Limits and consistency of non-local and graph approximations to the Eikonal equation

TL;DR

This paper investigates non-local approximations to the Eikonal equation, establishing existence, uniqueness, regularity, error bounds, and graph-based convergence to the local solution, advancing understanding of non-local and graph methods.

Contribution

It provides a comprehensive analysis of non-local and graph-based approximations to the Eikonal equation, including theoretical foundations, error estimates, and convergence results.

Findings

Existence and uniqueness of viscosity solutions for both local and non-local problems.

Error bounds between non-local and local solutions in continuous and discretized time.

Graph solutions converge to local solutions as graph size increases and kernel scales appropriately.

Abstract

In this paper, we study a non-local approximation of the time-dependent (local) Eikonal equation with Dirichlet-type boundary conditions, where the kernel in the non-local problem is properly scaled. Based on the theory of viscosity solutions, we prove existence and uniqueness of the viscosity solutions of both the local and non-local problems, as well as regularity properties of these solutions in time and space. We then derive error bounds between the solution to the non-local problem and that of the local one, both in continuous-time and Backward Euler time discretization. We then turn to studying continuum limits of non-local problems defined on random weighted graphs with $n$ vertices. In particular, we establish that if the kernel scale parameter decreases at an appropriate rate as $n$ grows, then almost surely, the solution of the problem on graphs converges uniformly to the viscosity solution of the local problem as the time step vanishes and the number vertices $n$ grows large.