Nonlinear integral equations and the iterative solution for an inverse boundary value problem

Kreß, Rainer; Rundell, William

doi:10.1088/0266-5611/21/4/002

Cited by 169 publications

(173 citation statements)

References 24 publications

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“…Even if σ ∈ Σ is further restricted to be piecewise continuous, little is known about the identifiability of the support of σ − 1; see Isakov [15] for a survey of the corresponding state of the art. On the other hand, this inverse problem has been approached numerically by various authors (cf., e.g., [5,12,16,17,18,21]) with quite encouraging results. With the exception of the one in [21], the corresponding numerical algorithms are iterative in nature.…”

Section: An Application: Electric Impedance Tomographymentioning

confidence: 99%

Convex Source Support and Its Application to Electric Impedance Tomography

Hanke¹,

Hyvönen²,

Reusswig

2008

SIAM J. Imaging Sci.

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Abstract. The aim in electric impedance tomography is to recover the conductivity inside a physical body from boundary measurements of current and voltage. In many situations of practical importance, the investigated object has known background conductivity but is contaminated by inhomogeneities. In this work, we try to extract all possible information about the support of such inclusions inside a twodimensional object from only one pair of measurements of impedance tomography. Our noniterative and computationally cheap method is based on the concept of the convex source support, which stems from earlier works of Kusiak, Sylvester, and the authors. The functionality of our algorithm is demonstrated by various numerical experiments.

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Section: An Application: Electric Impedance Tomographymentioning

confidence: 99%

Convex Source Support and Its Application to Electric Impedance Tomography

Hanke¹,

Hyvönen²,

Reusswig

2008

SIAM J. Imaging Sci.

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“…Extending a method proposed by Kress and Rundell [28] for an inverse Dirichlet problem for the Laplace equation, a second approach for iteratively solving the system (2.4) and (2.5) consists in simultaneously linearizing both equations with respect to both unknowns. In this case, given approximations p and ψ both for the boundary parameterization and the density we obtain the system of linear equations…”

Section: Simultaneous Linearization Of Both Equationsmentioning

confidence: 99%

“…This approach has been suggested by Johansson and Sleeman [23] and analyzed further by Ivanyshyn and Johansson [19,20]. In the second method, following ideas first developed for the Laplace equation by Kress and Rundell [28], one also can solve the system (1.6) and (1.7) simultaneously for ∂ D and ϕ by Newton iterations, that is, by linearizing both equations with respect to both unknowns. This approach has been intensively studied by Ivanyshyn, Johansson and Kress [15-18, 21, 22].…”

Section: Introductionmentioning

confidence: 99%

Huygens’ principle and iterative methods in inverse obstacle scattering

2009

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The inverse problem we consider in this paper is to determine the shape of an obstacle from the knowledge of the far field pattern for scattering of time-harmonic plane waves. In the case of scattering from a sound-soft obstacle, we will interpret Huygens' principle as a system of two integral equations, named data and field equation, for the unknown boundary of the scatterer and the induced surface flux, i.e., the unknown normal derivative of the total field on the boundary. Reflecting the ill-posedness of the inverse obstacle scattering problem these integral equations are ill-posed. They are linear with respect to the unknown flux and nonlinear with respect to the unknown boundary and offer, in principle, three immediate possibilities for their iterative solution via linearization and regularization. In addition to presenting new results on injectivity and dense range for the linearized operators, the main purpose of this paper is to establish and illuminate relations between these three solution methods based on Huygens' principle in inverse obstacle scattering. Furthermore, we will exhibit connections and differences to the Communicated by guest editors:

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“…As the complimentary approach, in [5] we derived another equivalent system of nonlinear and ill-posed integral equations based on Green's representation theorem. This second approach extends a method suggested by Kress and Rundell [16] to determine the shape of a perfectly conducting inclusion in a homogeneous background from a pair of Cauchy data on the accessible exterior boundary. The inverse problem to simultaneously recover the shape and impedance of an inclusion was recently considered by Rundell in [20] where, in particular, an algorithm was proposed which also can be considered as extension of [16].…”

mentioning

confidence: 90%

“…This second approach extends a method suggested by Kress and Rundell [16] to determine the shape of a perfectly conducting inclusion in a homogeneous background from a pair of Cauchy data on the accessible exterior boundary. The inverse problem to simultaneously recover the shape and impedance of an inclusion was recently considered by Rundell in [20] where, in particular, an algorithm was proposed which also can be considered as extension of [16]. It is the aim of the current paper to extend the analysis of [4,5] to the simultaneous inverse shape and impedance problem for the case of corrosion detection.…”

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confidence: 90%

Simultaneous Reconstruction of Shape and Impedance in Corrosion Detection

Cakoni¹,

Kreß²,

Schuft³

2010

Methods and Applications of Analysis

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Abstract. Corrosion detection can be modelled by the Laplace equation for an electric or a heat potential in a simply connected planar domain D with a homogeneous impedance boundary condition on a non-accessible part of the boundary ∂D. We consider the inverse problem to simultaneously recover the non-accessible part of the boundary and the impedance function from two pairs of Cauchy data on the accessible part of the boundary. Our approach extends the method proposed by Kress and Rundell [16] for the corresponding problem to recover the interior boundary curve of a doubly connected planar domain and is based on our previous work on reconstruction of the impedance function for a known shape or the shape for a known impedance function [4,5]. Based either on a potential approach or on a Green's integral formulation the inverse problem is equivalent to a system of nonlinear and ill-posed integral equations that can be solved iteratively by linearization. We will present the mathematical foundation of the method and, in particular, establish injectivity for the linearized system at the exact solution. Numerical reconstructions will show the feasibility of the method.

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Nonlinear integral equations and the iterative solution for an inverse boundary value problem

Cited by 169 publications

References 24 publications

Convex Source Support and Its Application to Electric Impedance Tomography

Convex Source Support and Its Application to Electric Impedance Tomography

Huygens’ principle and iterative methods in inverse obstacle scattering

Simultaneous Reconstruction of Shape and Impedance in Corrosion Detection

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