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Journal ArticleDOI

Monotonicity Principle in tomography of nonlinear conducting materials

01 Apr 2021-Inverse Problems (IOP Publishing)-Vol. 37, Iss: 4, pp 045012
TL;DR: In this article, the authors show that the Monotonicity principle for the Dirichlet Energy in nonlinear problems holds under mild assumptions, and they show that apart from linear and p-Laplacian cases, it is impossible to transfer this monotonicity result from the DtN operator to the Dn operator.
Abstract: We treat an inverse electrical conductivity problem which deals with the reconstruction of nonlinear electrical conductivity starting from boundary measurements in steady currents operations. In this framework, a key role is played by the Monotonicity Principle, which establishes a monotonic relation connecting the unknown material property to the (measured) Dirichlet-to-Neumann operator (DtN). Monotonicity Principles are the foundation for a class of non-iterative and real-time imaging methods and algorithms. In this article, we prove that the Monotonicity Principle for the Dirichlet Energy in nonlinear problems holds under mild assumptions. Then, we show that apart from linear and p-Laplacian cases, it is impossible to transfer this Monotonicity result from the Dirichlet Energy to the DtN operator. To overcome this issue, we introduce a new boundary operator, identified as an Average DtN operator.
Citations
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Journal ArticleDOI
TL;DR: In this paper, the authors prove a monotonic relationship between the electrical resistivity and the time constants characterizing the free-response for MQS systems, based on the analysis of an elliptic eigenvalue problem.
Abstract: The inverse problem treated in this article consists in reconstructing the electrical conductivity from the free response of the system in the magneto-quasi-stationary (MQS) limit. The MQS limit corresponds to a diffusion PDE. In this framework, a key role is played by the Monotonicity Principle, that is a monotone relation connecting the unknown material property to the (measured) free-response. MP is relevant as basis of noniterative and real-time imaging methods. Monotonicity Principles have been found in many different physical problems governed by PDEs of different nature. Despite its rather general nature, each different physical/mathematical context requires to discover the proper operator showing MP. For doing this, it is necessary to develop ad-hoc mathematical approaches tailored on the specific framework. In this article, we prove a monotonic relationship between the electrical resistivity and the time constants characterizing the free-response for MQS systems. The key result is the representation of the induced current density through a modal representation. The main result is based on the analysis of an elliptic eigenvalue problem, obtained from separation of variables.

3 citations

Posted Content
TL;DR: In this article, the authors consider the reconstruction of the support of an unknown perturbation to a known conductivity coefficient in Calderon's problem, and constructively characterise the outer shape of such a general perturbations based on a local Neumann-to-Dirichlet map defined on an open subset of the domain boundary.
Abstract: We consider the reconstruction of the support of an unknown perturbation to a known conductivity coefficient in Calderon's problem. In a previous result by the authors on monotonicity-based reconstruction, the perturbed coefficient is allowed to simultaneously take the values $0$ and $\infty$ in some parts of the domain and values bounded away from $0$ and $\infty$ elsewhere. We generalise this result by allowing the unknown coefficient to be the restriction of an $A_2$-Muckenhaupt weight in parts of the domain, thereby including singular and degenerate behaviour in the governing equation. In particular, the coefficient may tend to $0$ and $\infty$ in a controlled manner, which goes beyond the standard setting of Calderon's problem. Our main result constructively characterises the outer shape of the support of such a general perturbation, based on a local Neumann-to-Dirichlet map defined on an open subset of the domain boundary.

1 citations

Journal ArticleDOI
TL;DR: In this paper, a reconstruction method for piecewise constant layered conductivities (PCLC) from partial boundary measurements in electrical impedance tomography has been proposed, based on monotonicity-based reconstruction of extreme inclusions.
Posted Content
Henrik Garde1
TL;DR: In this paper, a reconstruction method for piecewise constant layered conductivities (PCLC) from partial boundary measurements in electrical impedance tomography is presented. But this method requires a priori lower and upper bounds to the unknown conductivity values.
Abstract: This short note considerably simplifies a reconstruction method by the author, for reconstructing piecewise constant layered conductivities (PCLC) from partial boundary measurements in electrical impedance tomography. Theory from monotonicity-based reconstruction of extreme inclusions eliminates most of the bookkeeping related to multiple components of each layer, and also simplifies the involved test operators. Moreover, the method no longer requires a priori lower and upper bounds to the unknown conductivity values.
Posted Content
TL;DR: In this paper, the authors prove a monotonic relationship between the electrical resistivity and the time constants characterizing the free-response for MQS systems, based on the analysis of an elliptic eigenvalue problem.
Abstract: The inverse problem treated in this article consists in reconstructing the electrical conductivity from the free response of the system in the magneto-quasi-stationary (MQS) limit. The MQS limit corresponds to a diffusion PDE. In this framework, a key role is played by the Monotonicity Principle, that is a monotone relation connecting the unknown material property to the (measured) free-response. MP is relevant as basis of noniterative and real-time imaging methods. Monotonicity Principles have been found in many different physical problems governed by PDEs of different nature. Despite its rather general nature, each different physical/mathematical context requires to discover the proper operator showing MP. For doing this, it is necessary to develop ad-hoc mathematical approaches tailored on the specific framework. In this article, we prove a monotonic relationship between the electrical resistivity and the time constants characterizing the free-response for MQS systems. The key result is the representation of the induced current density through a modal representation. The main result is based on the analysis of an elliptic eigenvalue problem, obtained from separation of variables.
References
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BookDOI
01 Mar 1988

4,584 citations

Book
01 Jan 1989
TL;DR: In this paper, the existence theorem for non-quasiconvex Integrands in the Scalar case has been established in the Vectorial case, where the objective function is to find the minimum of the minimum for a non-convex function.
Abstract: Introduction.- Convex Analysis and the Scalar Case.- Convex Sets and Convex Functions.- Lower Semicontinuity and Existence Theorems.- The one Dimensional Case.- Quasiconvex Analysis and the Vectorial Case.- Polyconvex, Quasiconvex and Rank one Convex Functions.- Polyconvex, Quasiconvex and Rank one Convex Envelopes.- Polyconvex, Quasiconvex and Rank one Convex Sets.- Lower Semi Continuity and Existence Theorems in the Vectorial Case.- Relaxation and Non Convex Problems.- Relaxation Theorems.- Implicit Partial Differential Equations.- Existence of Minima for Non Quasiconvex Integrands.- Miscellaneous.- Function Spaces.- Singular Values.- Some Underdetermined Partial Differential Equations.- Extension of Lipschitz Functions on Banach Spaces.- Bibliography.- Index.- Notations.

2,250 citations

Book
15 Jul 2009
TL;DR: Leoni as discussed by the authors takes a novel approach to the theory by looking at Sobolev spaces as the natural development of monotone, absolutely continuous, and BV functions of one variable.
Abstract: Sobolev spaces are a fundamental tool in the modern study of partial differential equations. In this book, Leoni takes a novel approach to the theory by looking at Sobolev spaces as the natural development of monotone, absolutely continuous, and BV functions of one variable. In this way, the majority of the text can be read without the prerequisite of a course in functional analysis. The first part of this text is devoted to studying functions of one variable. Several of the topics treated occur in courses on real analysis or measure theory. Here, the perspective emphasizes their applications to Sobolev functions, giving a very different flavor to the treatment. This elementary start to the book makes it suitable for advanced undergraduates or beginning graduate students. Moreover, the one-variable part of the book helps to develop a solid background that facilitates the reading and understanding of Sobolev functions of several variables. The second part of the book is more classical, although it also contains some recent results. Besides the standard results on Sobolev functions, this part of the book includes chapters on BV functions, symmetric rearrangement, and Besov spaces. The book contains over 200 exercises.

853 citations

Book ChapterDOI
TL;DR: In this paper, an inversion scheme for two-dimensional inverse scattering problems in the resonance region is proposed, which does not use nonlinear optimization methods and is relatively independent of the geometry and physical properties of the scatterer, assuming that the far field pattern corresponding to observation angle and plane waves incident at angle is known for all.
Abstract: This paper is concerned with the development of an inversion scheme for two-dimensional inverse scattering problems in the resonance region which does not use nonlinear optimization methods and is relatively independent of the geometry and physical properties of the scatterer It is assumed that the far field pattern corresponding to observation angle and plane waves incident at angle is known for all From this information, the support of the scattering obstacle is obtained by solving the integral equation where k is the wavenumber and is on a rectangular grid containing the scatterer The support is found by noting that is unbounded as approaches the boundary of the scattering object from inside the scatterer Numerical examples are given showing the practicality of this method

750 citations