Functional Derivative of the Data Misfit

Varying the weak Lagrangian in the conductivity, $\sigma \to \sigma +\epsilon \delta \sigma$:

$${\partial }_{\sigma }\mathcal{L}[\delta \sigma ]=-{\int }_{\Omega }\delta \sigma \nabla u\cdot \nabla \lambda \mathrm{d}x.$$

$\sigma$ enters the weak Lagrangian linearly and in a single term, so this is exact. No integration by parts is needed, no boundary terms arise, and no assumption on $\delta \sigma {|}_{\partial \Omega }$ is required.

Evaluated at the solutions $u$ of the State Equation and $\lambda$ of the Adjoint Equation, this is the derivative of the reduced functional (see Adjoint State Method). Its $L^2(\Omega )$ representative is

$$\boxed{{\nabla }_{\sigma }J(u(\sigma ))=-\nabla u\cdot \nabla \lambda \text{a.e. in }\Omega .}$$

For several current patterns, sum over the patterns: $\nabla J=-{\sum }_i\nabla u_i\cdot \nabla {\lambda }_i$. Each term is the sensitivity kernel contracted with the residual.

Discrete version. With ${\sigma }_h={\sum }_a{z}_a{\psi }_a$ the derivative with respect to the coefficients is

$$\frac{\partial J}{\partial z_a}=-{\int }_{\Omega }{\psi }_a\nabla u_h\cdot \nabla {\lambda }_h\mathrm{d}x,$$

assembled cell by cell with quadrature (see Numerical Quadrature and Assembly). For $P_0$ conductivities this is simply $-{\int }_{K_a}\nabla u_h\cdot \nabla {\lambda }_h$. The vector is a covector. Turning it into a function in the $\sigma$-space requires the Riesz map, for example the L2 Projection with the mass matrix (see Gradient Representation and the Riesz Map).

Adding a regulariser. If the objective contains $\beta \mathcal{R}(\sigma )$, the gradient becomes $\beta \nabla \mathcal{R}(\sigma )-{\sum }_i\nabla u_i\cdot \nabla {\lambda }_i$. The state and adjoint equations do not change.

In ModularEIT.jl: value_and_gradient!, tensor_gradient!.

References

1. M. Hinze, R. Pinnau, M. Ulbrich, S. Ulbrich (2009). Optimization with PDE Constraints. Springer. doi:10.1007/978-1-4020-8839-1
2. R.-E. Plessix (2006). A review of the adjoint-state method for computing the gradient of a functional with geophysical applications. Geophys. J. Int. 167(2), 495–503. doi:10.1111/j.1365-246X.2006.02978.x
3. F. J. Margotti (2015). On Inexact Newton Methods for Inverse Problems in Banach Spaces. PhD thesis, KIT. doi:10.5445/IR/1000048606