Neumann-to-Dirichlet Map

Real EIT devices inject currents and measure voltages. The matching operator is the Neumann-to-Dirichlet (NtD) map

$${\mathcal{R}}_{\gamma }={\Lambda }_{\gamma }^{-1}:H_{\diamond }^{-1/2}(\partial \Omega )\to H_{\diamond }^{1/2}(\partial \Omega ),g\mapsto u_g{|}_{\partial \Omega },$$

where $u_g$ solves the Neumann Problem with current $g$ and the grounding ${\int }_{\partial \Omega }{u}_g\mathrm{d}s=0$. The subscript $\diamond$ denotes zero-mean functions:

• the domain is restricted to zero-mean currents because of the compatibility condition $\int g=0$;
• the range is made unique by grounding, which removes the additive constant.

On these spaces the NtD map is the inverse of the Dirichlet-to-Neumann Map.

Properties (see Properties of the Boundary Operators):

• linear, self-adjoint, positive definite on $H_{\diamond }^{-1/2}$;
• compact as an operator on $L_{\diamond }^2(\partial \Omega )$: it smooths. For $\gamma \equiv 1$ on the unit disc, ${\mathcal{R}}_1{e}^{ik\theta }=|k{|}^{-1}{e}^{ik\theta }$ for $k\ne 0$, so high-frequency current patterns produce small voltages (see Current Patterns);
• weak form: $⟨g,{\mathcal{R}}_{\gamma }h⟩={\int }_{\Omega }\gamma \nabla u_g\cdot \nabla u_h\mathrm{d}x$.

With finitely many current patterns $G=[g_1,\dots ,g_N]$ and measured voltages $F=[f_1,\dots ,f_N]$, a discrete NtD matrix is estimated as in Discrete Boundary Operator.

In ModularEIT.jl: forward_neumann.

References

1. J. L. Mueller, S. Siltanen (2012). Linear and Nonlinear Inverse Problems with Practical Applications. SIAM. doi:10.1137/1.9781611972344
2. L. Borcea (2002). Electrical impedance tomography. Inverse Problems 18(6), R99–R136. doi:10.1088/0266-5611/18/6/201