Finite Difference Schemes for Hamilton–Jacobi Equation on the Wasserstein Space on Graphs

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 viscosity 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. Furthermore, we propose a new compatible boundary assumption on a compact subset of density space, and establish an [Formula: see text]-error estimate of order one-half. To the best of our knowledge, this is the first result on numerical schemes for HJEs on the Wasserstein space with a graph structure.