The Inverse Problem

Can the conductivity be recovered from boundary measurements, how stably, and by which methods with guarantees? This chapter collects the theory behind every reconstruction.

Reading order.

1. The question: Calderón Problem and Well-Posedness.
2. Uniqueness and stability: Uniqueness in the Calderón Problem, Boundary Determination, Stability of the Calderón Problem, and the limits for Anisotropic Conductivities.
3. What the data contain: Decay of Boundary Measurements and Linearized EIT and the Sensitivity Kernel.
4. Constructive methods: Complex Geometrical Optics Solutions, the D-bar Method and other Reconstruction Algorithms with Guarantees.
5. The statistical view: Bayesian Inversion.

Next. Stabilising the reconstruction is the task of Regularization; the practical iterative approach is outlined in Anatomy of an EIT Reconstruction.

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