I'm working on an Electrical Impedance Tomography (or EIT) problem i.e. I wish to reconstruct an image based on information obtained at the boundary. The problem consists of the generalised LaPlace equation with Neumann boundaries i.e. \begin{align} \nabla \sigma \nabla u & = 0, \quad \Omega \\ \sigma \frac{\partial u}{\partial \nu} & = g, \quad \partial \Omega. \end{align} From the solution $u$ one could compute the boundary values $$f = u|_{\partial \Omega},$$ which would correspond to the voltage (g is the current density). The ultimate goal is to determine the conductivity inside $\Omega$ i.e. to determine $\sigma$. The problem I consider is on a square [-1,1] x [-1,1] domain with a circle of radius $R$ placed at the origin. The circle will then represent some object inside the domain that we wish to reconstruct from the $f$'s obtained at the boundary. I have used the following conductivities (explained below) \begin{align} & \sigma_0 = 1 \quad in \; \Omega, \\ & \sigma_{true} = \left\{ \begin{array}{cc} 1 \quad for \; (x,y) > R, \\ 2 \quad for \; (x,y) \leq R \end{array} \right. \end{align}
Normally one would get some data (i.e. f) and one could use this for the reconstruction, but in my case no data was given, so I started by solving the forward problem for the true conductivity $\sigma_{true}$ to obtain an $f_{true}$. Similarly I computed the forward problem for a constant conductivity $\sigma_0$ (by this I mean that the conductivity is the same in the entire domain as stated above). This was done with a 5 point stencil i.e. I have only solved the problem numerically and have not looked at analytical solutions.
I then did this for a wide variety of boundary conditions (g's). I choose sine functions, which on the boundary added up to 0 (such that the total current density is zero). In this way I get an $f_{j, true}$ and an $f_{j,0}$ for let us say $j = 1,\dotsc,N$ different boundaries.
The actual problem -- the inverse conductivity reconstruction -- has given me quite some headaches. I was given the following inverse problem by the TA
\begin{equation} \langle f_{j,true}-f_{j,0}, g \rangle \approx - \int_\Omega (\sigma_{true}-\sigma_0) |\nabla u_{j,0}|^2 ds, \end{equation} where $u_{j,0}$ is the solution obtained for the $\sigma_0$ problem. I have tried applying a quadrature rule to the rhs thus arriving at an inverse problem, solving for $(\sigma_{true}-\sigma_0)$ with the kernel $K = |\nabla u_{j,0}|^2$ and rhs $\langle f_{j,true}-f_{j,0}, g \rangle$.
I have three main concerns:
1) I don't understand how the inner product on the lhs should be calculated. Is it just the dot product?
2) Is it correct that the term $|\nabla u_{j,0}|^2$ is given as $u_x^2+u_y^2$ and that I could just approximate this using e.g. a finite difference method?
3) How did the TA actually arrive at this equation? I can't seem to get the same.
Sorry if it is all a bit confusing - I guess I just am confused about the inverse problem and how to actually obtain $(\sigma_{true}-\sigma_0)$ from the data that I have generated.