Forward Map

The forward map of EIT takes a conductivity to the boundary operator it generates:

$$\mathcal{F}:\gamma \mapsto {\Lambda }_{\gamma }(\text{or }\gamma \mapsto {\mathcal{R}}_{\gamma }={\Lambda }_{\gamma }^{-1}).$$

For a finite set of current patterns $g_1,\dots ,g_N$ it becomes the finite-dimensional map $\gamma \mapsto ({\mathcal{R}}_{\gamma }{g}_i{)}_{i=1}^N$ into voltages.

Nonlinearity. $\mathcal{F}$ is positively homogeneous of degree one, ${\Lambda }_{c\gamma }=c{\Lambda }_{\gamma }$, but not additive. In general ${\Lambda }_{{\gamma }_1+{\gamma }_2}\ne {\Lambda }_{{\gamma }_1}+{\Lambda }_{{\gamma }_2}$, because the potential $u$ itself depends on $\gamma$. So $\mathcal{F}$ is nonlinear. It is, however, smooth (Fréchet differentiable, even analytic) on the set of conductivities bounded away from zero. Its derivative is described in Linearized EIT and the Sensitivity Kernel.

Smoothing. Small, fine-scale or deep changes in $\gamma$ produce very small changes in the boundary data. This is the source of the severe ill-posedness of the inverse direction (see Stability of the Calderón Problem).

Evaluation. Numerically, $\mathcal{F}$ is evaluated by solving one Neumann Problem (or Dirichlet Problem) per current pattern, usually with the finite element method. All patterns share the same system matrix $L_{\gamma }$ (see Weighted Stiffness Matrix), so they can be solved together as a block system (see Block Krylov Methods).

Inverting $\mathcal{F}$ is the Calderón Problem.

In ModularEIT.jl: ForwardModel, forward_neumann, forward_dirichlet.

References

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