partially constructing Force matrix from vector & optimization

[DanielBoigk]

Let's say I have a Bilinearform like this:

a(u, v) = ∫( γ * ∇(v) ⋅ ∇(u) )dΩ

As one can see the Forcematrix $K$ of this linear PDE is assembled from some function $\gamma$.
End goals:

• I want to be able to update the sparse matrix that is assembled via assemble_matrix_and_vector given that $\gamma$ changes only on some points. Want to construct it from a vector $\gamma$ prefereably not from a function.
• With this I want to solve an inverse Problem i.e. infer the underlying $\gamma$ with (partially) known $u$ or knowledge about the Dirichlet-to-Neumann map.
  For example via Regularization, something like:

$$ \underset{\gamma}{min} (\sum\limits_{i} || K_\gamma u_{i} - f_{i} || + \lambda \mathcal{R}(\gamma)) $$

for some data $(u_i,f_i)$ and some Regularizer $\mathcal{R}$.

Is there any hope of making such a map $\gamma \mapsto K_\gamma$ differentiable?

(I want to solve EIT (Electrical Impedance Tomography) right now. I could assemble everything myself but I find the weakform notation of Gridap convenient since it could allow one to easily assemble everything for another problem with other parameters.)
Sorry, that my question is not so specific right now. But I'm digging through your source code right now trying to get Gridap to serve my purpose.

• Is there anything of this sort already implemented
• Or am I expected to hit some roadblocks that I should know about before hand?
• (Why are certain functions not exported?)