Minisymposia Abstracts

Speaker: Jeremi Darde
Universit'e Paris 7, France
Title: The exterior approach to solve an inverse obstacle problem in electric impedance tomography
Abstract: We introduce a technique based on duality in optimization for approximating the Cauchy data of an electromagnetic potential from the corresponding electrode currents and potentials in the framework of the complete electrode model of impedance tomography. We then demonstrate that this approach can be combined with the so-called "exterior approach", a quasi-reversibility level set method to solve the inverse obstacle problem with ideally conducting inhomogeneity using (only a few) electrode measurements of electrical impedance tomography. We present numerical results highlighting the efficiency of the method.

Speaker: Martin Simon
Johannes Gutenberg-University of Mainz
Title: Probabilistic Interpretation of Current-to-Voltage-Maps within the Complete Electrode Model of EIT
Abstract: In this talk we consider current-to-voltage maps arising in the complete electrode model of EIT. A probabilistic representation formula in terms of reflected diffusion processes and their boundary local times is presented, which coincides with the well-known Feynman-Kac formula if the conductivity is constant. In the second part of the talk we discuss the numerical simulation of the corresponding processes. We will point out difficult ies in extending one-dimensional simulation schemes to the multidimensional case. In the third part of the presentation a formulation of the inverse problem of EIT in terms of the diffusion processes and their generators is given. This talk is based on joint work with Martin Hanke-Bourgeois, University of Mainz, Germany.

Speaker: Nuutti Hyvonen
Aalto University, Finland
Title: Electrical impedance tomography with two electrodes
Abstract: Electrical impedance tomography is a noninvasive imaging technique for recovering the conductivity distribution inside a body from boundary measurements of current and voltage. In this talk, we consider impedance tomography in the special case that the measurements are carried out with two electrodes that can be moved along the boundary of the (two-dimensional) object of interest. Two different types of data are considered: The backscatter data is obtained by using a small probe consisting of two electrodes for driving currents and measuring voltage differences subsequently at various neighboring locations on the boundary of the object. The scan data is gathered by fixing the location of one electrode and measuring the voltage difference required for maintaining a unit current as a function of the position of the other. A reconstruction algorithm for locating conductivity inhomogeneities in homogeneous background from such data sets is introduced and tested numerically.

Speaker: Armin Lechleiter Ecole Polytechnique, France
Title: Explicit Characterization of the Support of Non-Linear Inclusions
Abstract: We consider two inverse problems related to non-linear and penetrable media: An inverse scattering problem for a non-linear medium and the impedance tomography problem for a non-linear conductivity. The non-linear measurement operators of these two problems (that is, the far-field operator and the Dirichlet-to-Neumann operator) satisfy factorizations that are suitable to derive an explicit characterization of the support of a weakly non-linear inhomogeneity from the measurement operator. The technique is related to the so-called infimum criterion and the Factorization method for inverse problems.

Speaker: Peter Monk
University of Delaware
Title: The Linear Sampling method in the Time Domain
Abstract: We present an extension of the frequency domain linear sampling method to the time domain near-field inverse scattering problem for the wave equation of finding the shape of a Dirichlet scattering object from time domain measurements of scattered waves. The inversion algorithm works directly on time-domain data without using a Fourier transformation. This feature allows us to naturally incorporate multiple (in fact, a continuum of) frequencies in the inversion algorithm. Consequently, it offers the potential of improving the quality of the reconstruction compared to frequency domain methods working with a single frequency. We demonstrate this potential by several numerical examples.

Speaker: John Sylvester
University of Washington
Title: Transmission Eigenvalues and Upper Triangular Compactness
Abstract: The interior transmission eigenvalue problem can be formulated as a 2x2 system of pdes, where one of the two unknown functions must satisfy too many boundary conditions, and the other too few. The system is not self-adjoint and the resolvent is not compact.

Under the hypothesis that the contrast satisfies a coercivity condition on the boundary of the domain, we show that the corresponding operator has Upper Triangular Compact Resolvent and that the analytic Fredholm theorem holds for such operators.

As a corollary, we can show that the set of (complex) interior transmission eigenvalues is a (possibly empty) discrete set. This is different from previous results because the contrast need not have a constant sign (or be real valued) in the interior of the domain.

Speaker: Marcel Ullrich
University of Mainz, Germany
Title: A Monotony Based Method for EIT
Abstract: The aim of Electrical Impedance Tomography (EIT) is to reconstruct the spatially dependent conductivity σ: Ω → R in the inside of a conductive object Ω ⊆ Rn on the basis of a sequence of current-voltage measurements on the outside of that object.

The relation between the electric potential u: Ω → R and the conductivity σ is modeled by the equation
div(σ ∇ u) = 0.
The Neumann boundary condition vσ = σ ∂ν u |∂ Ω describes the current at the boundary and the Dirichlet boundary condition uσ = u |∂ Ω describes the voltage at the boundary, which is uniquely related to the respective current. This defines the Neumann-to-Dirichlet Operator
Λ(σ):vσ↦ uσ.
In this talk we discuss a special case of EIT, in which the conductivity is inhomogeneous only at certain areas and constant elsewhere. We content ourselves with the localisation of those areas, which we also call inclusions.

A well known monotony relation for the dependence of Λ on σ is given by:
σ1≥ σ2⇒ Λ(σ1)≤ Λ(σ2)
By using the concept of localized potentials, we will show, that a (partly) converse of the monotony relation can derived.

From both relations we then obtain a monotony based method that is capable of exactly reconstructing the outer support of a conductivity change. We show that this method holds even for the indefinite case, i.e. for conductivities, which are respectively higher and lower than the background conductivity on separated areas.

Speaker: Ting Zhou
University of Washington
Title: Reconstructing Electromagnetic Obstacles by the Enclosure Method
Abstract: We show that one can determine Perfectly Magnetic Conductor obstacles, Perfectly Electric Conductor obstacles and obstacles satisfying impedance boundary condition, embedded in a known electromagnetic medium, by making electromagnetic measurements at the boundary of the medium. The boundary measurements are encoded in the impedance map that sends the tangential component of the electric field to the tangential component of the magnetic field. We do this by probing the medium with complex geometrical optics solutions to the corresponding Maxwells equations and extend the enclosure method to this case. Moreover, using complex spherical waves, constructed by the inversion transformation with respect to a sphere, the enclosure method can recover some non-convex part of the obstacle.

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