Linearized EIT and the Sensitivity Kernel

The Forward Map $\gamma \mapsto {\Lambda }_{\gamma }$ is Fréchet differentiable. Its derivative in direction $\delta \gamma$ is given by the linearisation identity

$$⟨{\Lambda }_{\gamma }^{\prime }[\delta \gamma ]f,h⟩={\int }_{\Omega }\delta \gamma \nabla u_f\cdot \nabla u_h\mathrm{d}x,$$

where $u_f,u_h$ solve the Dirichlet Problem with data $f,h$. For the Neumann-to-Dirichlet Map the sign flips:

$$⟨g,{\mathcal{R}}_{\gamma }^{\prime }[\delta \gamma ]k⟩=-{\int }_{\Omega }\delta \gamma \nabla u_g\cdot \nabla u_k\mathrm{d}x.$$

The product $\nabla u_g\cdot \nabla u_k$ is the sensitivity kernel. It tells how much the measurement “drive with $k$, read out with $g$” reacts to a local change of conductivity. It is large near the electrodes and decays towards the interior, which is another view of the ill-posedness.

Derivation of the first identity. Let $\dot{u}$ be the derivative of $u_f$ with respect to $\gamma$. Differentiating $\nabla \cdot (\gamma \nabla u_f)=0$ gives $\nabla \cdot (\gamma \nabla \dot{u})=-\nabla \cdot (\delta \gamma \nabla u_f)$ with $\dot{u}{|}_{\partial \Omega }=0$. Testing with $u_h$ and using Green’s Identities yields the formula.

Uses.

• Calderón proved injectivity of this linear map at $\gamma \equiv$ const, the first uniqueness result.
• One-step linear reconstruction methods (NOSER, GREIT; see EIT Software and Solvers) solve a regularised version of ${\Lambda }_{\gamma }^{\prime }[\delta \gamma ]\approx {\Lambda }_{\text{meas}}-{\Lambda }_{{\gamma }_0}$.
• The same product $\nabla u\cdot \nabla \lambda$ appears as the gradient in the Adjoint State Method. There $\lambda$ plays the role of the second field, driven by the data residual.
• Stacking the kernels for all pattern pairs gives the Jacobian used in the Gauss-Newton Method.

In ModularEIT.jl: residual_and_jacobian!.

References

1. A. P. Calderón (1980/2006). On an inverse boundary value problem. Comput. Appl. Math. 25(2–3), 133–138. doi:10.1590/S0101-82052006000200002
2. W. R. B. Lionheart (2004). EIT reconstruction algorithms: pitfalls, challenges and recent developments. Physiol. Meas. 25(1), 125–142. doi:10.1088/0967-3334/25/1/021
3. L. Borcea (2002). Electrical impedance tomography. Inverse Problems 18(6), R99–R136. doi:10.1088/0266-5611/18/6/201