Reconstruction Algorithms with Guarantees

Some methods come with a mathematical guarantee that they recover $\gamma$ from ${\Lambda }_{\gamma }$, under assumptions. This is unlike generic optimisation, which only finds local minima.

• Nachman (1988), $n\ge 3$. A constructive procedure based on Complex Geometrical Optics Solutions. A boundary integral equation gives the traces of the CGO solutions, these give the scattering transform of $q=\Delta \sqrt{\gamma }/\sqrt{\gamma }$, and from that $q$ and then $\gamma$ are recovered. Novikov (1988) obtained related results independently.
• Nachman (1996), $n=2$. A constructive proof for $\gamma \in W^{2,p}$ through a $\bar{\partial }$ (“D-bar”) equation in the complex spectral parameter. This became the practical D-bar Method. Siltanen, Mueller and Isaacson (2000) gave the first numerical implementation. Knudsen, Lassas, Mueller and Siltanen (2009) proved that a truncated version is a regularisation strategy for noisy data.
• Linearised and one-step methods. Calderón’s linearisation, NOSER, and series-reversion approaches expand the Forward Map around a reference conductivity. Garde, Hyvönen and Kuutela (2023) derived series reversion with local convergence guarantees, including modelling errors.
• Monotonicity methods recover the shape of inclusions with guarantees, using the order properties of ${\Lambda }_{\gamma }$ (see Properties of the Boundary Operators).

Iterative, regularised, optimisation-based methods are covered in Iterative Reconstruction Loop. They have no global guarantee but are flexible about electrodes, noise and prior information.

References

1. A. I. Nachman (1988). Reconstructions From Boundary Measurements. Ann. of Math. 128(3), 531–576. doi:10.2307/1971435
2. R. G. Novikov (1988). Multidimensional inverse spectral problem for the equation −Δψ + (v(x) − Eu(x))ψ = 0. Funct. Anal. Appl. 22(4), 263–272. doi:10.1007/BF01077418
3. A. I. Nachman (1996). Global Uniqueness for a Two-Dimensional Inverse Boundary Value Problem. Ann. of Math. 143(1), 71–96. doi:10.2307/2118653
4. S. Siltanen, J. Mueller, D. Isaacson (2000). An implementation of the reconstruction algorithm of A Nachman for the 2D inverse conductivity problem. Inverse Problems 16(3), 681–699. doi:10.1088/0266-5611/16/3/310
5. K. Knudsen, M. Lassas, J. L. Mueller, S. Siltanen (2009). Regularized D-bar method for the inverse conductivity problem. Inverse Probl. Imaging 3(4), 599–624. doi:10.3934/ipi.2009.3.599
6. H. Garde, N. Hyvönen, T. Kuutela (2023). Series reversion for electrical impedance tomography with modeling errors. Inverse Problems 39(8), 085007. doi:10.1088/1361-6420/acdab8
7. B. Harrach, M. Ullrich (2013). Monotonicity-based shape reconstruction in electrical impedance tomography. SIAM J. Math. Anal. 45(6), 3382–3403. doi:10.1137/120886984