Limits of non-local approximations to the Eikonal equation on manifolds

TL;DR

This paper investigates the accuracy of non-local approximations to the Eikonal equation on manifolds, establishing convergence and error bounds, and applying results to graph-based problems.

Contribution

It provides a rigorous analysis of non-local Eikonal approximations on manifolds, including well-posedness, regularity, error estimates, and convergence on random graphs.

Findings

Non-local and local Eikonal problems are well-posed as viscosity solutions.

Error bounds are derived for the approximation under proper kernel scaling.

Solutions on graphs converge almost surely to the continuum solution as parameters vary.

Abstract

In this paper, we consider a non-local approximation of the time-dependent Eikonal equation defined on a Riemannian manifold. We show that the local and the non-local problems are well-posed in the sense of viscosity solutions and we prove regularity properties of these solutions in time and space. If the kernel is properly scaled, we then derive error bounds between the solution to the non-local problem and the one to the local problem, both in continuous-time and Forward Euler discretization. Finally, we apply these results to a sequence of random weighted graphs with n vertices. In particular, we establish that the solution to the problem on graphs converges almost surely uniformly to the viscosity solution of the local problem as the kernel scale parameter decreases at an appropriate rate when the number of vertices grows and the time step vanishes.