Adjoint Equation

Varying the weak Lagrangian in the state, $u\to u+\epsilon \delta u$ with $\delta u\in V_0$:

$${\partial }_u\mathcal{L}[\delta u]=J^{\prime }(u)[\delta u]-{\int }_{\Omega }\sigma \nabla \delta u\cdot \nabla \lambda \mathrm{d}x=0\forall \delta u\in V_0.$$

Split the derivative of the misfit into interior and boundary Riesz representers,

$$J^{\prime }(u)[\delta u]={\int }_{\Omega }({\partial }_{u}J{)}_{\Omega }\delta u\mathrm{d}x+{\int }_{\partial \Omega }({\partial }_{u}J{)}_{\partial \Omega }\delta u\mathrm{d}s.$$

Weak adjoint equation. Find $\lambda \in V_0$ with

$${\int }_{\Omega }\sigma \nabla \lambda \cdot \nabla v\mathrm{d}x={\int }_{\Omega }({\partial }_{u}J{)}_{\Omega }v\mathrm{d}x+{\int }_{\partial \Omega }({\partial }_{u}J{)}_{\partial \Omega }v\mathrm{d}s\forall v\in V_0.$$

It has the same bilinear form as the state equation, because the conductivity operator is self-adjoint. So the same matrix $L_{\sigma }$, factorisation or preconditioner is reused, and only the right-hand side changes.

Strong form (integrating by parts and using the two-stage argument as in State Equation):

$$\nabla \cdot (\sigma \nabla \lambda )=-({\partial }_{u}J{)}_{\Omega }\text{in }\Omega ,\sigma {\partial }_{\nu }\lambda =({\partial }_{u}J{)}_{\partial \Omega }\text{on }\partial \Omega .$$

Standard EIT misfit. For $J(u)=\Vert u-f{\Vert }_{L^2(\partial \Omega )}^2$ (no interior term):

$$\nabla \cdot (\sigma \nabla \lambda )=0\text{in }\Omega ,\sigma {\partial }_{\nu }\lambda =2(u-f)\text{on }\partial \Omega .$$

The adjoint is itself an EIT forward problem, driven by the voltage residual as boundary current. For a Neumann adjoint the compatibility condition requires the right-hand side to have zero mean. This holds when both $u$ and $f$ are grounded to zero boundary mean. $\lambda$ is determined up to a constant, which does not affect $\nabla \lambda$.

Sign convention. With the Lagrangian written as $J+\int \sigma \nabla u\cdot \nabla \lambda -\dots$ instead, $\lambda$ changes sign, and so does the formula for the gradient. The product $-\nabla u\cdot \nabla \lambda$ in Functional Derivative of the Data Misfit belongs to the convention used here.

Dirichlet case. $V_0=H_0^1$, so $\lambda =0$ on $\partial \Omega$. The measured quantity is the current, and the misfit enters through the flux instead.

In ModularEIT.jl: AdjointStateObjective.

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. D. Lahaye, W. Mulckhuyse (2012). Adjoint sensitivity in PDE constrained least squares problems as a multiphysics problem. COMPEL 31(3), 895–903. doi:10.1108/03321641211209780
3. F. J. Margotti (2015). On Inexact Newton Methods for Inverse Problems in Banach Spaces. PhD thesis, KIT, Sec. 5.2.2. doi:10.5445/IR/1000048606