Optimal parameter and state estimation in 1D magnetohydrodynamic flow system

In this paper, we consider an optimal parameter and state estimation problem arising in an one-dimensional (1D) magnetohydrodynamic (MHD) flow system, whose dynamics can be modeled by a coupled partial differential equations (PDEs). In this model, the coefficients of the Reynolds number and initial conditions as well as state variables are supposed to be unknown and need to be estimated. An adjoint-based optimization method is employed to estimate the unknown coefficients and states arising in the flow model. We first employ the Lagrange multiplier method to connect the dynamics of the 1D MHD system and the cost functional defined as the least square errors between the measurements in the experiment and the numerical simulation values. Then, we use the adjoint-based method to the augmented Lagrangian cost functional to get an adjoint coupled PDEs system. The exact gradients of the defined cost functional with respect to these unknown parameters and states are derived. Numerical simulation results are illustrated to verify the effectiveness of our adjoint-based estimation approach.