Factorization method and irregular inclusions in electrical impedance tomography

In electrical impedance tomography, one tries to recover the conductivity inside a physical body from boundary measurements of current and voltage. In many practically important situations, the investigated object has known background conductivity but it is contaminated by inhomogeneities. The factorization method of Andreas Kirsch provides a tool for locating such inclusions. Earlier, it has been shown that under suitable regularity conditions positive (or negative) inhomogeneities can be characterized by the factorization technique if the conductivity or one of its higher normal derivatives jumps on the boundaries of the inclusions. In this work, we use a monotonicity argument to generalize these results: We show that the factorization method provides a characterization of an open inclusion (modulo its boundary) if each point inside the inhomogeneity has an open neighbourhood where the perturbation of the conductivity is strictly positive (or negative) definite. In particular, we do not assume any regularity of the inclusion boundary or set any conditions on the behaviour of the perturbed conductivity at the inclusion boundary. Our theoretical findings are verified by two-dimensional numerical experiments.

Sensitivity analysis of a parabolic-elliptic problem

We consider the heat flux through a domain with subregions in which the thermal capacity approaches zero. In these subregions the parabolic heat equation degenerates to an elliptic one. We show the well-posedness of such parabolic-elliptic differential equations for general non-negative L-infinity-capacities and study the continuity of the solutions with respect to the capacity, thus giving a rigorous justification for modeling a small thermal capacity by setting it to zero. We also characterize weak directional derivatives of the temperature with respect to capacity as solutions of related parabolic-elliptic problems.

Detecting interfaces in a parabolic-elliptic problem from surface measurements

Assuming that the heat capacity of a body is negligible outside certain inclusions the heat equation degenerates to a parabolic-elliptic interface problem. In this work we aim to detect these interfaces from thermal measurements on the surface of the body. We deduce an equivalent variational formulation for the parabolic-elliptic problem and give a new proof of the unique solvability based on Lions’s projection lemma. For the case that the heat conductivity is higher inside the inclusions, we develop an adaptation of the factorization method to this time-dependent problem. In particular this shows that the locations of the interfaces are uniquely determined by boundary measurements. The method also yields to a numerical algorithm to recover the inclusions and thus the interfaces. We demonstrate how measurement data can be simulated numerically by a coupling of a finite element method with a boundary element method, and finally we present some numerical results for the inverse problem.

A sampling method for detecting buried objects using electromagnetic scattering

We consider a simple (but fully three-dimensional) mathematical model for the electromagnetic exploration of buried, perfect electrically conducting objects within the soil underground. Moving an electric device parallel to the ground at constant height in order to generate a magnetic field, we measure the induced magnetic field within the device, and factor the underlying mathematics into a product of three operations which correspond to the primary excitation, some kind of reflection on the surface of the buried object(s) and the corresponding secondary excitation, respectively. Using this factorization we are able to give a justification of the so-called sampling method from inverse scattering theory for this particular set-up.

Search for prompt neutrinos with AMANDA-II

A search for prompt neutrinos is performed with an analysis of the atmospheric neutrino data recorded by the AMANDA-II detector at the geographical South Pole in the years 2000-2003. The spectrum is reconstructed and limits on prompt production models spectrum are set according to our measurements.

Mutually catalytic branching at infinite rate

The purpose of this doctoral thesis is to prove existence for a mutually catalytic random walk with infinite branching rate on countably many sites. The process is defined as a weak limit of an approximating family of processes. An approximating process is constructed by adding jumps to a deterministic migration on an equidistant time grid. As law of jumps we need to choose the invariant probability measure of the mutually catalytic random walk with a finite branching rate in the recurrent regime. This model was introduced by Dawson and Perkins (1998) and this thesis relies heavily on their work. Due to the properties of this invariant distribution, which is in fact the exit distribution of planar Brownian motion from the first quadrant, it is possible to establish a martingale problem for the weak limit of any convergent sequence of approximating processes. We can prove a duality relation for the solution to the mentioned martingale problem, which goes back to Mytnik (1996) in the case of finite rate branching, and this duality gives rise to weak uniqueness for the solution to the martingale problem. Using standard arguments we can show that this solution is in fact a Feller process and it has the strong Markov property. For the case of only one site we prove that the model we have constructed is the limit of finite rate mutually catalytic branching processes as the branching rate approaches infinity. Therefore, it seems naturalto refer to the above model as an infinite rate branching process. However, a result for convergence on infinitely many sites remains open.

Entwicklung, Aufbau und Messungen eines schnellen Eiskeimzählers

Im Rahmen dieser Arbeit wurde ein neuer Eiskeimzähler FINCH (Fast Ice Nucleus CHamber) entwickelt und erste Messungen von verschiedenen Testaerosolen im Labor und atmosphärischem Aerosol durchgeführt. Die Aerosolpartikel bzw. Ice Nuclei IN werden bei Temperaturen unter dem Gefrierpunkt und Übersättigungen in Bezug auf Eis zum Anwachsen zu Eiskristallen gebracht, um sie mittels optischer Detektion zu erfassen. In FINCH ist dies durch das Prinzip der Mischung realisiert, wodurch eine kontinuierliche Messung der IN-Anzahlkonzentration gewährleistet ist. Hierbei kann mit sehr hohen Sammelflussraten von bis zu 10 l/min gemessen werden. Ebenso ist ein schnelles Abfahren von verschiedenen Sättigungsverhältnissen in Bezug auf Eis in einem weiten Bereich von 0.9 - 1.7 bei konstanten Temperaturen bis zu −23 °C möglich. Die Detektion der Eiskristalle und damit der Bestimmung der IN-Anzahlkonzentration erfolgt über einen neu entwickelten optischen Sensor basierend auf der unterschiedlichen Depolarisation des zurückgestreuten Lichtes von Eiskristallen und unterkühlten Tropfen. In Labermessungen wurden Aktivierungstemperatur und -sättigungsverhältnis von Silberjodid AgI und Kaolinit vermessen. Die Resultate zeigten gute Übereinstimmungen mit Ergebnissen aus der Literatur sowie Parallelmessungen mit FRIDGE (FRankfurt Ice Deposition freezinG Experiment). FRIDGE ist eine statische Diffusionskammer zur Aktivierung und Auszählung von Eiskeimen, die auf einem Filter gesammelt wurden. Bei atmosphärischen Messungen auf dem Jungfraujoch(Schweiz) lagen die IN-Anzahlkonzentrationen mit bis zu 4 l−1 im Rahmen der aus der Literatur bekannten Werte. Messungen der Eiskristallresiduen von Mischwolken zeigten hingegen, dass nur jedes tausendste als Eiskeim im Depositionsmode aktiv ist. Hier scheinen andere Gefrierprozesse und sekundäre Eiskristallbildung von sehr großer Bedeutung für die Anzahlkonzentration der Eiskristallresiduen zu sein. Eine weitere Messung von atmosphärischem Aerosol in Frankfurt zeigte IN-Anzahlkonzentrationen bis zu 30 l−1 bei Aktivierungstemperaturen um −14 °C. Die parallele Probenahme auf Siliziumplättchen für die Messungen der IN-Anzahlkonzentration in FRIDGE ergaben Werte im gleichen Anzahlkonzentrationsbereich.