# American Institute of Mathematical Sciences

October  2018, 14(4): 1579-1594. doi: 10.3934/jimo.2018022

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

 1 School of Automation, Guangdong University of Technology, Guangzhou, Guangdong, China 2 School of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang, China 3 School of Control Science and Engineering, Shandong University, Jinan, Shandong, China

* Corresponding author: Zongze Wu

Received  April 2017 Revised  October 2017 Published  January 2018

Fund Project: This work is supported by the National Natural Science Foundation of China grants (61703114,61673126,61473253) and the Open Research Project of the State Key Laboratory of Industrial Control Technology, Zhejiang University, China (ICT170288, ICT170301)

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.

Citation: Zhigang Ren, Shan Guo, Zhipeng Li, Zongze Wu. Adjoint-based parameter and state estimation in 1-D magnetohydrodynamic (MHD) flow system. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1579-1594. doi: 10.3934/jimo.2018022
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##### References:
Estimation of initial condition $u(x,0)$ (scenario 1).
Estimation of initial condition $B(x,0)$ (scenario 1).
Iteration process of cost function value (scenario 1).
Estimation of state $u(x,t)$ (scenario 1).
Estimation of state $B(x,t)$ (scenario 1).
Estimation of initial condition $u(x,0)$ (scenario 2).
Estimation of initial condition $B(x,0)$ (scenario 2).
Iteration process of cost function value (scenario 2).
Estimation of state $u(x,t)$ (scenario 2).
Estimation of state $B(x,t)$ (scenario 2).
 Algorithm 1. Gradient-based optimization procedure for solving Problem P$_0$. Step 1: Choose the initial guess $\nu$, $\nu_m$ and $u_{0}^{in}(x)$, $B_{0}^{in}(x)$. Step 2: Solve the 1-D MHD model (1) forward in time corresponding to initial guess values from $t=0$ to $t=T$, and solve the adjoint coupled PDE systems (7) backward in time corresponding to initial guess values from $t=T$ to $t=0$ to obtain $u(x, t)$, $B(x, t)$, $\lambda_1(x, t)$, $\lambda_2(x, t)$. Step 3: Compute cost functional and its gradients according to (4) and (5)-(6). Step 4: Use the gradients information obtained in $\mathbf{Step 3}$ to perform an optimality test. If $\nu$, $\nu_m$ and $u_{0}^{in}(x), B_0^{in}(x)$ are optimal, then exit; Otherwise, go to $\mathbf{Step 5}$. Step 5: Use the gradient information obtained in $\mathbf{Step 3}$ to calculate a searching direction. Step 6: Perform a line search to determine the optimal step length. Step 7: Compute the new points $\nu$, $\nu_m$ and $u_{0}^{in}(x)$, $B_{0}^{in}(x)$ and return to $\mathbf{Step 2}$.
 Algorithm 1. Gradient-based optimization procedure for solving Problem P$_0$. Step 1: Choose the initial guess $\nu$, $\nu_m$ and $u_{0}^{in}(x)$, $B_{0}^{in}(x)$. Step 2: Solve the 1-D MHD model (1) forward in time corresponding to initial guess values from $t=0$ to $t=T$, and solve the adjoint coupled PDE systems (7) backward in time corresponding to initial guess values from $t=T$ to $t=0$ to obtain $u(x, t)$, $B(x, t)$, $\lambda_1(x, t)$, $\lambda_2(x, t)$. Step 3: Compute cost functional and its gradients according to (4) and (5)-(6). Step 4: Use the gradients information obtained in $\mathbf{Step 3}$ to perform an optimality test. If $\nu$, $\nu_m$ and $u_{0}^{in}(x), B_0^{in}(x)$ are optimal, then exit; Otherwise, go to $\mathbf{Step 5}$. Step 5: Use the gradient information obtained in $\mathbf{Step 3}$ to calculate a searching direction. Step 6: Perform a line search to determine the optimal step length. Step 7: Compute the new points $\nu$, $\nu_m$ and $u_{0}^{in}(x)$, $B_{0}^{in}(x)$ and return to $\mathbf{Step 2}$.
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