Calderón Problem

The Calderón problem (inverse conductivity problem) asks:

Given the Dirichlet-to-Neumann Map ${\Lambda }_{\gamma }$, which is all boundary voltage–current pairs, determine the conductivity $\gamma$ in $\Omega$.

It was posed by A. P. Calderón in 1980. He also proved that the linearised problem around a constant conductivity is injective. The problem is the mathematical idealisation of Electrical Impedance Tomography: it assumes noise-free data and a continuum of electrodes.

The theory is organised around three questions, which mirror Hadamard’s well-posedness conditions:

1. Uniqueness: does ${\Lambda }_{{\gamma }_1}={\Lambda }_{{\gamma }_2}$ imply ${\gamma }_1={\gamma }_2$? See Uniqueness in the Calderón Problem and Boundary Determination.
2. Reconstruction: is there a constructive algorithm that maps ${\Lambda }_{\gamma }$ to $\gamma$? See Reconstruction Algorithms with Guarantees and the D-bar Method.
3. Stability: how strongly do errors in ${\Lambda }_{\gamma }$ affect $\gamma$? See Stability of the Calderón Problem.

The answers depend on the dimension, the regularity class of $\gamma$, and whether $\gamma$ is isotropic (see Anisotropic Conductivities). Central tools are Complex Geometrical Optics Solutions and the linearisation identity.

In practice, noise and finitely many measurements make the inverse problem severely ill-posed. It is therefore solved as a regularised optimisation problem (see Variational Regularization and Iterative Reconstruction Loop).

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