2023/03/19 by Marianne Akian, Stéphane Gaubert, Akian, Marianne +3
Engineering · Mathematics · Physics and Astronomy · #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Model Reduction and Neural Networks #Numerical methods for differential equations #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2303.10705
openalex publication_date 2023/03/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a new numerical method to approximate the solutions of a class of stationary Hamilton-Jacobi (HJ) partial differential equations arising from minimum time optimal control problems. We rely on nested grid approximations, and look for the optimal trajectories by using the coarse grid approximations to reduce the search space in fine grids. This provides an infinitesimal version of the ``highway hierarchy'' method which has been developed to solve shortest path problems (with discrete time and discrete state). We obtain, for each level, an approximate value function on a sub-domain of the state space. We show that the sequence obtained in this way does converge to the viscosity solution of the HJ equation. Moreover, for our multi-level algorithm, if 0<γ≤ 1 is the convergence rate of the classical numerical scheme, then the number of arithmetic operations needed to obtain an error in O(ε) is in \widetildeO(ε-θ), with θ< \fracdγ, to be compared with \widetildeO(ε-d/ γ) for ordinary grid-based methods. Here d is the dimension of the problem, θ depends on d,γ and on the ``stiffness" of the value function around optimal trajectories, and the notation \widetildeO ignores logarithmic factors. In particular, in typical smooth cases, one has γ=1 and θ=(d+1)/2.