollary 1
  • theorem 4
  • lemma 1
  • ...and 1 more