Convergence and Error Estimates of A Semi-Lagrangian scheme for the Minimum Time Problem

TL;DR

This paper analyzes a semi-Lagrangian scheme for the minimum time problem, establishing convergence rates of order 1 and applying results to fast-marching methods with state constraints.

Contribution

It provides new convergence rate results for a semi-Lagrangian scheme using both deterministic and stochastic control interpretations.

Findings

Convergence rate of order 1 in time for the scheme.

Convergence rate of order 1 in time and space for the full discretization.

Application to analyze the complexity of fast-marching methods.

Abstract

We consider a semi-Lagrangian scheme for solving the minimum time problem, with a given target, and the associated eikonal type equation. We first use a discrete time deterministic optimal control problem interpretation of the time discretization scheme, and show that the discrete time value function is semiconcave under regularity assumptions on the dynamics and the boundary of target set. We establish a convergence rate of order $1$ in terms of time step based on this semiconcavity property. Then, we use a discrete time stochastic optimal control interpretation of the full discretization scheme, and we establish a convergence rate of order $1$ in terms of both time and spatial steps using certain interpolation operators, under further regularity assumptions. We extend our convergence results to problems with particular state constraints. We apply our results to analyze the convergence rate and computational complexity of the fast-marching method. We also consider the multi-level fast-marching method recently introduced by the authors.