Adjoint Gradients

How the derivative of a reconstruction functional with respect to the conductivity is computed efficiently, independently of the number of unknowns, and how it is represented.

Reading order.

1. The setting: PDE-Constrained Optimization, State Equation, Lagrangian Formulation, KKT Conditions, with the tools of the Rules of the Calculus of Variations.
2. The method: Adjoint State Method, Adjoint Equation, Functional Derivative of the Data Misfit, and the Adjoint Method for the Dirichlet Problem.
3. An alternative functional without adjoint: Kohn-Vogelius Functional.
4. Representation and discretisation: Gradient Representation and the Riesz Map, Discretize-then-Optimize vs Optimize-then-Discretize, Automatic Differentiation vs Adjoint Methods.
5. Practice: Gradient Testing and the Iterative Reconstruction Loop.

Next. The gradients feed the methods of Optimization; the overall picture is Anatomy of an EIT Reconstruction.

References

1. M. Hinze, R. Pinnau, M. Ulbrich, S. Ulbrich (2009). Optimization with PDE Constraints. Springer. doi:10.1007/978-1-4020-8839-1