Finite difference schemes for Hamilton--Jacobi equation on Wasserstein space on graphs

TL;DR

This work develops finite-difference schemes for Hamilton–Jacobi equations on the Wasserstein space over graphs with graph-structured noise, formulating a discretization that couples discrete graph geometry with Wasserstein calculus. It introduces a Euclidean-embedded meshspace and skew-symmetric difference matrices to approximate the Wasserstein gradient, and implements two schemes: a monotone explicit method with a CFL condition and a fully implicit method that is unconditionally stable. The authors prove uniform convergence of the schemes to the viscosity solution in the interior and obtain a local $L^{\infty}$ error bound of order $1/2$ on a compact density subset, leveraging a boundary barrier and a boundary monotonicity assumption. Numerical experiments corroborate the theoretical results, demonstrate first-order temporal convergence, and reveal the effects of graph-structured noise and Hamiltonians on solution dynamics, highlighting boundary extrapolation as crucial for stability and accuracy.

Abstract

This work proposes and studies numerical schemes for initial value problems of Hamilton--Jacobi equations (HJEs) with a graph individual noise on the Wasserstein space on graphs. Numerically solving such equations is particularly challenging due to the structural complexity caused by discrete geometric derivatives and logarithmic geometry. Our numerical schemes are constructed using finite difference approximations that are adapted to both the discrete geometry of graphs and the differential structure of Wasserstein spaces. To ensure numerical stability and accuracy of numerical behavior, we use extrapolation-type techniques to simulate the numerical solution on the boundary of density space. By analyzing approximation error of Wasserstein gradient of the viscosity solution, we prove the uniform convergence of the schemes to the original initial value problem, and establish an $L^{\infty}_{\mathrm{loc}}$-error estimate of order one-half. Several numerical experiments are presented to illustrate our theoretical findings and to study the effect of individual noise and Hamiltonians on graphs. To the best of our knowledge, this is the first result on numerical schemes for HJEs on the Wasserstein space with a graph structure.

Figures (5)

• Figure 1: Numerical solution for $\mathcal{H}(\xi,p)=(\sum_{i=1}^3\xi_i^{-1})^{-2}\|p\|_{\xi}^2$, $\frac{\tau}{h}=0.1,h=0.1\times 2^{-4},\epsilon=0.01,$ initial value $\mathcal{U}_0(\xi)=\|\xi\|^2_{l^2}$.
• Figure 2: (A): ${\mathcal{P}}^h_{\epsilon}(G)$ and (B): $\widetilde{\mathcal{P}}_{\epsilon}^h(G)$, $\epsilon=0.01,h=0.025.$
• Figure 3: Numerical solution for $\mathcal{H}(\xi,p)=\mathcal{I}_{0.1}^{-2}(\xi)\|p\|_{\xi}^{2},\; \mathcal{U}_0(\xi)=\min\{\xi_1,\xi_2,\xi_3\}\cos(\|\xi\|^2_{l^2}), \tau/h=0.1,h=0.1\times 2^{-3},\epsilon=0.001$.
• Figure 4: Numerical solution for $\mathcal{H}(\xi,p)= \mathcal{I}_{1}^{-2}(\xi)\|p\|^2_{\xi},\;\mathcal{U}_0(\xi)=-\min(\xi_i)\cos(\sum_{i=1}^3\xi^2_i),h=0.0125,\epsilon=0.001,\frac{\tau}{h}=0.05,T=2.$
• Figure 5: Numerical solution for $\mathcal{H}(\xi,p)=\mathfrak a(\xi)\|p\|_{\xi}^2$, $\frac{\tau}{h}=0.1,h=0.0125,\epsilon=0.001,$ (A): initial value $\mathcal{U}_0(\xi)=\|\xi\|^2_{l^2}$, and (B)--(D): $T=5$.

Theorems & Definitions (29)

• Example 1
• Example 2
• Proposition 3.1
• Lemma 4.1
• Example 3
• Example 4
• Lemma 5.1
• Proposition 5.2
• proof
• Remark 5.3