Optimal Bounds for Numerical Approximations of Infinite Horizon Problems Based on Dynamic Programming Approach
Javier de Frutos, Julia Novo · SIAM Journal on Control and Optimization · 2023
Abstract. In this paper we get error bounds for fully discrete approximations of infinite horizon problems via the dynamic programming approach. It is well known that, considering a time discretization with a positive step size [Formula: see text], an error bound of size [Formula: see text] can be proved for the difference between the value function (viscosity solution of the Hamilton–Jacobi–Bellman equation corresponding to the infinite horizon) and the value function of the discrete time problem. However, including also a spatial discretization based on elements of size [Formula: see text], an error bound of size [Formula: see text] can be found in the literature for the error between the value functions of the continuous problem and the fully discrete problem. In this paper we revise the error bound of the fully discrete method and prove, under assumptions similar to those of the time discrete case, that the error of the fully discrete case is in fact [Formula: see text], which gives first order in time and space for the method. This error bound matches the numerical experiments of many papers in the literature in which the behavior [Formula: see text] from the bound [Formula: see text] has not been observed.