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.

The factorization method for real elliptic problems

The Factorization Method localizes inclusions inside a body from measurements on its surface. Without a priori knowing the physical parameters inside the inclusions, the points belonging to them can be characterized using the range of an auxiliary operator. The method relies on a range characterization that relates the range of the auxiliary operator to the measurements and is only known for very particular applications. In this work we develop a general framework for the method by considering symmetric and coercive operators between abstract Hilbert spaces. We show that the important range characterization holds if the difference between the inclusions and the background medium satisfies a coerciveness condition which can immediately be translated into a condition on the coefficients of a given real elliptic problem. We demonstrate how several known applications of the Factorization Method are covered by our general results and deduce the range characterization for a new example in linear elasticity.

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.

Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem

In various imaging problems the task is to use the Cauchy data of the solutions to an elliptic boundary value problem to reconstruct the coefficients of the corresponding partial differential equation. Often the examined object has known background properties but is contaminated by inhomogeneities that cause perturbations of the coefficient functions. The factorization method of Kirsch provides a tool for locating such inclusions. In this paper, the factorization technique is studied in the framework of coercive elliptic partial differential equations of the divergence type: Earlier it has been demonstrated that the factorization algorithm can reconstruct the support of a strictly positive (or negative) definite perturbation of the leading order coefficient, or if that remains unperturbed, the support of a strictly positive (or negative) perturbation of the zeroth order coefficient. In this work we show that these two types of inhomogeneities can, in fact, be located simultaneously. Unlike in the earlier articles on the factorization method, our inclusions may have disconnected complements and we also weaken some other a priori assumptions of the method. Our theoretical findings are complemented by two-dimensional numerical experiments that are presented in the framework of the diffusion approximation of optical tomography.

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.

Mixed finite-element methods for elliptic convection-dominated problems arising in semiconductor physics

In this work we develop and analyze an adaptive numerical scheme for simulating a class of macroscopic semiconductor models. At first the numerical modelling of semiconductors is reviewed in order to classify the Energy-Transport models for semiconductors that are later simulated in 2D. In this class of models the flow of charged particles, that are negatively charged electrons and so-called holes, which are quasi-particles of positive charge, as well as their energy distributions are described by a coupled system of nonlinear partial differential equations. A considerable difficulty in simulating these convection-dominated equations is posed by the nonlinear coupling as well as due to the fact that the local phenomena such as "hot electron effects" are only partially assessable through the given data. The primary variables that are used in the simulations are the particle density and the particle energy density. The user of these simulations is mostly interested in the current flow through parts of the domain boundary - the contacts. The numerical method considered here utilizes mixed finite-elements as trial functions for the discrete solution. The continuous discretization of the normal fluxes is the most important property of this discretization from the users perspective. It will be proven that under certain assumptions on the triangulation the particle density remains positive in the iterative solution algorithm. Connected to this result an a priori error estimate for the discrete solution of linear convection-diffusion equations is derived. The local charge transport phenomena will be resolved by an adaptive algorithm, which is based on a posteriori error estimators. At that stage a comparison of different estimations is performed. Additionally a method to effectively estimate the error in local quantities derived from the solution, so-called "functional outputs", is developed by transferring the dual weighted residual method to mixed finite elements. For a model problem we present how this method can deliver promising results even when standard error estimator fail completely to reduce the error in an iterative mesh refinement process.

Inverse problems in classical and quantum physics

The subject of this thesis is in the area of Applied Mathematics known as Inverse Problems. Inverse problems are those where a set of measured data is analysed in order to get as much information as possible on a model which is assumed to represent a system in the real world. We study two inverse problems in the fields of classical and quantum physics: QCD condensates from tau-decay data and the inverse conductivity problem. Despite a concentrated effort by physicists extending over many years, an understanding of QCD from first principles continues to be elusive. Fortunately, data continues to appear which provide a rather direct probe of the inner workings of the strong interactions. We use a functional method which allows us to extract within rather general assumptions phenomenological parameters of QCD (the condensates) from a comparison of the time-like experimental data with asymptotic space-like results from theory. The price to be paid for the generality of assumptions is relatively large errors in the values of the extracted parameters. Although we do not claim that our method is superior to other approaches, we hope that our results lend additional confidence to the numerical results obtained with the help of methods based on QCD sum rules. EIT is a technology developed to image the electrical conductivity distribution of a conductive medium. The technique works by performing simultaneous measurements of direct or alternating electric currents and voltages on the boundary of an object. These are the data used by an image reconstruction algorithm to determine the electrical conductivity distribution within the object. In this thesis, two approaches of EIT image reconstruction are proposed. The first is based on reformulating the inverse problem in terms of integral equations. This method uses only a single set of measurements for the reconstruction. The second approach is an algorithm based on linearisation which uses more then one set of measurements. A promising result is that one can qualitatively reconstruct the conductivity inside the cross-section of a human chest. Even though the human volunteer is neither two-dimensional nor circular, such reconstructions can be useful in medical applications: monitoring for lung problems such as accumulating fluid or a collapsed lung and noninvasive monitoring of heart function and blood flow.

Sampling methods for low-frequency electromagnetic imaging

For the detection of hidden objects by low-frequency electromagnetic imaging the Linear Sampling Method works remarkably well despite the fact that the rigorous mathematical justification is still incomplete. In this work, we give an explanation for this good performance by showing that in the low-frequency limit the measurement operator fulfills the assumptions for the fully justified variant of the Linear Sampling Method, the so-called Factorization Method. We also show how the method has to be modified in the physically relevant case of electromagnetic imaging with divergence-free currents. We present numerical results to illustrate our findings, and to show that similar performance can be expected for the case of conducting objects and layered backgrounds.

Direct and inverse transient eddy current problems

Die vorliegende Arbeit behandelt Vorwärts- sowie Rückwärtstheorie transienter Wirbelstromprobleme. Transiente Anregungsströme induzieren elektromagnetische Felder, welche sogenannte Wirbelströme in leitfähigen Objekten erzeugen. Im Falle von sich langsam ändernden Feldern kann diese Wechselwirkung durch die Wirbelstromgleichung, einer Approximation an die Maxwell-Gleichungen, beschrieben werden. Diese ist eine lineare partielle Differentialgleichung mit nicht-glatten Koeffizientenfunktionen von gemischt parabolisch-elliptischem Typ. Das Vorwärtsproblem besteht darin, zu gegebener Anregung sowie den umgebungsbeschreibenden Koeffizientenfunktionen das elektrische Feld als distributionelle Lösung der Gleichung zu bestimmen. Umgekehrt können die Felder mit Messspulen gemessen werden. Das Ziel des Rückwärtsproblems ist es, aus diesen Messungen Informationen über leitfähige Objekte, also über die Koeffizientenfunktion, die diese beschreibt, zu gewinnen. In dieser Arbeit wird eine variationelle Lösungstheorie vorgestellt und die Wohlgestelltheit der Gleichung diskutiert. Darauf aufbauend wird das Verhalten der Lösung für verschwindende Leitfähigkeit studiert und die Linearisierbarkeit der Gleichung ohne leitfähiges Objekt in Richtung des Auftauchens eines leitfähigen Objektes gezeigt. Zur Regularisierung der Gleichung werden Modifikationen vorgeschlagen, welche ein voll parabolisches bzw. elliptisches Problem liefern. Diese werden verifiziert, indem die Konvergenz der Lösungen gezeigt wird. Zuletzt wird gezeigt, dass unter der Annahme von sonst homogenen Umgebungsparametern leitfähige Objekte eindeutig durch die Messungen lokalisiert werden können. Hierzu werden die Linear Sampling Methode sowie die Faktorisierungsmethode angewendet.

Gebietserkennung mit der Faktorisierungsmethode

In der vorliegenden Arbeit wird die Faktorisierungsmethode zur Erkennung von Gebieten mit sprunghaft abweichenden Materialparametern untersucht. Durch eine abstrakte Formulierung beweisen wir die der Methode zugrunde liegende Bildraumidentität für allgemeine reelle elliptische Probleme und deduzieren bereits bekannte und neue Anwendungen der Methode. Für das spezielle Problem, magnetische oder perfekt elektrisch leitende Objekte durch niederfrequente elektromagnetische Strahlung zu lokalisieren, zeigen wir die eindeutige Lösbarkeit des direkten Problems für hinreichend kleine Frequenzen und die Konvergenz der Lösungen gegen die der elliptischen Gleichungen der Magnetostatik. Durch Anwendung unseres allgemeinen Resultats erhalten wir die eindeutige Rekonstruierbarkeit der gesuchten Objekte aus elektromagnetischen Messungen und einen numerischen Algorithmus zur Lokalisierung der Objekte. An einem Musterproblem untersuchen wir, wie durch parabolische Differentialgleichungen beschriebene Einschlüsse in einem durch elliptische Differentialgleichungen beschriebenen Gebiet rekonstruiert werden können. Dabei beweisen wir die eindeutige Lösbarkeit des zugrunde liegenden parabolisch-elliptischen direkten Problems und erhalten durch eine Erweiterung der Faktorisierungsmethode die eindeutige Rekonstruierbarkeit der Einschlüsse sowie einen numerischen Algorithmus zur praktischen Umsetzung der Methode.

Ein inverses Problem für die degeneriert parabolische Richardsgleichung

In der vorliegenden Arbeit werden zwei physikalischeFließexperimente an Vliesstoffen untersucht, die dazu dienensollen, unbekannte hydraulische Parameter des Materials, wiez. B. die Diffusivitäts- oder Leitfähigkeitsfunktion, ausMeßdaten zu identifizieren. Die physikalische undmathematische Modellierung dieser Experimente führt auf einCauchy-Dirichlet-Problem mit freiem Rand für die degeneriertparabolische Richardsgleichung in derSättigungsformulierung, das sogenannte direkte Problem. Ausder Kenntnis des freien Randes dieses Problems soll dernichtlineare Diffusivitätskoeffizient derDifferentialgleichung rekonstruiert werden. Für diesesinverse Problem stellen wir einOutput-Least-Squares-Funktional auf und verwenden zu dessenMinimierung iterative Regularisierungsverfahren wie dasLevenberg-Marquardt-Verfahren und die IRGN-Methode basierendauf einer Parametrisierung des Koeffizientenraumes durchquadratische B-Splines. Für das direkte Problem beweisen wirunter anderem Existenz und Eindeutigkeit der Lösung desCauchy-Dirichlet-Problems sowie die Existenz des freienRandes. Anschließend führen wir formal die Ableitung desfreien Randes nach dem Koeffizienten, die wir für dasnumerische Rekonstruktionsverfahren benötigen, auf einlinear degeneriert parabolisches Randwertproblem zurück.Wir erläutern die numerische Umsetzung und Implementierungunseres Rekonstruktionsverfahrens und stellen abschließendRekonstruktionsergebnisse bezüglich synthetischer Daten vor.

Die Faktorisierungsmethode für die elektrische Impedanztomographie im Halbraum

In der vorliegenden Arbeit wird die Faktorisierungsmethode zur Erkennung von Inhomogenitäten der Leitfähigkeit in der elektrischen Impedanztomographie auf unbeschränkten Gebieten - speziell der Halbebene bzw. dem Halbraum - untersucht. Als Lösungsräume für das direkte Problem, d.h. die Bestimmung des elektrischen Potentials zu vorgegebener Leitfähigkeit und zu vorgegebenem Randstrom, führen wir gewichtete Sobolev-Räume ein. In diesen wird die Existenz von schwachen Lösungen des direkten Problems gezeigt und die Gültigkeit einer Integraldarstellung für die Lösung der Laplace-Gleichung, die man bei homogener Leitfähigkeit erhält, bewiesen. Mittels der Faktorisierungsmethode geben wir eine explizite Charakterisierung von Einschlüssen an, die gegenüber dem Hintergrund eine sprunghaft erhöhte oder erniedrigte Leitfähigkeit haben. Damit ist zugleich für diese Klasse von Leitfähigkeiten die eindeutige Rekonstruierbarkeit der Einschlüsse bei Kenntnis der lokalen Neumann-Dirichlet-Abbildung gezeigt. Die mittels der Faktorisierungsmethode erhaltene Charakterisierung der Einschlüsse haben wir in ein numerisches Verfahren umgesetzt und sowohl im zwei- als auch im dreidimensionalen Fall mit simulierten, teilweise gestörten Daten getestet. Im Gegensatz zu anderen bekannten Rekonstruktionsverfahren benötigt das hier vorgestellte keine Vorabinformation über Anzahl und Form der Einschlüsse und hat als nicht-iteratives Verfahren einen vergleichsweise geringen Rechenaufwand.

Detection of small buried objects: asymptotic factorization and MUSIC

In this work, we consider a simple model problem for the electromagnetic exploration of small perfectly conducting objects buried within the lower halfspace of an unbounded two–layered background medium. In possible applications, such as, e.g., humanitarian demining, the two layers would correspond to air and soil. Moving a set of electric devices parallel to the surface of ground to generate a time–harmonic field, the induced field is measured within the same devices. The goal is to retrieve information about buried scatterers from these data. In mathematical terms, we are concerned with the analysis and numerical solution of the inverse scattering problem to reconstruct the number and the positions of a collection of finitely many small perfectly conducting scatterers buried within the lower halfspace of an unbounded two–layered background medium from near field measurements of time–harmonic electromagnetic waves. For this purpose, we first study the corresponding direct scattering problem in detail and derive an asymptotic expansion of the scattered field as the size of the scatterers tends to zero. Then, we use this expansion to justify a noniterative MUSIC–type reconstruction method for the solution of the inverse scattering problem. We propose a numerical implementation of this reconstruction method and provide a series of numerical experiments.

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.

Localized potentials in electrical impedance tomography

In this work we study localized electric potentials that have an arbitrarily high energy on some given subset of a domain and low energy on another. We show that such potentials exist for general L-infinity-conductivities (with positive infima) in almost arbitrarily shaped subregions of a domain, as long as these regions are connected to the boundary and a unique continuation principle is satisfied. From this we deduce a simple, but new, theoretical identifiability result for the famous Calderon problem with partial data. We also show how to construct such potentials numerically and use a connection with the factorization method to derive a new non-iterative algorithm for the detection of inclusions in electrical impedance tomography.