PDE-Constrained Optimization

EIT reconstruction is a PDE-constrained optimisation problem:

$$\underset{\sigma \in \Sigma ,u}{\min }J(u)+\beta \mathcal{R}(\sigma )\text{subject to}e(\sigma ,u)=0,$$

where $e(\sigma ,u)=0$ is the (weak) Conductivity Equation with its boundary condition. $\sigma$ is the control or parameter and $u$ the state. The admissible set

$$\Sigma =\{\sigma \in L^{\infty }(\Omega ):{\sigma }_{\min }\le \sigma \le {\sigma }_{\max }\text{a.e.}\},0<{\sigma }_{\min }\le {\sigma }_{\max }<\infty ,$$

guarantees a unique state for every $\sigma$ (see Lax-Milgram Theorem).

Two viewpoints.

• All-at-once: treat $(\sigma ,u)$ as joint unknowns and enforce $e=0$ through Lagrange multipliers (see Lagrangian Formulation and KKT Conditions).
• Reduced (black-box): eliminate the state through the control-to-state map $u=u(\sigma )$ and minimise the reduced functional $\hat{J}(\sigma )=J(u(\sigma ))+\beta \mathcal{R}(\sigma )$ over $\sigma$ alone.

The Adjoint State Method computes the gradient of the reduced functional at the cost of one extra linear solve per state, independent of the number of parameters. That is what makes pixel- or element-wise conductivity reconstructions feasible.

Discretisation can happen before or after deriving optimality conditions (see Discretize-then-Optimize vs Optimize-then-Discretize).

References

1. M. Hinze, R. Pinnau, M. Ulbrich, S. Ulbrich (2009). Optimization with PDE Constraints. Springer. doi:10.1007/978-1-4020-8839-1
2. F. Tröltzsch (2010). Optimal Control of Partial Differential Equations. AMS GSM 112. doi:10.1090/gsm/112