Dirichlet-to-Neumann Map

For a conductivity $\gamma$, the Dirichlet-to-Neumann (DtN) map sends a boundary voltage to the boundary current it produces:

$${\Lambda }_{\gamma }:H^{1/2}(\partial \Omega )\to H^{-1/2}(\partial \Omega ),f\mapsto \gamma {\partial }_{\nu }{u}_f{|}_{\partial \Omega },$$

where $u_f$ solves the Dirichlet Problem with boundary value $f$.

It is the complete idealised measurement of the continuum model: knowing ${\Lambda }_{\gamma }$ means knowing the current response to every possible voltage pattern. Its weak definition avoids normal derivatives altogether:

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

where $u_h$ is any $H^1$ extension of $h$ (for example the solution with boundary value $h$).

Key facts, proved in Properties of the Boundary Operators:

• ${\Lambda }_{\gamma }$ is linear, bounded, self-adjoint and positive semidefinite;
• its kernel is the constants, so ${\Lambda }_{\gamma }1=0$;
• its range has zero mean, and it is inverted on mean-zero functions by the Neumann-to-Dirichlet Map;
• the dependence $\gamma \mapsto {\Lambda }_{\gamma }$ is nonlinear (see Forward Map).

Example. For $\gamma \equiv 1$ on the unit disc, ${\Lambda }_1{e}^{ik\theta }=|k|e^{ik\theta }$. The DtN map is a first-order operator: it amplifies high spatial frequencies on the boundary.

Recovering $\gamma$ from ${\Lambda }_{\gamma }$ is the Calderón Problem.

In ModularEIT.jl: forward_dirichlet.

References

1. A. P. Calderón (1980/2006). On an inverse boundary value problem. Reprinted in Comput. Appl. Math. 25(2–3), 133–138. doi:10.1590/S0101-82052006000200002
2. G. Uhlmann (2009). Electrical impedance tomography and Calderón’s problem. Inverse Problems 25(12), 123011. doi:10.1088/0266-5611/25/12/123011
3. J. L. Mueller, S. Siltanen (2012). Linear and Nonlinear Inverse Problems with Practical Applications. SIAM. doi:10.1137/1.9781611972344