Stability of the Calderón Problem

Question. If ${\Lambda }_{{\gamma }_1}$ and ${\Lambda }_{{\gamma }_2}$ are close, are ${\gamma }_1$ and ${\gamma }_2$ close?

We look for an estimate of the form

$$\Vert {\gamma }_1-{\gamma }_2{\Vert }_{L^{\infty }(\Omega )}\le \omega (\Vert {\Lambda }_{{\gamma }_1}-{\Lambda }_{{\gamma }_2}{\Vert }_{H^{1/2}\to H^{-1/2}})$$

with a modulus of continuity $\omega (t)\to 0$ as $t\to 0$.

Logarithmic stability. Alessandrini (1988) proved, for $n\ge 3$ and conductivities satisfying a priori bounds (for example in $H^s$, $s>n/2+2$), that

$$\omega (t)=C|\log t{|}^{-\delta }(t<1)$$

for some $\delta \in (0,1)$. Mandache (2001) showed that this rate cannot be improved in general. Halving the reconstruction error requires the data error to shrink to about its $2^{1/\delta }$-th power. This is why EIT is called severely ill-posed (see Well-Posedness).

Better stability under stronger priors. If $\gamma$ is known to lie in a finite-dimensional set, for example piecewise constant on a known partition into finitely many cells, then the estimate is Lipschitz: $\Vert {\gamma }_1-{\gamma }_2\Vert \le C\Vert {\Lambda }_{{\gamma }_1}-{\Lambda }_{{\gamma }_2}\Vert$ (Alessandrini–Vessella 2005). The constant $C$, however, grows exponentially with the number of unknowns. This is the theoretical case for strong prior information, both hand-crafted and learned (see Variational Regularization and Learned Regularization).

Practical meaning. Features deep inside the domain, or small ones, produce boundary signals that are exponentially small. With finite measurement precision, no algorithm can resolve them without prior assumptions.

References

1. G. Alessandrini (1988). Stable determination of conductivity by boundary measurements. Applicable Analysis 27(1–3), 153–172. doi:10.1080/00036818808839730
2. N. Mandache (2001). Exponential instability in an inverse problem for the Schrödinger equation. Inverse Problems 17(5), 1435–1444. doi:10.1088/0266-5611/17/5/313
3. G. Alessandrini, S. Vessella (2005). Lipschitz stability for the inverse conductivity problem. Adv. Appl. Math. 35(2), 207–241. doi:10.1016/j.aam.2004.12.002
4. J. L. Mueller, S. Siltanen (2012). Linear and Nonlinear Inverse Problems with Practical Applications. SIAM. doi:10.1137/1.9781611972344