Dirichlet and Thomson Principles

Both forward problems have energy characterisations.

Dirichlet principle. The solution $u_f$ of the Dirichlet Problem minimises the dissipated power among all potentials with the prescribed boundary voltage:

$$⟨f,{\Lambda }_{\gamma }f⟩=\underset{v\in H^1,v{|}_{\partial \Omega }=f}{\min }{\int }_{\Omega }\gamma |\nabla v{|}^2\mathrm{d}x={\int }_{\Omega }\gamma |\nabla u_f{|}^2\mathrm{d}x.$$

Thomson principle. The current density $\mathbf{J}=-\gamma \nabla u_g$ of the Neumann Problem minimises the dissipated power among all divergence-free current fields with the prescribed boundary flux:

$$⟨g,{\Lambda }_{\gamma }^{-1}g⟩=\underset{\begin{array}{c}\nabla \cdot \mathbf{k}=0\\ -\mathbf{k}\cdot \nu =g\end{array}}{\min }{\int }_{\Omega }{\gamma }^{-1}|\mathbf{k}{|}^2\mathrm{d}x={\int }_{\Omega }\gamma |\nabla u_g{|}^2\mathrm{d}x.$$

Consequences.

• Monotonicity: since the integrands increase in $\gamma$ (Dirichlet) or in ${\gamma }^{-1}$ (Thomson), larger conductivity means larger ${\Lambda }_{\gamma }$ and smaller ${\Lambda }_{\gamma }^{-1}$ (see Properties of the Boundary Operators).
• Derivatives: by the envelope theorem, only the explicit $\gamma$-dependence survives differentiation. The derivative of $⟨f,{\Lambda }_{\gamma }f⟩$ in direction $\delta \gamma$ is $\int \delta \gamma |\nabla u_f{|}^2$, and that of $⟨g,{\Lambda }_{\gamma }^{-1}g⟩$ is $-\int \delta \gamma |\nabla u_g{|}^2$. These formulas give the gradient of the Kohn-Vogelius Functional without any adjoint solve.

References

1. L. Borcea (2002). Electrical impedance tomography. Inverse Problems 18(6), R99–R136. doi:10.1088/0266-5611/18/6/201
2. R. V. Kohn, M. Vogelius (1987). Relaxation of a variational method for impedance computed tomography. Comm. Pure Appl. Math. 40(6), 745–777. doi:10.1002/cpa.3160400605