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.
翻译:在本文中,我们研究了时间(本地) Eikonal 等式与Drichlet 型边界条件的非本地近似值, 即非本地问题的内核得到适当的缩放。 根据粘度解决方案理论, 我们证明本地和非本地问题的粘度解决方案的存在和独特性, 以及这些解决方案在时间和空间上的规律性特性。 然后我们从非本地问题的解决方案和本地问题的解决方案之间得出错误的界限, 包括连续时间和后向欧尔时间分解。 然后我们转而研究以随机加权图表和美元脊椎界定的非本地问题的连续限制。 特别是, 我们确定, 如果内核值参数以适当速度下降为美元增长, 那么几乎可以肯定地说, 图表上问题的解决方案随着时间步骤的消失和数额的大幅增长, 与本地问题的粘度解决方案一致。