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Generalized linearization techniques in electrical impedance tomography

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Abstract

Electrical impedance tomography aims at reconstructing the interior electrical conductivity from surface measurements of currents and voltages. As the current–voltage pairs depend nonlinearly on the conductivity, impedance tomography leads to a nonlinear inverse problem. Often, the forward problem is linearized with respect to the conductivity and the resulting linear inverse problem is regarded as a subproblem in an iterative algorithm or as a simple reconstruction method as such. In this paper, we compare this basic linearization approach to linearizations with respect to the resistivity or the logarithm of the conductivity. It is numerically demonstrated that the conductivity linearization often results in compromised accuracy in both forward and inverse computations. Inspired by these observations, we present and analyze a new linearization technique which is based on the logarithm of the Neumann-to-Dirichlet operator. The method is directly applicable to discrete settings, including the complete electrode model. We also consider Fréchet derivatives of the logarithmic operators. Numerical examples indicate that the proposed method is an accurate way of linearizing the problem of electrical impedance tomography.

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Acknowledgements

This work was supported by the Academy of Finland (Decision 267789), the Finnish Foundation for Technology Promotion TES, and the Foundation for Aalto University Science and Technology.

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Correspondence to Nuutti Hyvönen.

An equivalent norm for \(H^s_\diamond (\partial \varOmega )\)

An equivalent norm for \(H^s_\diamond (\partial \varOmega )\)

This appendix is based on the assumptions and definitions of Sects. 2 and 3. The proof of Corollary 1 requires the following lemma. Observe that the lemma could be extended (with obvious modifications) for \(-1/2 \le s \le 1/2\) by duality and for a wider scale of smoothness indices by utilizing some integer power of \(\varLambda (\sigma )^{-1}\) in place of \(\varLambda (\sigma )^{-1/2}\) in the following proof, assuming \(\partial \varOmega \) is smooth enough.

Lemma 1

For any fixed \(\sigma \in L^\infty _+(\varOmega )\), it holds that

$$\begin{aligned} H^{s}_{\diamond }(\partial \varOmega ) = \big \{ g \in L^2_\diamond (\partial \varOmega ) : \Vert g\Vert _{{s}} < \infty \big \}, \quad 0 \le s \le \frac{1}{2}, \end{aligned}$$

where

$$\begin{aligned} \Vert g\Vert _{s} := \left( \sum _{k=1}^\infty \frac{1}{\lambda _k^{2s}} |(g,\phi _k)|^2 \right) ^{1/2} \end{aligned}$$

defines an equivalent norm for \(H^{s}_{\diamond }(\partial \varOmega )\).

Proof

Let us introduce the positive powers of \(\varLambda (\sigma ) :L^2_\diamond (\partial \varOmega ) \rightarrow L^2_\diamond (\partial \varOmega )\) in the natural way, that is,

$$\begin{aligned} \varLambda ^s(\sigma ) :f \mapsto \sum _{k=1}^\infty \lambda _k^{s} (f,\phi _k) \, \phi _k, \quad L^2_\diamond (\partial \varOmega ) \rightarrow L^2_\diamond (\partial \varOmega ), \end{aligned}$$

which defines a compact, injective, self-adjoint operator with a dense range for any \(s > 0\). The negative powers are the corresponding inverse operators:

$$\begin{aligned} \varLambda ^{-s}(\sigma ) :f \mapsto \sum _{k=1}^\infty \frac{1}{\lambda _k^{s}} (f,\phi _k) \, \phi _k, \quad {\mathcal {D}}\big (\varLambda ^{-s}(\sigma )\big ) \rightarrow L^2_\diamond (\partial \varOmega ), \quad s > 0, \end{aligned}$$

where (cf., e.g., [10, Theorem 2.8])

$$\begin{aligned} {\mathcal {D}}\big (\varLambda ^{-s}(\sigma )\big ) := {\mathcal {R}}\big (\varLambda ^{s}(\sigma )\big ) = \left\{ g \in L^2_\diamond (\partial \varOmega ) \ : \ \Vert \varLambda ^{-s}(\sigma )g\Vert _{L^2(\partial \varOmega )} = \Vert g \Vert _{s} < \infty \right\} . \end{aligned}$$

Using the same arguments as for \(\log \!\varLambda (\sigma )\) in Sect. 3.1, it is easy to show that \(\varLambda ^{-s}(\sigma ) :{\mathcal {D}}(\varLambda ^{-s}(\sigma )) \rightarrow L^2_\diamond (\partial \varOmega )\) is self-adjoint and becomes an isomorphism between Hilbert spaces if its domain is equipped with the graph norm defined via

$$\begin{aligned} \Vert g \Vert _{s} \le \Vert g \Vert _{{\mathcal {G}}(\varLambda ^{-s}(\sigma ))} := \Big (\Vert g \Vert _{L^2(\partial \varOmega )}^2 + \Vert \varLambda ^{-s} (\sigma ) g \Vert _{L^2(\partial \varOmega )}^2\Big )^{1/2} \le C \Vert g \Vert _{s} \end{aligned}$$
(34)

for \(g \in {\mathcal {D}}(\varLambda ^{-s}(\sigma ))\) and with \(C>0\).

Since \(\varLambda (\sigma ) :H^{-1/2}_\diamond (\partial \varOmega ) \rightarrow H^{1/2}_\diamond (\partial \varOmega )\) is positive and self-adjoint, the (positive) square root \(\varLambda ^{1/2}(\sigma )\) is, in fact, an isomorphism from \(L^2_\diamond (\partial \varOmega )\) to \(H^{1/2}_\diamond (\partial \varOmega )\); see, e.g., [2, Lemma 3.4] for a simple proof. In particular, \({\mathcal {D}}(\varLambda ^{-1/2}(\sigma )) = H^{1/2}_\diamond (\partial \varOmega )\). According to [28, p. 10, Definition 2.1 and Remark 2.3] and the definition of \(H^{s}(\partial \varOmega )\) as a (complex) interpolation space (see, e.g., [28, p. 36, Theorem 7.7]), we thus have

$$\begin{aligned} H^{s}_\diamond (\partial \varOmega )= & {} \left[ H^{1/2}_\diamond (\partial \varOmega ), L^2_{\diamond }(\partial \varOmega ) \right] _{1-2s} = {\mathcal {D}}\big (\varLambda ^{(1-(1-2s))/2}(\sigma )\big ) = {\mathcal {D}}\big ( \varLambda ^{-s}(\sigma )\big ),\\&0 \le s \le 1/2, \end{aligned}$$

with the graph norm of \({\mathcal {D}}( \varLambda ^{-s}(\sigma ))\) being equivalent to that of \(H^{s}_\diamond (\partial \varOmega )\). The claim now follows from (34).\(\square \)

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Hyvönen, N., Mustonen, L. Generalized linearization techniques in electrical impedance tomography. Numer. Math. 140, 95–120 (2018). https://doi.org/10.1007/s00211-018-0959-1

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