Interior Control of the Poisson Equation with the Conjugate Gradient Method in OpenFOAM

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In this work we solve the optimal control problem

$\min _{u \in L^2 \left( \Omega \right)} \mathcal{J}\left( u\right) = \min _{u \in L^2 \left( \Omega \right)} \frac{1}{2} \int_{\Omega} \left( y - y_d \right) ^2 \mathrm{d} \Omega + \frac{\beta}{2} \int_{\Omega} u ^2 \mathrm{d} \Omega,$

where $u$ is the control variable, $y$ the state variable and $y_d$ a target function. The minimization problem is subject to the elliptic partial differential equation

$\begin{cases} -\Delta y = f + u & \text{in } \Omega, \\ y = 0 & \text{on } \Gamma. \end{cases}$

In order to use the conjugate gradient method, the state variable is separated in two terms as

$y = y_u + y_{f},$

where $y_u$ solves the state equation with zero Dirichlet boundary conditions,

$\begin{cases} -\Delta y_u = u & \text{in } \Omega, \\ y_u = 0 & \text{on } \Gamma, \end{cases}$

and $y_f$ is the control-free solution to the state equation,

$\begin{cases} -\Delta y_{f} = f & \text{in } \Omega, \\ y_{f} = 0 & \text{on } \Gamma. \end{cases}$

With the above separation of the state variable, the cost functional can be expressed as

$\mathcal{J} \left( u \right) = \frac{1}{2} \left( y_u + y_f - y_d, y_u + y_f - y_d \right)_{L^2\left( \Omega \right)} + \frac{\beta}{2} \left( u , u \right) _{L^2\left( \Omega \right)}.$

We define a linear operator

$\begin{align*} \Lambda: L^2\left( \Omega \right) & \rightarrow L^2\left( \Omega \right) \\ u & \rightarrow y_u \end{align*}2$

and its adjoint

$\begin{align*} \Lambda: L^2\left( \Omega \right) & \rightarrow L^2\left( \Omega \right) \\ \phi & \rightarrow \lambda \end{align*}$

with $\lambda$ solution to

$\begin{cases} - \Delta \lambda = \phi & \text{in } \Omega, \\ \lambda = 0 & \text{on } \Gamma. \end{cases}$

The directional derivative of the cost function then reads as

$\mathcal{D}_{\delta u} \mathcal{J}\left( u \right) = \left( \underbrace{ \left( \Lambda^* \Lambda + \beta I \right)}_{A_{cg}} u - \underbrace{ \Lambda^* \left( y_d - y_f \right)}_{b_{cg}}, \delta u \right) _{L^2\left( \Omega \right)}.$

After having identified $A_{cg}$ and $b_{cg}$ we can use the conjugate gradient method to reach the optimal control faster.

Getting Started

The solver must be compiled in the terminal. It is advisable to first clean previous compilations with

wclean

and then use

wmake

Prerequisites

OpenFOAM C++ library must be installed in order to compile the code.

The OpenFOAM distribution provided by the OpenFOAM Foundation was used.

Running a Case

In order to run the solver move to the case folder poissonCGAdjoinFoamCase and type in the command line

./Allprepare

poissonCGAdjointFoam

The poissonCGAdjointFoam solver has been tested in a square domain $[0, 1] \times [0, 1]$ with zero Dirichlet boundary conditions and $\beta = 10^{-3},10^{-4},10^{-5},10^{-6}$ . The target function is $y_d = xy \sin \left( \pi x \right) \sin \left( \pi y \right)$ .