Thank you for your quick response!
Firstly, let me clarify that I am solving an unsteady problem. Please consider the following discretised domain:
I define the control variable to be the time dependent Dirichlet BC at the right end. I then parametrise the control vector as the value of the BC at each time step: m = \left[u_{R_{1}}, u_{R_{2}}, \dots, u_{R_{N}}\right]^{T}, where N is the number of time steps. For simplicity, I want u_{R_{k}} to be constant across the boundary (i.e., the value of u at all the DOFs associated to \partial\Omega_{c} is the same), so that at each time step I just have one single parameter.
Therefore, as you say, I first solve the direct problem by forward integration in time, and then I solve the adjoint problem by backward integration in time. During the solution of the adjoint problem, I compute the gradient as you stated:
for k = Num\_steps : 0
\hspace{15pt} \left(\nabla J\right)_{k} = -\left(\partial F/\partial m_{k}\right)^{\ast} \cdot \lambda_{k}
end
If we focus on one single time step, I need to compute \left(\partial F/\partial m_{k}\right)^{\ast} \cdot \lambda_{k} as you suggest:
assemble(-action(adjoint(derivative(F, u_R)), lmbda))
But note that m_{k} is a scalar, so \left(\partial F/\partial m_{k}\right)^{\ast} is a vector and after all the time integration I will have a gradient vector whose size is equal to the number of time steps (which is consistent with my problem).
Consequently, my problem is that, in order to use derivative(F, u_R), I need to define u_R as a dolfin function, which is actually a vector defined at all the DOFs of the mesh. Then, derivative(F, u_R) is a matrix whose columns contain the partial derivative of F with respect to each u_R defined at every DOF of the mesh. However, I just need the column associated to the control boundary.
Is there a way of isolating the column of derivative(F, u_R) associated to the Boundary \partial\Omega_{c}?
Please, let me know if something is unclear. Also, I can supply the code if you want.
Thank you very much again!