There are various ways to regularize ill-posed operator equations in Hilbert space. If the underlying operator is accretive then Lavrentiev regularization (singular perturbation) is an immediate choice. The corresponding convergence rates for the regularization error depend on the given smoothness assumptions, and for general accretive operators these may be both with respect to the operator or its adjoint. Previous analysis revealed different convergence rates, and their optimality was unclear, specifically for adjoint source conditions. Based on the fundamental study by T. Kato, Fractional powers of dissipative operators. J. Math. Soc. Japan, 13(3):247--274, 1961, we establish power type convergence rates for this case. By measuring the optimality of such rates in terms on limit orders we exhibit optimality properties of the convergence rates, for general accretive operators under direct and adjoint source conditions, but also for the subclass of nonnegative selfadjoint operators.

In Brokate-Sprekels 1996 wurde ein Darstellungsresultat für Hysterese-Operatoren, die auf skalaren, stetigen, stückweise monotonen Funktionen definiert sind, hergeleitet. Dieses erlaubt es, die Operatoren eindeutig aus Funktionalen auf Strings reeller Zahlen, d.h. auf Tupeln aus reellen Zahlen mit beliebiger Länge, bei denen die Vorzeichen der Differenzen aufeinander folgender Paare alternieren, zu gewinnen. In dieser Habilitation wird nun ein Ansatz vorgestellt, um auch für Hysterese-Operatoren mit vektoriellen Input-Funktionen eine Darstellung durch Abbildungen auf einer String-Menge zu ermöglichen. Dabei wird die Rolle der monotonen Funktionen von den neu eingeführten sogenannten monotaffinen Funktionen übernommen, die man erhalten kann, indem man die Ausgabe einer monotonen Funktion auf den reellen Zahlen in eine affine Funktion auf den reellen Zahlen einsetzt. Die Rolle der alternierender Strings wird von den sogenannten Konvexitätstripel-freien Strings mit Elementen aus Vektorraum übernommen, d.h. von den Tupel aus Elementen aus dem Vektorraum, so dass sich kein Eintrag als Konvexkombination seines Vorgängers und seines Nachfolgers schreiben lässt. Das Darstellungsresultat erlaubt es, die Hysterese-Operatoren auf den stetigen, stückweise monotaffinen Input-Funktionen eindeutig aus den Abbildungen auf den Konvexitätstripel-freien Strings zu gewinnen. Damit können dann Eigenschaften des Hysterese-Operators durch Untersuchung der entsprechenden Abbildung bestimmt werden. Es wird ein weiteres Darstellungsresultat vorgestellt, bei dem die Hysterese-Operatoren für Funktionen definiert sind, die endlich viele Sprungstellen haben. Die betrachteten Strings sind jetzt Tupel von Quintupeln. In diesen Quintupeln werden für jede Sprung- und Richtungswechselstelle der Funktion der Funktionswert, der links- und der rechtsseitigen Grenzwert und Informationen über das Verhalten der Funktion in der Nähe dieser Stelle gespeichert. / In Brokate-Sprekels 1996 a representation result for hysteresis operators acting on scalar-valued continuous piecewise monotone functions was derived. Thanks to this result, the operators can be derived in a unique way from the functionals on alternating strings, i.e. on the tuple of real numbers of arbitrary length, such that the sign of the differences between the elements alternates. In this habilitation an ansatz will be presented that allows also to represent hysteresis operators with vector valued input functions by a mapping defined on a set of strings. Here,the monotone functions are replaced by the monotaffine functions, intruded by the author. One can describe these functions by considering the composition of a monotone function on the real numbers with an affine function from the real number to the vector space such that the monotone function is applied first. Instead of alternating strings of real numbers, the so called convexity triple free string are considered. These strings are tuple of elements of the vector space such that no element in the tuple can be written as the convex combination of its predecessor and its successor. Thanks to representation result, one can generate uniquely hysteresis operator on continuous piecewise monotaffine input functions from mappings on the set of all convexity triple free string. This allows to investigate properties of operators by investigating properties of the corresponding mapping. Moreover, a further representation result is presented for hysteresis operators with input functions having a finite number of jumps. The corresponding strings are now tuple of quintuples. In the quintuple for each position of a change of the direction of the input function and for each jump, the corresponding value of the function, its limits from the right and from the left and information about the behavior of the function near to this position are stored.

Um in einer Menge von Daten nach bestimmten Informationen suchen und filtern zu können, verwenden Suchmaschinen und Datenbanksysteme Queries (Suchanfragen). Diese Queries sind häufig durch eine eigene Sprache definiert, welche die Bildung von komplexen Ausdrücken erlaubt. Die Systeme antworten auf die Suchanfrage in Form einer Ergebnismenge. Komplexe Suchanfragen ermöglichen dabei das Auffinden von präzisen Ergebnissen. Faceted Browsing ist ein Benutzerschnittstellen-Paradigma zum Suchen und Filtern von Daten. Dabei können Suchanfragen visuell erstellt und sukzessiv verfeinert werden, ohne die spezielle Anfragesprache kennen zu müssen. Die einfache und intuitive Benutzbarkeit der Oberfläche bildet das Erfolgsrezept, sodass Faceted Browsing in vielen Anwendungen, wie beispielsweise auch in Online-Shops, zum Einsatz kommt. Bisher sind die Systeme überwiegend so konzipiert, dass Queries, welche aus Konjunktionen von Disjunktionen bestehen, gebildet werden können. Es stellt sich nun die Frage, ob auch komplexere Suchanfragen mittels Faceted Browsing erstellt werden können und welche Veränderungen der Oberfläche dafür notwendig sind. Reichen die Veränderungen dabei so weit, dass zu Gunsten der Komplexität der Suchanfrage auf die Einfachheit der Oberfläche verzichtet werden muss oder existieren Möglichkeiten, komplexere Queries zu bilden und dabei die Einfachheit der Oberfläche zu bewahren? Ziel der Arbeit ist es, zu ermitteln, welche Komplexität die Suchanfragen, die mittels Faceted Browsing gebildet werden, aufweisen können, ohne dabei die einfache Benutzbarkeit der Facettenbrowseroberfläche zu verlieren. Dazu wird die bisherige Mächtigkeit von Facettenbrowseroberflächen hinsichtlich der Querybildung analysiert. Weiterhin werden komplexere Suchanfragen auf ihre Umsetzbarkeit mit Hilfe des Faceted Browsing untersucht. Es wird betrachtet, auf welche Weise sich bisherige Facettenbrowseroberflächen verändern müssen, um die visuelle Erstellung solcher Suchanfragen zu ermöglichen. Durch die prototypische Erweiterung eines bestehenden Facettenbrowsers um notwendige Oberflächenelemente soll die Möglichkeit bestehen, komplexere Suchanfragen, als bisher mittels Faceted Browsing möglich, zu bilden.

This book is devoted to the study of operators on discrete structures. The operators are supposed to be self-adjoint and obey a certain translation invariance property. The discrete structures are given as Cayley graphs via finitely generated groups. Here, sofic groups and amenable groups are in the center of our considerations. Note that every finitely generated amenable group is sofic. We investigate the spectrum of a discrete self-adjoint operator by studying a sequence of finite dimensional analogues of these operators. In the setting of amenable groups we obtain these approximating operators by restricting the operator in question to finite subsets Qn , n ∈ N. These finite dimensional operators are self-adjoint and therefore admit a well-defined normalized eigenvalue counting function. The limit of the normalized eigenvalue counting functions when |Qn | → ∞ (if it exists) is called the integrated density of states (IDS). It is a distribution function of a probability measure encoding the distribution of the spectrum of the operator in question on the real axis. We prove the existence of the IDS in various geometric settings and for different types of operators. The models we consider include deterministic as well as random situations. Depending on the specific setting, we prove existence of the IDS as a weak limit of distribution functions or even as a uniform limit. Moreover, in certain situations we are able to express the IDS via a semi-explicit formula using the trace of the spectral projection of the original operator. This is sometimes referred to as the validity of the Pastur-Shubin trace formula. In the most general geometric setting we study, the operators are defined on Cayley graphs of sofic groups. Here we prove weak convergence of the eigenvalue counting functions and verify the validity of the Pastur-Shubin trace formula for random and non-random operators . These results apply to operators which not necessarily bounded or of finite hopping range. The methods are based on resolvent techniques. This theory is established without having an ergodic theorem for sofic groups at hand. Note that ergodic theory is the usual tool used in the proof of convergence results of this type. Specifying to operators on amenable groups we are able to prove stronger results. In the discrete case, we show that the IDS exists uniformly for a certain class of finite hopping range operators. This is obtained by using a Banach space-valued ergodic theorem. We show that this applies to eigenvalue counting functions, which implies their convergence with respect to the Banach space norm, in this case the supremum norm. Thus, the heart of this theory is the verification of the Banach space-valued ergodic theorem. Proceeding in two steps we first prove this result for so-called ST-amenable groups. Then, using results from the theory of ε-quasi tilings, we prove a version of the Banach space-valued ergodic theorem which is valid for all amenable groups. Focusing on random operators on amenable groups, we prove uniform existence of the IDS without the assumption that the operator needs to be of finite hopping range or bounded. Moreover, we verify the Pastur-Shubin trace formula. Here we present different techniques. First we show uniform convergence of the normalized eigenvalue counting functions adapting the technique of the Banach space-valued ergodic theorem from the deterministic setting. In a second approach we use weak convergence of the eigenvalue counting functions and additionally obtain control over the convergence at the jumps of the IDS. These ingredients are applied to verify uniform existence of the IDS. In both situations we employ results from the theory of large deviations, in order to deal with long-range interactions.

There are various ways to regularize ill-posed operator equations in Hilbert space. If the underlying operator is accretive then Lavrentiev regularization (singular perturbation) is an immediate choice. The corresponding convergence rates for the regularization error depend on the given smoothness assumptions, and for general accretive operators these may be both with respect to the operator or its adjoint. Previous analysis revealed different convergence rates, and their optimality was unclear, specifically for adjoint source conditions. Based on the fundamental study by T. Kato, Fractional powers of dissipative operators. J. Math. Soc. Japan, 13(3):247--274, 1961, we establish power type convergence rates for this case. By measuring the optimality of such rates in terms on limit orders we exhibit optimality properties of the convergence rates, for general accretive operators under direct and adjoint source conditions, but also for the subclass of nonnegative selfadjoint operators.

We study a class of differential inclusions involving maximal monotone relations, which cover a huge class of problems in mathematical physics. For this purpose we introduce the time derivative as a continuously invertible operator in a suitable Hilbert space. It turns out that this realization is a strictly monotone operator and thus, the question on existence and uniqueness can be answered by well-known results in the theory of maximal monotone relations. Furthermore, we show that the resulting solution operator is Lipschitz-continuous and causal, which is a natural property of evolutionary processes. Finally, the results are applied to a system of partial differential equations and inclusions, which describes the diffusion of a compressible fluid through a saturated, porous, plastically deforming media, where certain hysteresis phenomena are modeled by maximal montone relations.

In dieser Arbeit untersuchen wir zwei Modelle, die das Laden und Entladen einer Lithium-Ionen Batterie beschreiben. Beide Modelle spiegeln eine Hysterese in dem Spannungs-Ladungs-Verlauf wider. Wir skizzieren den Modellierungsprozess von einem diskreten vielteilchen Modell sowie einem kontinuierlichen vielteilchen Modell. Das erste führt zu einer axiomatischen Beschreibung der Evolution makroskopischer Größen, während das zweite in eine nichtlineare Fokker-Planck Gleichung mündet. Wir zeigen die Existenz und Eindeutigkeit von Lösungen der nichtlinearen Fokker-Planck Gleichung und untersuchen deren qualitative Eigenschaften. Wir benutzen Interpolationsräume und Halbgruppen sektorieller Operatoren um den semilinearen Charakter der partiellen Differentialgleichung auszunutzen. Um globale Existenz zu erhalten, schätzen wir die Dissipation einer mit dem Modell verknüpften Energie ab. Diese Energie ist verwandt mit der L-log-L Norm, welche wir mithilfe einer Gagliardo-Nirenberg Ungleichung zu der L^2 Norm in Verbindung setzen können. Die notwendigen und hinreichenden Bedingungen zur globalen Existenz von Lösungen sind aus physikalischer Sicht plausibel. Der Ladezustand der Batterie muss innerhalb der Werte Voll und Leer sein. In numerischen Experimenten untersuchen wir das qualitative Verhalten von Lösungen. Wir zeigen die Konvergenz der numerischen Lösungen zu den exakten Lösungen. Dafür nutzen wir ähnliche Techniken wie bei der lokalen Existenztheorie. Wir beobachten die Tendenz von Lösungen sich um bestimmte Punkte zu konzentrieren. Unterstützt durch die formale Asymptotik zeigt dies für eine bestimmte Wahl von Parameter-Skalierungen, dass Lösungen gegen Dirac-Maße konvergieren. In diesem Grenzverhalten wird das System durch die Evolution von makroskopischen Größen beschrieben, welche wir auch in dem diskreten vielteilchen Modell wiederfinden. In diesen makroskopischen Größen lässt sich eine Hysterese beobachten. / We discuss two models which describe the charging and discharging of a lithium-ion battery and especially the hysteretical behaviour therein. We give an overview on the modelling process for a discrete many particle model and a continuous many particle model. The former results in an axiomatic description of macroscopic quantities while the latter gives a nonlinear Fokker-Planck equation. The nonlinear Fokker-Planck equation is analysed with respect to existence and uniqueness of solutions as well as qualitative behaviour of solutions. The nonlinearity in this partial differential equation stems from a coefficient which depends on the solution first non-local and second in a higher order. We use interpolation spaces and semigroups generated from sectorial operators to show the existence and uniqueness of solutions locally in time. The global existence in time relies on estimates for the dissipation of an energy. The suitable energy is related to the L-log-L norm and so a Gagliardo-Nirenberg inequality is needed to connect this back to L^2 estimates. It turns out that the conditions for global in time existence of solutions are physical reasonable. One needs that the loading state of the battery shall stay between totally empty and totally full. In numerical experiments we investigate the qualitative behaviour of solutions to the nonlinear Fokker-Planck equation. We are able to show convergence of the numerical solutions to the exact solution. We observe that solutions tend to concentrate at certain points. Supported by results from formal asymptotic expansions, we document the limiting behaviour in a certain scaling of the appearing parameters, which is the formation of Dirac measures. The evolution of the global quantities, which we observe in numerical simulations, is the same as what results from the discrete many particle model and one observes hysteretic behaviour in macroscopic quantities.

This thesis is devoted to the study of operators on discrete structures. The operators are supposed to be self-adjoint and obey a certain translation invariance property. The discrete structures are given as Cayley graphs via finitely generated groups. Here, sofic groups and amenable groups are in the center of our considerations. Note that every finitely generated amenable group is sofic. We investigate the spectrum of a discrete self-adjoint operator by studying a sequence of finite dimensional analogues of these operators. In the setting of amenable groups we obtain these approximating operators by restricting the operator in question to finite subsets Qn , n ∈ N. These finite dimensional operators are self-adjoint and therefore admit a well-defined normalized eigenvalue counting function. The limit of the normalized eigenvalue counting functions when |Qn | → ∞ (if it exists) is called the integrated density of states (IDS). It is a distribution function of a probability measure encoding the distribution of the spectrum of the operator in question on the real axis. In this thesis, we prove the existence of the IDS in various geometric settings and for different types of operators. The models we consider include deterministic as well as random situations. Depending on the specific setting, we prove existence of the IDS as a weak limit of distribution functions or even as a uniform limit. Moreover, in certain situations we are able to express the IDS via a semi-explicit formula using the trace of the spectral projection of the original operator. This is sometimes referred to as the validity of the Pastur-Shubin trace formula. In the most general geometric setting we study, the operators are defined on Cayley graphs of sofic groups. Here we prove weak convergence of the eigenvalue counting functions and verify the validity of the Pastur-Shubin trace formula for random and non-random operators . These results apply to operators which not necessarily bounded or of finite hopping range. The methods are based on resolvent techniques. This theory is established without having an ergodic theorem for sofic groups at hand. Note that ergodic theory is the usual tool used in the proof of convergence results of this type. Specifying to operators on amenable groups we are able to prove stronger results. In the discrete case, we show that the IDS exists uniformly for a certain class of finite hopping range operators. This is obtained by using a Banach space-valued ergodic theorem. We show that this applies to eigenvalue counting functions, which implies their convergence with respect to the Banach space norm, in this case the supremum norm. Thus, the heart of this theory is the verification of the Banach space-valued ergodic theorem. Proceeding in two steps we first prove this result for so-called ST-amenable groups. Then, using results from the theory of ε-quasi tilings, we prove a version of the Banach space-valued ergodic theorem which is valid for all amenable groups. Focusing on random operators on amenable groups, we prove uniform existence of the IDS without the assumption that the operator needs to be of finite hopping range or bounded. Moreover, we verify the Pastur-Shubin trace formula. Here we present different techniques. First we show uniform convergence of the normalized eigenvalue counting functions adapting the technique of the Banach space-valued ergodic theorem from the deterministic setting. In a second approach we use weak convergence of the eigenvalue counting functions and additionally obtain control over the convergence at the jumps of the IDS. These ingredients are applied to verify uniform existence of the IDS. In both situations we employ results from the theory of large deviations, in order to deal with long-range interactions.

Um in einer Menge von Daten nach bestimmten Informationen suchen und filtern zu können, verwenden Suchmaschinen und Datenbanksysteme Queries (Suchanfragen). Diese Queries sind häufig durch eine eigene Sprache definiert, welche die Bildung von komplexen Ausdrücken erlaubt. Die Systeme antworten auf die Suchanfrage in Form einer Ergebnismenge. Komplexe Suchanfragen ermöglichen dabei das Auffinden von präzisen Ergebnissen. Faceted Browsing ist ein Benutzerschnittstellen-Paradigma zum Suchen und Filtern von Daten. Dabei können Suchanfragen visuell erstellt und sukzessiv verfeinert werden, ohne die spezielle Anfragesprache kennen zu müssen. Die einfache und intuitive Benutzbarkeit der Oberfläche bildet das Erfolgsrezept, sodass Faceted Browsing in vielen Anwendungen, wie beispielsweise auch in Online-Shops, zum Einsatz kommt. Bisher sind die Systeme überwiegend so konzipiert, dass Queries, welche aus Konjunktionen von Disjunktionen bestehen, gebildet werden können. Es stellt sich nun die Frage, ob auch komplexere Suchanfragen mittels Faceted Browsing erstellt werden können und welche Veränderungen der Oberfläche dafür notwendig sind. Reichen die Veränderungen dabei so weit, dass zu Gunsten der Komplexität der Suchanfrage auf die Einfachheit der Oberfläche verzichtet werden muss oder existieren Möglichkeiten, komplexere Queries zu bilden und dabei die Einfachheit der Oberfläche zu bewahren? Ziel der Arbeit ist es, zu ermitteln, welche Komplexität die Suchanfragen, die mittels Faceted Browsing gebildet werden, aufweisen können, ohne dabei die einfache Benutzbarkeit der Facettenbrowseroberfläche zu verlieren. Dazu wird die bisherige Mächtigkeit von Facettenbrowseroberflächen hinsichtlich der Querybildung analysiert. Weiterhin werden komplexere Suchanfragen auf ihre Umsetzbarkeit mit Hilfe des Faceted Browsing untersucht. Es wird betrachtet, auf welche Weise sich bisherige Facettenbrowseroberflächen verändern müssen, um die visuelle Erstellung solcher Suchanfragen zu ermöglichen. Durch die prototypische Erweiterung eines bestehenden Facettenbrowsers um notwendige Oberflächenelemente soll die Möglichkeit bestehen, komplexere Suchanfragen, als bisher mittels Faceted Browsing möglich, zu bilden.

This book is devoted to the study of operators on discrete structures. The operators are supposed to be self-adjoint and obey a certain translation invariance property. The discrete structures are given as Cayley graphs via finitely generated groups. Here, sofic groups and amenable groups are in the center of our considerations. Note that every finitely generated amenable group is sofic. We investigate the spectrum of a discrete self-adjoint operator by studying a sequence of finite dimensional analogues of these operators. In the setting of amenable groups we obtain these approximating operators by restricting the operator in question to finite subsets Qn , n ∈ N. These finite dimensional operators are self-adjoint and therefore admit a well-defined normalized eigenvalue counting function. The limit of the normalized eigenvalue counting functions when |Qn | → ∞ (if it exists) is called the integrated density of states (IDS). It is a distribution function of a probability measure encoding the distribution of the spectrum of the operator in question on the real axis. We prove the existence of the IDS in various geometric settings and for different types of operators. The models we consider include deterministic as well as random situations. Depending on the specific setting, we prove existence of the IDS as a weak limit of distribution functions or even as a uniform limit. Moreover, in certain situations we are able to express the IDS via a semi-explicit formula using the trace of the spectral projection of the original operator. This is sometimes referred to as the validity of the Pastur-Shubin trace formula. In the most general geometric setting we study, the operators are defined on Cayley graphs of sofic groups. Here we prove weak convergence of the eigenvalue counting functions and verify the validity of the Pastur-Shubin trace formula for random and non-random operators . These results apply to operators which not necessarily bounded or of finite hopping range. The methods are based on resolvent techniques. This theory is established without having an ergodic theorem for sofic groups at hand. Note that ergodic theory is the usual tool used in the proof of convergence results of this type. Specifying to operators on amenable groups we are able to prove stronger results. In the discrete case, we show that the IDS exists uniformly for a certain class of finite hopping range operators. This is obtained by using a Banach space-valued ergodic theorem. We show that this applies to eigenvalue counting functions, which implies their convergence with respect to the Banach space norm, in this case the supremum norm. Thus, the heart of this theory is the verification of the Banach space-valued ergodic theorem. Proceeding in two steps we first prove this result for so-called ST-amenable groups. Then, using results from the theory of ε-quasi tilings, we prove a version of the Banach space-valued ergodic theorem which is valid for all amenable groups. Focusing on random operators on amenable groups, we prove uniform existence of the IDS without the assumption that the operator needs to be of finite hopping range or bounded. Moreover, we verify the Pastur-Shubin trace formula. Here we present different techniques. First we show uniform convergence of the normalized eigenvalue counting functions adapting the technique of the Banach space-valued ergodic theorem from the deterministic setting. In a second approach we use weak convergence of the eigenvalue counting functions and additionally obtain control over the convergence at the jumps of the IDS. These ingredients are applied to verify uniform existence of the IDS. In both situations we employ results from the theory of large deviations, in order to deal with long-range interactions.