State and parameter estimation in 1-D hyperbolic PDEs based on an adjoint method

An optimal estimation method for state and distributed parameters in 1-D hyperbolic system based on adjoint method is proposed in this paper. A general form of the partial differential equations governing the dynamics of system is first introduced. In this equation, the initial condition or state variable as well as some empirical parameters are supposed to be unknown and need to be estimated. The Lagrangian multiplier method is used to connect the dynamics of the system and the cost function defined as the least square error between the simulation values and the measurements. The adjoint state method is applied to the objective functional in order to get the adjoint system and the gradients with respect to parameters and initial state. The objective functional is minimized by Broyden–Fletcher–Goldfarb–Shanno (BFGS) method. Due to the non-linearity of both direct and adjoint system, the nonlinear explicit Lax–Wendroff scheme is used to solve them numerically. The presented optimal estimation approach is validated by two illustrative examples, the first one about state and parameter estimation in a traffic flow, and the second one in an overland flow system.