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Monotonicity-Based Reconstruction of Extreme Inclusions in Electrical Impedance Tomography
SIAM Journal on Mathematical Analysis ( IF 2 ) Pub Date : 2020-12-15 , DOI: 10.1137/19m1299219 Valentina Candiani , Jérémi Dardé , Henrik Garde , Nuutti Hyvönen
SIAM Journal on Mathematical Analysis ( IF 2 ) Pub Date : 2020-12-15 , DOI: 10.1137/19m1299219 Valentina Candiani , Jérémi Dardé , Henrik Garde , Nuutti Hyvönen
SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 6234-6259, January 2020.
The monotonicity-based approach has become one of the fundamental methods for reconstructing inclusions in the inverse problem of electrical impedance tomography. Thus far, the method has not been proven to be able to handle extreme inclusions that correspond to some parts of the studied domain becoming either perfectly conducting or perfectly insulating. The main obstacle has arguably been establishing suitable monotonicity principles for the corresponding Neumann-to-Dirichlet boundary maps. In this work, we tackle this shortcoming by first giving a convergence result in the operator norm for the Neumann-to-Dirichlet map when the conductivity coefficient decays to zero and/or grows to infinity in some given parts of the domain. This allows passing the necessary monotonicity principles to the limiting case. Subsequently, we show how the monotonicity method generalizes to not only the definite case of reconstructing either perfectly conducting or perfectly insulating inclusions but also the indefinite case where the perturbed conductivity can take any values between, and including, zero and infinity.
中文翻译:
基于单调性的电阻抗层析成像中极端夹杂物的重建
SIAM数学分析杂志,第52卷,第6期,第6234-6259页,2020年1月。
基于单调性的方法已成为重构电阻抗层析成像反问题中夹杂物的基本方法之一。迄今为止,尚未证明该方法能够处理极端夹杂物,这些夹杂物对应于所研究领域的某些部分变得完全导电或完全绝缘。可以说,主要障碍是为相应的Neumann-to-Dirichlet边界图建立合适的单调性原则。在这项工作中,我们首先在电导率系数衰减为零和/或增长到无穷大时,首先在Neumann-to-Dirichlet映射的算子范数中给出收敛结果,从而解决了该缺陷。这允许将必要的单调性原理传递给极限情况。后来,
更新日期:2020-12-16
The monotonicity-based approach has become one of the fundamental methods for reconstructing inclusions in the inverse problem of electrical impedance tomography. Thus far, the method has not been proven to be able to handle extreme inclusions that correspond to some parts of the studied domain becoming either perfectly conducting or perfectly insulating. The main obstacle has arguably been establishing suitable monotonicity principles for the corresponding Neumann-to-Dirichlet boundary maps. In this work, we tackle this shortcoming by first giving a convergence result in the operator norm for the Neumann-to-Dirichlet map when the conductivity coefficient decays to zero and/or grows to infinity in some given parts of the domain. This allows passing the necessary monotonicity principles to the limiting case. Subsequently, we show how the monotonicity method generalizes to not only the definite case of reconstructing either perfectly conducting or perfectly insulating inclusions but also the indefinite case where the perturbed conductivity can take any values between, and including, zero and infinity.
中文翻译:
基于单调性的电阻抗层析成像中极端夹杂物的重建
SIAM数学分析杂志,第52卷,第6期,第6234-6259页,2020年1月。
基于单调性的方法已成为重构电阻抗层析成像反问题中夹杂物的基本方法之一。迄今为止,尚未证明该方法能够处理极端夹杂物,这些夹杂物对应于所研究领域的某些部分变得完全导电或完全绝缘。可以说,主要障碍是为相应的Neumann-to-Dirichlet边界图建立合适的单调性原则。在这项工作中,我们首先在电导率系数衰减为零和/或增长到无穷大时,首先在Neumann-to-Dirichlet映射的算子范数中给出收敛结果,从而解决了该缺陷。这允许将必要的单调性原理传递给极限情况。后来,