Refined stability estimates in electrical impedance tomography with multi-layer structure

Haigang Li; Jenn-Nan Wang; Ling Wang

doi:10.3934/ipi.2021048

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2022, Volume 16, Issue 1: 229-249. Doi: 10.3934/ipi.2021048

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Refined stability estimates in electrical impedance tomography with multi-layer structure

1.
School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing Normal University, Beijing 100875, China
2.
Institute of Applied Mathematical Sciences, National Taiwan University, Taipei 106, Taiwan
3.
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

^* Corresponding author: Jenn-Nan Wang
^* Corresponding author: Jenn-Nan Wang

Received: December 31, 2020

Revised: April 30, 2021

Early access: July 2021

Published: February 2022

H.G. Li was partially supported by BJNSF (1202013) and NSFC (11631002, 11971061). J.N. Wang was supported in part by MOST 108-2115-M-002-002-MY3 and 109-2115-M-002-001-MY3

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Abstract

In this paper we study the inverse problem of determining an electrical inclusion in a multi-layer composite from boundary measurements in 2D. We assume the conductivities in different layers are different and derive a stability estimate for the linearized map with explicit formulae on the conductivity and the thickness of each layer. Intuitively, if an inclusion is surrounded by a highly conductive layer, then, in view of "the principle of the least work", the current will take a path in the highly conductive layer and disregard the existence of the inclusion. Consequently, a worse stability of identifying the hidden inclusion is expected in this case. Our estimates indeed show that the ill-posedness of the problem increases as long as the conductivity of some layer becomes large. This work is an extension of the previous result by Nagayasu-Uhlmann-Wang[15], where a depth-dependent estimate is derived when an inclusion is deeply hidden in a conductor. Estimates in this work also show the influence of the depth of the inclusion.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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