Adjoint-based parameter and state estimation in 1-D magnetohydrodynamic (MHD) flow system

Abstract

In this paper, an adjoint-based optimization method is employed to estimate the unknown coefficients and states arising in an one-dimensional (1-D) magnetohydrodynamic (MHD) flow, whose dynamics can be modeled by a coupled partial differential equations (PDEs) under some suitable assumptions. 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. We first employ the Lagrange multiplier method to connect the dynamics of the 1-D 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, and the exact gradients of the defined cost functional with respect to these unknown parameters and initial states are further derived. The existed gradient-based optimization technique such as sequential quadratic programming (SQP) is employed for minimizing the cost functional in the optimization process. Finally, we illustrate the numerical examples to verify the effectiveness of our adjoint-based estimation approach.