Kohn-Vogelius Functional

Instead of comparing boundary voltages, the Kohn–Vogelius functional compares two interior fields driven by the same measured Cauchy pair $(f,g)$:

• $u_f$ solves the Dirichlet Problem with $u_f=f$ on $\partial \Omega$;
• $u_g$ solves the Neumann Problem with $\sigma {\partial }_{\nu }{u}_g=g$ on $\partial \Omega$.

If $\sigma$ is the true conductivity, $u_f=u_g$ up to a constant. Otherwise their difference measures the inconsistency everywhere in $\Omega$:

$$J_{\mathrm{KV}}(\sigma )=\frac{1}{2}{\int }_{\Omega }\sigma |\nabla u_f-\nabla u_g{|}^2\mathrm{d}x.$$

It depends only on $\nabla (u_f-u_g)$, so the undetermined constant in $u_g$ drops out.

Energy identity. With the Dirichlet and Thomson Principles and the measured power $P={\int }_{\partial \Omega }fg\mathrm{d}s$ (independent of $\sigma$):

$$2J_{\mathrm{KV}}(\sigma )=⟨f,{\Lambda }_{\sigma }f⟩+⟨g,{\Lambda }_{\sigma }^{-1}g⟩-2P.$$

The cross term follows from testing the Neumann weak form with $v=u_f$: $\int \sigma \nabla u_f\cdot \nabla u_g={\int }_{\partial \Omega }gf=P$.

Gradient without adjoints. Both energies are minima over fields constrained only by the data, so by the envelope theorem only the explicit $\sigma$-dependence contributes:

$${\nabla }_{\sigma }{J}_{\mathrm{KV}}=\frac{1}{2}(|\nabla u_f{|}^2-|\nabla u_g{|}^2).$$

One iteration costs two forward solves (one Dirichlet, one Neumann) and no adjoint solve. At the true conductivity, the misfit corresponds to a data fit in the natural energy ($H^{1/2}$-type) norm rather than in $L^2(\partial \Omega )$.

Discrete version. With the current-driven state ${\mathbf{x}}_N$ ($A{\mathbf{x}}_N=PI$) and the voltage-driven state ${\mathbf{x}}_D$ (measured voltages prescribed, see Discrete Electrode Models),

$$J_{\mathrm{KV}}=\frac{1}{2}\sum _s({\mathbf{x}}_{N,s}-{\mathbf{x}}_{D,s}{)}^{\top }A(\sigma )({\mathbf{x}}_{N,s}-{\mathbf{x}}_{D,s}),\frac{\partial J_{\mathrm{KV}}}{\partial {\sigma }_a}=\frac{1}{2}\sum _s({\mathbf{x}}_{D,s}^{\top }\frac{\partial A}{\partial {\sigma }_a}{\mathbf{x}}_{D,s}-{\mathbf{x}}_{N,s}^{\top }\frac{\partial A}{\partial {\sigma }_a}{\mathbf{x}}_{N,s}),$$

two contractions with the Conductivity Tensor. The cross term ${\mathbf{x}}_N^{\top }A{\mathbf{x}}_D=(PI{)}^{\top }{\mathbf{x}}_D$ depends only on the data if the voltage-driven problem prescribes the measured voltages exactly where the current enters. This holds for the continuum model, point electrodes and the Complete Electrode Model with voltages on all electrodes, and then $J_{\mathrm{KV}}$ vanishes at the true conductivity. For the Gap Model, the voltage-driven counterpart is the Shunt Model, a different model, so $J_{\mathrm{KV}}>0$ even for exact data.

Relaxation and regularisation. Minimising sequences of the unregularised functional can oscillate finer and finer. Kohn and Vogelius studied its relaxation (homogenisation), which leads to anisotropic, non-unique limits (see Anisotropic Conductivities). In practice the unrelaxed functional is used with bounds ${\sigma }_{\min }\le \sigma \le {\sigma }_{\max }$ and an explicit regulariser. Kohn and McKenney reported that early termination has a desirable smoothing effect.

In ModularEIT.jl: KohnVogeliusObjective.

References

1. 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
2. R. V. Kohn, A. McKenney (1990). Numerical implementation of a variational method for electrical impedance tomography. Inverse Problems 6(3), 389–414. doi:10.1088/0266-5611/6/3/009
3. L. Borcea (2002). Electrical impedance tomography. Inverse Problems 18(6), R99–R136, Sec. 7.2. doi:10.1088/0266-5611/18/6/201
4. L. Borcea (2003). Addendum to “Electrical impedance tomography”. Inverse Problems 19(4), 997–998. doi:10.1088/0266-5611/19/4/501