A Macroscopic Traffic Data-Assimilation Framework Based on the Fourier–Galerkin Method and Minimax Estimation

In this paper, we propose a new framework for macroscopic traffic state estimation. Our approach is a robust “discretize” then “optimize” strategy, based on the Fourier-Galerkin projection method and minimax state estimation. We assign a Fourier-Galerkin reduced model to a macroscopic model of traffic flow, described by a hyperbolic partial differential equation. Taking into account a priori estimates for the projection error, we apply the minimax method to construct the state estimate for the reduced model that gives us, in turn, the estimate of the Fourier-Galerkin coefficients associated with a solution of the original macroscopic model. We illustrate our approach with a numerical example that demonstrates its shock capturing capability using only sparse measurements and under high uncertainty in initial conditions. We present implementation details for our algorithm, as well as a comparison of our method against the ensemble Kalman filter applied to a “local” discretization of the same traffic flow model.