Well-Posedness

Following Hadamard (1902; see Engl, Hanke & Neubauer, Ch. 1), a problem $F(x)=y$ is well-posed if

1. a solution exists for every admissible $y$ (existence);
2. the solution is unique (uniqueness);
3. the solution depends continuously on $y$ (stability).

A problem that violates any of these is ill-posed. Inverse problems are typically ill-posed because the forward operator $F$ smooths: it is compact, or at least its inverse is unbounded.

For EIT:

• Existence fails for noisy data. A measured noisy operator is generally not the DtN map of any conductivity.
• Uniqueness holds for isotropic conductivities under mild regularity assumptions (see Uniqueness in the Calderón Problem). It fails for anisotropic ones (see Anisotropic Conductivities).
• Stability is only logarithmic (see Stability of the Calderón Problem). EIT is therefore called severely (exponentially) ill-posed. Mildly ill-posed problems, such as the Radon transform of CT, have algebraic decay of singular values.

For linear problems, the degree of ill-posedness is measured by the decay rate of the singular values of $F$. Unless the problem is restricted to a finite-dimensional or compact set, regularization is needed to obtain stable approximate solutions (see Variational Regularization).

References

1. H. W. Engl, M. Hanke, A. Neubauer (1996). Regularization of Inverse Problems. Kluwer. doi:10.1007/978-94-009-1740-8
2. J. L. Mueller, S. Siltanen (2012). Linear and Nonlinear Inverse Problems with Practical Applications. SIAM. doi:10.1137/1.9781611972344