2024/07/09 by Marianne Akian, Shanqing Liu, Akian, Marianne +1
Decision Sciences · Economics, Econometrics and Finance · #FOS: Mathematics #Optimization and Control (math.OC) #Risk and Portfolio Optimization #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2407.06969
openalex publication_date 2024/07/09 · openalex created_date 2024/07/11 · openalex updated_date 2026/07/28
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.