No description available.

In dieser Arbeit wird eine detaillierte Analysis des konsistenten kinetischen Modells zum Kornwachstum von Fradkov durchgeführt. Dieses Modell beschreibt - basierend auf dem von Neumann--Mullins Gesetz - die Flächenänderung eines Korns abhängig von seiner Topologieklasse, d.h. der Anzahl der Kanten. Topologieänderungen werden durch Kopplungsterme zwischen den Gleichungen für die Anzahldichten der verschiedenen Topologieklassen beschrieben. Daraus resultiert ein unendlich-dimensionales System von Transportgleichungen mit tridiagonaler Kopplungsstruktur. Durch eine spezielle Wahl des Kopplungsgewichts, welche die Gleichungen nichtlinear und räumlich nichtlokal macht, wird das Modell konsistent. Nach einer Einführung wird das Modell von Fradkov im zweiten Kapitel hergeleitet; formale Rechnungen zeigen die Konsistenz des Modells auf. Im dritten Kapitel wird das Kopplungsgewicht a priori beschränkt. Dadurch kann im ersten Teil des vierten Kapitels Existenz und Eindeutigkeit von Lösungen für endlich-dimensionale Systeme gezeigt werden. Weitere Schranken an die Anzahldichten im fünften Kapitel ermöglichen den Grenzübergang hinsichtlich der Anzahl der Gleichungen im zweiten Teil des vierten Kapitels. Die Existenz von Lösungen des unendlich-dimensionalen Systems wird somit über eine geeignete Approximation gezeigt. Energiemethoden liefern Eindeutigkeit und stetige Abhängigkeit von den Daten. Im sechsten Kapitel wird das Langzeitverhalten untersucht. Besonderes Augenmerk liegt dabei auf stationären Lösungen eines reskalierten Systems als Kandidaten für selbstähnliche Lösungen. Abschließend wird das Lewis''sche Gesetz asymptotisch verifiziert. / The subject matter of this thesis is a detailed analysis of the self--consistent kinetic model for grain growth introduced by Fradkov. The model is based on the von Neumann--Mullins law describing the change of area of grains according to their topological class, i.e. the number of edges they have. Topological events are performed by coupling terms between equations for the number densities of different topological classes. The resulting system of transport equations is infinite-dimensional with a tridiagonal coupling structure. Self-consistency of this kinetic model is achieved by introducing a coupling''s weight making the equations nonlinear and nonlocal in space. We start with an introduction in the first chapter. Afterwards in the second chapter we derive Fradkov''s model and carry out formal calculations to illustrate self-consistency. In the third chapter we present a priori calculations mainly allowing us to bound the nonlinearity. This enables us to prove existence and uniqueness of solutions to finite-dimensional systems in the first part of the fourth chapter. Further bounds on the number densities established in the fifth chapter allow for passing to the limit concerning the number of equations in the second part of the fourth chapter. Therefore we prove existence of solutions to the infinite-dimensional system by a suitable approximation procedure. Uniqueness and continuous dependence on the data is then provided by energy methods. The sixth chapter focusses on long-time behaviour and mainly on stationary solutions of a rescaled system as candidates for self-similar solutions. Finally we prove Lewis'' law asymptotically.

Feynman path integrals introduced heuristically in the 1940s are a powerful tool used in many areas of physics, but also an intriguing mathematical challenge. In this work we used techniques of infinite dimensional integration (i.e. the infinite dimensional oscillatory integrals) in two different, but strictly connected, directions. On the one hand we construct a functional integral representation for solutions of a general high-order heat-type equations exploiting a recent generalization of infinite dimensional Fresnel integrals; in this framework we prove a a Girsanov-type formula, which is related, in the case of Schrödinger equation, to the Feynman path integral representation for the solution in presence of a magnetic field; eventually a new phase space path integral solution for higher-order heat-type equations is also presented. On the other hand for the three dimensional Schrödinger equation with magnetic field we provide a rigorous mathematical Feynman path integral formula still in the context of infinite dimensional oscillatory integrals; moreover, the requirement of independence of the integral on the approximation procedure forces the introduction of a counterterm, which has to be added to the classical action functional (this is done by the example of a linear vector potential). Thanks to that, it is possible to give a natural explanation for the appearance of the Stratonovich integral in the path integral formula for both the Schrödinger and the heat equation with magnetic field.

This thesis focuses on optimization on a generic objective function based on reflected Brownian motion (RBM). We investigate in several approaches including the partial differential equation approach where we write our objective function in terms of a Hamilton-Jacobi-Bellman equation using the dynamic programming principle and the gradient descent approach where we use two different gradient estimators. We provide extensive numerical results with the gradient descent approach and we discuss the difficulties and future study opportunities for this problem.

<p>In 1989 Jun Kigami made an analytic construction of a Laplacian on the Sierpiński gasket, a construction that he extended to post critically finite fractals. Since then, this field has evolved into a proper theory of analysis on fractals. The new results obtained in this thesis are all in the setting of Kigami's theory. They are presented in three papers.</p><p>Strichartz recently showed that there are first order linear differential equations, based on the Laplacian, that are not solvable on the Sierpiński gasket. In the first paper we give a characterization on the polynomial p so that the differential equation p(Δ)u=f is solvable on any open subset of the Sierpiński gasket for any f continuous on that subset. For general p we find the open subsets on which p(Δ)u=f is solvable for any continuous f.</p><p>In the second paper we describe the infinitesimal geometric behavior of a large class of smooth functions on the Sierpiński gasket in terms of the limit distribution of their local eccentricity, a generalized direction of gradient. The distribution of eccentricities is codified as an infinite dimensional perturbation problem for a suitable iterated function system, which has the limit distribution as an invariant measure. We extend results for harmonic functions found by Öberg, Strichartz and Yingst to larger classes of functions.</p><p>In the third paper we define and study intrinsic first order derivatives on post critically finite fractals and prove differentiability almost everywhere for certain classes of fractals and functions. We apply our results to extend the geography is destiny principle, and also obtain results on the pointwise behavior of local eccentricities. Our main tool is the Furstenberg-Kesten theory of products of random matrices.</p>

<p>In this thesis, we give two constructions for Riemannian metrics on Seiberg-Witten moduli spaces. Both these constructions are naturally induced from the L2-metric on the configuration space. The construction of the so called quotient L2-metric is very similar to the one construction of an L2-metric on Yang-Mills moduli spaces as given by Groisser and Parker. To construct a Riemannian metric on the total space of the Seiberg-Witten bundle in a similar way, we define the reduced gauge group as a subgroup of the gauge group. We show, that the quotient of the premoduli space by the reduced gauge group is isomorphic as a U(1)-bundle to the quotient of the premoduli space by the based gauge group. The total space of this new representation of the Seiberg-Witten bundle carries a natural quotient L2-metric, and the bundle projection is a Riemannian submersion with respect to these metrics. We compute explicit formulae for the sectional curvature of the moduli space in terms of Green operators of the elliptic complex associated with a monopole. Further, we construct a Riemannian metric on the cobordism between moduli spaces for different perturbations. The second construction of a Riemannian metric on the moduli space uses a canonical global gauge fixing, which represents the total space of the Seiberg-Witten bundle as a finite dimensional submanifold of the configuration space.</p> <p>We consider the Seiberg-Witten moduli space on a simply connected K&auml;uhler surface. We show that the moduli space (when nonempty) is a complex projective space, if the perturbation does not admit reducible monpoles, and that the moduli space consists of a single point otherwise. The Seiberg-Witten bundle can then be identified with the Hopf fibration. On the complex projective plane with a special Spin-C structure, our Riemannian metrics on the moduli space are Fubini-Study metrics. Correspondingly, the metrics on the total space of the Seiberg-Witten bundle are Berger metrics. We show that the diameter of the moduli space shrinks to 0 when the perturbation approaches the wall of reducible perturbations. Finally we show, that the quotient L2-metric on the Seiberg-Witten moduli space on a K&auml;hler surface is a K&auml;hler metric.</p> / <p>In dieser Dissertationsschrift geben wir zwei Konstruktionen Riemannscher Metriken auf Seiberg-Witten-Modulr&auml;umen an. Beide Metriken werden in nat&uuml;rlicher Weise durch die L2-Metrik des Konfiguartionsraumes induziert. Die Konstruktion der sogenannten Quotienten-L2-Metrik entspricht der durch Groisser und Parker angegebenen Konstruktion einer L2-Metrik auf Yang-Mills-Modulr&auml;umen. Zur Konstruktion einer Quotienten-Metrik auf dem Totalraum des Seiberg-Witten-B&uuml;ndels f&uuml;hren wir die sogenannte reduzierte Eichgruppe ein. Wir zeigen, dass der Quotient des Pr&auml;modulraumes nach der reduzierten Eichgruppe als U(1)-B&uuml;ndel isomorph ist zu dem Quotienten nach der basierten Eichgruppe. Dadurch tr&auml;gt der Totalraum des Seiberg-Witten B&uuml;ndels eine nat&uuml;rliche Quotienten-L2-Metrik, bzgl. derer die B&uuml;ndelprojektion eine Riemannsche Submersion ist. Wir berechnen explizite Formeln f&uuml;r die Schnittr&uuml;mmung des Modulraumes in Ausdr&uuml;cken der Green-Operatoren des zu einem Monopol geh&ouml;rigen elliptischen Komplexes. Ferner konstruieren wir eine Riemannsche Metrik auf dem Kobordismus zwischen Modulr&auml;umen zu verschiedenen St&ouml;rungen. Die zweite Konstruktion einer Riemannschen Metrik auf Seiberg-Witten-Modulr&auml;umen benutzt eine kanonische globale Eichfixierung, verm&ouml;ge derer der Totalraum des Seiberg-Witten-B&uuml;ndels als endlich-dimensionale Untermannigfaltigkeit des Konfigurationsraumes dargestellt werden kann.</p> <p>Wir betrachten speziell die Seiberg-Witten-Modulr&auml;ume auf einfach zusammenh&auml;ngenden K&auml;hler-Mannigfaltigkeiten. Wir zeigen, dass der Seiberg-Witten-Modulraum (falls nicht-leer) im irreduziblen Fall ein komplex projektiver Raum its und im reduziblen Fall aus einem einzelnen Punkt besteht. Das Seiberg-Witten-B&uuml;ndel l&auml;&szlig;t sich mit der Hopf-Faserung identifizieren. Die L2-Metrik des Modulraumes auf der komplex projektiven Fl&auml;che CP2 (mit einer speziellen Spin-C-Struktur) ist die Fubini-Study-Metrik; entsprechend sind die Metriken auf dem Totalraum Berger-Metriken. Wir zeigen, dass der Durchmesser des Modulraumes gegen 0 konvergiert, wenn die St&ouml;rung sich dem reduziblen Fall n&auml;hert. Schlie&szlig;lich zeigen wir, dass die Quotienten-L2-Metrik auf dem Seiberg-Witten-Modulraum einer K&auml;hlerfl&auml;che eine K&auml;hler-Metrik ist.</p>

In 1989 Jun Kigami made an analytic construction of a Laplacian on the Sierpiński gasket, a construction that he extended to post critically finite fractals. Since then, this field has evolved into a proper theory of analysis on fractals. The new results obtained in this thesis are all in the setting of Kigami's theory. They are presented in three papers. Strichartz recently showed that there are first order linear differential equations, based on the Laplacian, that are not solvable on the Sierpiński gasket. In the first paper we give a characterization on the polynomial p so that the differential equation p(Δ)u=f is solvable on any open subset of the Sierpiński gasket for any f continuous on that subset. For general p we find the open subsets on which p(Δ)u=f is solvable for any continuous f. In the second paper we describe the infinitesimal geometric behavior of a large class of smooth functions on the Sierpiński gasket in terms of the limit distribution of their local eccentricity, a generalized direction of gradient. The distribution of eccentricities is codified as an infinite dimensional perturbation problem for a suitable iterated function system, which has the limit distribution as an invariant measure. We extend results for harmonic functions found by Öberg, Strichartz and Yingst to larger classes of functions. In the third paper we define and study intrinsic first order derivatives on post critically finite fractals and prove differentiability almost everywhere for certain classes of fractals and functions. We apply our results to extend the geography is destiny principle, and also obtain results on the pointwise behavior of local eccentricities. Our main tool is the Furstenberg-Kesten theory of products of random matrices.

Studying the zeros of a parameter dependent operator F defined on a Hilbert space H is a fundamental problem in mathematics. When the Hilbert space is finite dimensional, continuation provides, via predictor-corrector algorithms, efficient techniques to numerically follow the zeros of F as we move the parameter. In the case of infinite dimensional Hilbert spaces, this procedure must be applied to some finite dimensional approximation which of course raises the question of validity of the output. We introduce a new technique that combines the information obtained from the predictor-corrector steps with ideas from rigorous computations and verifies that the numerically produced zero for the finite dimensional system can be used to explicitly define a set which contains a unique zero for the infinite dimensional problem F: HxR->Im(F). We use this new validated continuation to study equilibrium solutions of partial differential equations, to prove the existence of chaos in ordinary differential equations and to follow branches of periodic solutions of delay differential equations. In the context of partial differential equations, we show that the cost of validated continuation is less than twice the cost of the standard continuation method alone.

Quantifying uncertainties in large-scale forward and inverse PDE simulations has emerged as a central challenge facing the field of computational science and engineering. The promise of modeling and simulation for prediction, design, and control cannot be fully realized unless uncertainties in models are rigorously quantified, since this uncertainty can potentially overwhelm the computed result. While statistical inverse problems can be solved today for smaller models with a handful of uncertain parameters, this task is computationally intractable using contemporary algorithms for complex systems characterized by large-scale simulations and high-dimensional parameter spaces. In this dissertation, I address issues regarding the theoretical formulation, numerical approximation, and algorithms for solution of infinite-dimensional Bayesian statistical inverse problems, and apply the entire framework to a problem in global seismic wave propagation. Classical (deterministic) approaches to solving inverse problems attempt to recover the “best-fit” parameters that match given observation data, as measured in a particular metric. In the statistical inverse problem, we go one step further to return not only a point estimate of the best medium properties, but also a complete statistical description of the uncertain parameters. The result is a posterior probability distribution that describes our state of knowledge after learning from the available data, and provides a complete description of parameter uncertainty. In this dissertation, a computational framework for such problems is described that wraps around the existing forward solvers, as long as they are appropriately equipped, for a given physical problem. Then a collection of tools, insights and numerical methods may be applied to solve the problem, and interrogate the resulting posterior distribution, which describes our final state of knowledge. We demonstrate the framework with numerical examples, including inference of a heterogeneous compressional wavespeed field for a problem in global seismic wave propagation with 10⁶ parameters.

Incomplete markets provide many challenges for both investment decisions and valuation problems. While both problems have received extensive attention in complete markets, there remain many open areas in the theory of incomplete markets. We present the results in three parts. In the first essay we consider the Merton investment problem of optimal portfolio choice when the traded instruments are the set of zero-coupon bonds. Working within a Markovian Heath-Jarrow-Morton framework of the interest rate term structure driven by an infinite dimensional Wiener process, we give sufficient conditions for the existence and uniqueness of an optimal investment strategy. When there is uniqueness, we provide a characterization of the optimal portfolio. Furthermore, we show that a specific Gauss-Markov random field model can be treated within this framework, and explicitly calculate the optimal portfolio. We show that the optimal portfolio in this case can be identified with the discontinuities of a certain function of the market parameters. In the second essay we price a claim, using the indifference valuation methodology, in the model presented in the first section. We appeal to the indifference pricing framework instead of the classic Black-Scholes method due to the natural incompleteness in such a market model. Because we price time-sensitive interest rate claims, the units in which we price are very important. This will require us to take care in formulating the investor’s utility function in terms of the units in which we express the wealth function. This leads to new results, namely a general change-of-numeraire theorem in incomplete markets via indifference pricing. Lastly, in the third essay, we propose a method to price credit derivatives, namely collateralized debt obligations (CDOs) using indifference. We develop a numerical algorithm for pricing such CDOs. The high illiquidity of the CDO market coupled with the allowance of default in the underlying traded assets creates a very incomplete market. We explain the market-observed prices of such credit derivatives via the risk aversion of investors. In addition to a general algorithm, several approximation schemes are proposed. / text