Incorporating knowledge on the measurement noise in electrical impedance tomography
ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 84.

The present article is concerned with the identification of an obstacle or void of different conductivity which is included in a two-dimensional domain by measurements of voltage and currents at the boundary. In general, the voltage distribution is prescribed and hence deterministic. Whereas, the current distribution is measured and contains measurement errors. We assume that some information is given on these measurement errors which can be described by means of a random field. We exploit this extra knowledge by minimizing a linear combination of the expectation and the variance of the Kohn–Vogelius functional. It is shown how these ideas can be realized in numerical computations. By numerical results, the applicability and feasibility of our approach is demonstrated.

DOI : 10.1051/cocv/2018010
Dambrine, Marc 1 ; Harbrecht, Helmut 1 ; Puig, Benedicte 1

1 
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Dambrine, Marc; Harbrecht, Helmut; Puig, Benedicte. Incorporating knowledge on the measurement noise in electrical impedance tomography. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 84. doi : 10.1051/cocv/2018010. http://www.numdam.org/articles/10.1051/cocv/2018010/

[1] L. Afraites, M. Dambrine and D. Kateb, Conformal mapping and inverse conductivity problem with one measurement. ESAIM: COCV. 13 (2007) 163–177. | Numdam | MR | Zbl

[2] L. Afraites, M. Dambrine and D. Kateb, Shape methods for the transmission problem with a single measurement. Numer. Funct. Anal. Optim. 28 (2007) 519–551. | DOI | MR | Zbl

[3] L. Afraites, M. Dambrine and D. Kateb, On second order shape optimization methods for electrical impedance tomography. SIAM J. Control Optim. 47 (2008) 1556–1590. | DOI | MR | Zbl

[4] I. Akduman and R. Kress, Electrostatic imaging via conformal mapping. Inverse Probl. 18 (2002) 1659–1672. | DOI | MR | Zbl

[5] G. Alessandrini, V. Isakov and J. Powell, Local uniqueness in the inverse problem with one measurement. Trans. Am. Math. Soc. 347 (1995) 3031–3041. | DOI | MR | Zbl

[6] M. Badra, F. Caubet and M. Dambrine, Detecting an obstacle immersed in a fluid by shape optimization methods. Math. Models Methods Appl. Sci. 21 (2011) 2069–2101. | DOI | MR | Zbl

[7] M. Brühl, Explicit characterization of inclusions in electrical impedance tomography. SIAM J. Math. Anal. 32 (2001) 1327–1341. | DOI | MR | Zbl

[8] M. Brühl and M. Hanke, Numerical implementation of two noniterative methods for locating inclusions by impedance tomography. Inverse Probl. 16 (2000) 1029–1042. | DOI | MR | Zbl

[9] R. Chapko and R. Kress, A hybrid method for inverse boundary value problems in potential theory, J. Inv. Ill-Posed Probl. 13 (2005) 27–40. | DOI | MR | Zbl

[10] M. Dambrine, C. Dapogny and H. Harbrecht, Shape optimization for quadratic functionals and states with random right-hand sides. SIAM J. Control Optim. 53 (2015) 3081–3103. | DOI | MR | Zbl

[11] M. Dambrine, H. Harbrecht, M. Peters and B. Puig, On Bernoulli’s free boundary problem with a random boundary. Int. J. Uncertain. Quantif . 7 (2017) 335–353. | DOI | MR | Zbl

[12] M. Delfour and J.-P. Zolesio, Shapes and Geometries. SIAM, Philadelphia (2001). | MR | Zbl

[13] K. Eppler and H. Harbrecht, A regularized Newton method in electrical impedance tomography using shape Hessian information. Control Cybernet. 34 (2005) 203–225. | MR | Zbl

[14] K. Eppler and H. Harbrecht, Shape optimization for 3D electrical impedance tomography. In R. Glowinski and J. Zolesio, editors, Free and Moving Boundaries: Analysis, Simulation and Control. Vol. 252 of Lecture Notes in Pure and Applied Mathematics. Chapman & Hall/CRC, Boca Raton, FL (2007) 165–184. | DOI | MR

[15] A.V. Fiacco and G.P. Mccormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques. Wiley, New York (1968). | MR

[16] R. Fletcher, Practical Methods for Optimization. Wiley, New York (1980). | MR | Zbl

[17] A. Friedman and V. Isakov, On the uniqueness in the inverse conductivity problem with one measurement. Indiana Univ. Math. J. 38 (1989) 563–579. | DOI | MR | Zbl

[18] R. Ghanem and P. Spanos, Stochastic finite elements. A Spectral Approach. Springer, New York (1991). | DOI | MR | Zbl

[19] C. Grossmann and J. Terno, Numerik der Optimierung. B.G. Teubner, Stuttgart (1993). | MR | Zbl

[20] H. Harbrecht and R. Schneider, Wavelet Galerkin schemes for 2D-BEM. Problems and Methods in Mathematical Physics (Chemnitz, 1999). Edited by J. Elschner, et al. In Vol. 121 of Operator Theory: Advances and Applications. Birkhäuser, Basel (2001) 221–260. | MR | Zbl

[21] F. Hettlich and W. Rundell, The determination of a discontinuity in a conductivity from a single boundary measurement, Inverse Probl. 14 (1998) 67–82. | DOI | MR | Zbl

[22] A. Henrot and M. Pierre, Shape Variation and Optimization. In Vol. 28 of Tracts in Mathematics. European Mathematical Society (2017). | MR

[23] G. Hsiao and W. Wendland, A finite element method for some equations of first kind. J. Math. Anal. Appl. 58 (1977) 449–481. | DOI | MR | Zbl

[24] R. Kohn and M. Vogelius, Determining conductivity by boundary measurements. Comm. Pure Appl. Math 37 (1984) 289–298. | DOI | MR | Zbl

[25] R. Kress, Inverse Dirichlet problem and conformal mapping. Math. Comp. Simul. 66 (2004) 255–265. | DOI | MR | Zbl

[26] W. Lionheart, N. Polydorides and A. Borsic, Electrical impedance tomography: methods, history and applications, edited by D.S. Holder. IOP Series in Medical Physics and Biomedical Engineering. Institute of Physics Publishing (2005) 3–64.

[27] M. Loève, Probability theory. I+II. Fourth edn. In Vol. 45 of Graduate Texts in Mathematics. Springer, New York (1977). | DOI | MR | Zbl

[28] O. Pironneau, Optimal Shape Design for Elliptic Systems. Springer, New York (1984). | DOI | MR

[29] J.-R. Roche and J. Sokolowski, Numerical methods for shape identification problems. Control Cybern. 25 (1996) 867–894. | MR | Zbl

[30] J. Sokolowski and J.-P. Zolesio, Introduction to Shape Optimization. Springer, Berlin (1992). | DOI | MR | Zbl

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