Adjoint Method for the Dirichlet Problem

In voltage-driven EIT, voltages $f$ are prescribed on the boundary (or on the electrodes) and the currents are measured. The data are samples of the Dirichlet-to-Neumann Map ${\Lambda }_{\sigma }$. The least-squares misfit is

$$J(\sigma )=\frac{1}{2}\sum _s\Vert {\Lambda }_{\sigma }{f}_s-g_s{\Vert }^2.$$

Derivative of the DtN map. Let $u_f$, $u_v$ solve the Dirichlet Problem with data $f$ and $v$. Testing the equation for $u_f$ with $u_v$ gives $⟨v,{\Lambda }_{\sigma }f⟩={\int }_{\Omega }\sigma \nabla u_f\cdot \nabla u_v\mathrm{d}x$. Differentiate in $\sigma$: the derivatives of $u_f$ and $u_v$ vanish on the boundary, and both fields satisfy the equation, so their contributions drop out. What remains is

$$⟨v,{\Lambda }_{\sigma }^{\prime }[\delta \sigma ]f⟩={\int }_{\Omega }\delta \sigma \nabla u_f\cdot \nabla u_v\mathrm{d}x.$$

For the Neumann-to-Dirichlet Map the sign is opposite: $⟨h,{\mathcal{R}}_{\sigma }^{\prime }[\delta \sigma ]g⟩=-\int \delta \sigma \nabla u_g\cdot \nabla u_h$. More conductivity means more current for a given voltage, and less voltage for a given current.

Gradient. Take $v=r_s={\Lambda }_{\sigma }{f}_s-g_s$. Then

$$\nabla J=\sum _s\nabla u_{f_s}\cdot \nabla {\lambda }_s,{\lambda }_s=\text{Dirichlet solution with boundary data }{r}_s.$$

The adjoint is another Dirichlet problem with the residual as boundary data. Its operator is the same as the state’s, so one factorisation serves both.

Discrete version. With the free and Dirichlet degrees of freedom $F$, $B$ (see Discrete Electrode Models), the discrete DtN map is the Schur complement $S(\sigma )=A_{BB}-A_{BF}{A}_{FF}^{-1}{A}_{FB}$. The identity ${\mathbf{v}}^{\top }S\mathbf{f}={\tilde{\mathbf{x}}}_{\mathbf{v}}^{\top }A{\mathbf{x}}_{\mathbf{f}}$ holds for the discrete harmonic extensions ${\mathbf{x}}_{\mathbf{f}}$, ${\tilde{\mathbf{x}}}_{\mathbf{v}}$. The same stationarity argument then gives

$$\frac{\partial }{\partial {\sigma }_a}({\mathbf{v}}^{\top }S(\sigma )\mathbf{f})={\tilde{\mathbf{x}}}_{\mathbf{v}}^{\top }\frac{\partial A}{\partial {\sigma }_a}{\mathbf{x}}_{\mathbf{f}},$$

a contraction with the Conductivity Tensor. If the currents are compared in another representation $C^{-1}{E}^{\top }(A\mathbf{x}{)}_B$, the adjoint boundary data become $C^{-1}\mathbf{r}$. For a weighted misfit with $W=U^{\top }U$, they become $C^{-1}{U}^{\top }\mathbf{r}$.

Jacobian. Replacing $\mathbf{r}$ by the unit vectors of the measurements gives one Dirichlet solve per measurement. These are independent of the pattern $s$, so a single block solve gives the Jacobian for all patterns.

In ModularEIT.jl: AdjointStateObjective.

References

1. L. Borcea (2002). Electrical impedance tomography. Inverse Problems 18(6), R99–R136. doi:10.1088/0266-5611/18/6/201
2. M. Hinze, R. Pinnau, M. Ulbrich, S. Ulbrich (2009). Optimization with PDE Constraints. Springer. doi:10.1007/978-1-4020-8839-1
3. J. L. Mueller, S. Siltanen (2012). Linear and Nonlinear Inverse Problems with Practical Applications. SIAM. doi:10.1137/1.9781611972344