State Equation

Varying the weak Lagrangian

$$\mathcal{L}(\sigma ,u,\lambda )=J(u)-{\int }_{\Omega }\sigma \nabla u\cdot \nabla \lambda +{\int }_{\Omega }h\lambda +{\int }_{\partial \Omega }g\lambda$$

in the multiplier, $\lambda \to \lambda +\epsilon \delta \lambda$, and letting $\epsilon \to 0$:

$${\partial }_{\lambda }\mathcal{L}[\delta \lambda ]=-{\int }_{\Omega }\sigma \nabla u\cdot \nabla \delta \lambda \mathrm{d}x+{\int }_{\Omega }h\delta \lambda \mathrm{d}x+{\int }_{\partial \Omega }g\delta \lambda \mathrm{d}s=0\forall \delta \lambda \in V_0.$$

This is exactly the weak forward problem. The state equation is the forward problem.

Recovering the strong form (two-stage argument).

1. Test with $\delta \lambda \in C_c^{\infty }(\Omega )$, which vanishes near the boundary, and integrate by parts. This gives $\nabla \cdot (\sigma \nabla u)+h=0$ in $\Omega$.
2. Subtract the interior identity and allow $\delta \lambda$ with arbitrary trace. This gives the natural boundary condition $\sigma {\partial }_{\nu }u=g$ on $\partial \Omega$ (Neumann case). In the Dirichlet case the condition $u=f$ is built into the trial space.

Testing with $\delta \lambda \equiv 1$ gives the compatibility condition ${\int }_{\Omega }h+{\int }_{\partial \Omega }g=0$.

Solving the state equations for all Current Patterns already gives the objective value $J(u(\sigma ))$. The gradient additionally needs the Adjoint Equation.

In ModularEIT.jl: forward_neumann, 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. L. C. Evans (2010). Partial Differential Equations, 2nd ed. AMS GSM 19. doi:10.1090/gsm/019