The present thesis addresses both theoretical as well as numerical aspects of the treatment of nonlinear inverse problems. The first part considers Tikhonov regularization for nonlinear ill-posed operator equations in Hilbert scales with oversmoothing penalties. Sufficient as well as necessary conditions to establish convergence are introduced and convergence rate results are given for various parameter choice rules under a two sided nonlinearity constraint. Ultimately, both a posteriori as well as certain a priori parameter choice rules lead to identical converce rates. The theoretical results are supported and augmented by extensive numerical case studies. In particular it is shown, that the localization of the above mentioned nonlinearity constraint is not trivial. Incorrect localization will prevent convergence of the regularized to the exact solution. The second part of the thesis considers two open problems in inverse option pricing and electrical impedance tomography. While regularization through discretization is sufficient to overcome ill-posedness of the latter, the first requires a more sophisticated approach. It is shown, that the recovery of time dependent volatility and interest rate functions from observed option prices is everywhere locally ill-posed. This motivates Tikhonov-type or variational regularization with two parameters and penalty terms to simultaneously recover these functions. Two parameter choice rules using the L-hypersurface as well as a combination of L-curve and quasi-optimality are introduced. The results are again supported by extensive numerical case studies.

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Optimierungsaufgaben unter Partiellen Differentialgleichungen (PDGLs) tauchen in verschiedensten Anwendungen der Wissenschaft und Technik auf. Wenn wir ein PDGL Problem formulieren, kann es aufgrund seiner Größe unmöglich werden, das Problem mit konventionellen Methoden zu lösen. Zusätzlich noch eine Optimierung auszuführen birgt zusätzliche Schwierigkeiten. In vielen Fällen können wir das PDGL Problem in einem kompakteren Format formulieren indem wir der zugrundeliegenden Kronecker-Produkt Struktur zwischen Raum- und Zeitdimension Aufmerksamkeit schenken. Wenn die PDGL zusätzlich mit Isogeometrischer Analysis diskretisiert wurde, können wir zusätlich eine Niedrig-Rang Approximation zwischen den einzelnen Raumdimensionen erzeugen. Diese Niedrig-Rang Approximation lässt uns die Systemmatrizen schnell und speicherschonend aufstellen. Das folgende PDGL-Problem lässt sich als Summe aus Kronecker-Produkten beschreiben, welche als eine Niedrig-Rang Tensortrain Formulierung interpretiert werden kann. Diese kann effizient im Niedrig-Rang Format gelöst werden. Wir illustrieren dies mit unterschiedlichen, anspruchsvollen Beispielproblemen.:Introduction Tensor Train Format Isogeometric Analysis PDE-constrained Optimization Bayesian Inverse Problems A low-rank tensor method for PDE-constrained optimization with Isogeometric Analysis A low-rank matrix equation method for solving PDE-constrained optimization problems A low-rank tensor method to reconstruct sparse initial states for PDEs with Isogeometric Analysis Theses and Summary Bibilography / Optimization problems governed by Partial Differential Equations (PDEs) arise in various applications of science and engineering. If we formulate a discretization of a PDE problem, it may become infeasible to treat the problem with conventional methods due to its size. Solving an optimization problem on top of the forward problem poses additional difficulties. Often, we can formulate the PDE problem in a more compact format by paying attention to the underlying Kronecker product structure between the space and time dimension of the discretization. When the PDE is discretized with Isogeometric Analysis we can additionally formulate a low-rank representation with Kronecker products between its individual spatial dimensions. This low-rank formulation gives rise to a fast and memory efficient assembly for the system matrices. The PDE problem represented as a sum of Kronecker products can then be interpreted as a low-rank tensor train formulation, which can be efficiently solved in a low-rank format. We illustrate this for several challenging PDE-constrained problems.:Introduction Tensor Train Format Isogeometric Analysis PDE-constrained Optimization Bayesian Inverse Problems A low-rank tensor method for PDE-constrained optimization with Isogeometric Analysis A low-rank matrix equation method for solving PDE-constrained optimization problems A low-rank tensor method to reconstruct sparse initial states for PDEs with Isogeometric Analysis Theses and Summary Bibilography

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Ziel dieser Arbeit ist theoretische Eigenschaften verschiedener Bootstrap Methoden zu untersuchen. Als Ergebnis führen wir die Konvergenzraten des Bootstrap-Verfahrens ein, die sich auf die Differenz zwischen der tatsächlichen Verteilung einer Statistik und der Resampling-Näherung beziehen. In dieser Arbeit analysieren wir die Verteilung der l2-Norm der Summe unabhängiger Vektoren, des Summen Maximums in hoher Dimension, des Wasserstein-Abstands zwischen empirischen Messungen und Wassestein-Barycenters. Um die Bootstrap-Konvergenz zu beweisen, verwenden wir die Gaussche Approximations technik. Das bedeutet dass man in der betrachteten Statistik eine Summe unabhängiger Vektoren finden muss, so dass Bootstrap eine erneute Abtastung dieser Summe ergibt. Ferner kann diese Summe durch Gaussche Verteilung angenähert und mit der Neuabtastung Verteilung als Differenz zwischen Kovarianzmatrizen verglichen werden. Im Allgemeinen scheint es sehr schwierig zu sein, eine solche Summe unabhängiger Vektoren aufzudecken, da einige Statistiken (zum Beispiel MLE) keine explizite Gleichung haben und möglicherweise unendlich dimensional sind. Um mit dieser Schwierigkeit fertig zu werden, verwenden wir einige neuartige Ergebnisse aus der statistischen Lerntheorie. Darüber hinaus wenden wir Bootstrap bei Methoden zur Erkennung von Änderungspunkten an. Im parametrischen Fall analysieren wir den statischen Likelihood Ratio Test (LRT). Seine hohen Werte zeigen Änderungen der Parameter Verteilung in der Datensequenz an. Das Maximum von LRT hat eine unbekannte Verteilung und kann mit Bootstrap kalibriert werden. Wir zeigen die Konvergenzraten zur realen maximalen LRT-Verteilung. In nicht parametrischen Fällen verwenden wir anstelle von LRT den Wasserstein-Abstand zwischen empirischen Messungen. Wir testen die Genauigkeit von Methoden zur Erkennung von Änderungspunkten anhand von synthetischen Zeitreihen und Elektrokardiographiedaten. Letzteres zeigt einige Vorteile des nicht parametrischen Ansatzes gegenüber komplexen Modellen und LRT. / The objective of this thesis is to explore theoretical properties of various bootstrap methods. We introduce the convergence rates of the bootstrap procedure which corresponds to the difference between real distribution of some statistic and its resampling approximation. In this work we analyze the distribution of Euclidean norm of independent vectors sum, maximum of sum in high dimension, Wasserstein distance between empirical measures, Wassestein barycenters. In order to prove bootstrap convergence we involve Gaussian approximation technique which means that one has to find a sum of independent vectors in the considered statistic such that bootstrap yields a resampling of this sum. Further this sum may be approximated by Gaussian distribution and compared with the resampling distribution as a difference between variance matrices. In general it appears to be very difficult to reveal such a sum of independent vectors because some statistics (for example, MLE) don't have an explicit equation and may be infinite-dimensional. In order to handle this difficulty we involve some novel results from statistical learning theory, which provide a finite sample quadratic approximation of the Likelihood and suitable MLE representation. In the last chapter we consider the MLE of Wasserstein barycenters model. The regularised barycenters model has bounded derivatives and satisfies the necessary conditions of quadratic approximation. Furthermore, we apply bootstrap in change point detection methods. In the parametric case we analyse the Likelihood Ratio Test (LRT) statistic. Its high values indicate changes of parametric distribution in the data sequence. The maximum of LRT has a complex distribution but its quantiles may be calibrated by means of bootstrap. We show the convergence rates of the bootstrap quantiles to the real quantiles of LRT distribution. In non-parametric case instead of LRT we use Wasserstein distance between empirical measures. We test the accuracy of change point detection methods on synthetic time series and electrocardiography (ECG) data. Experiments with ECG illustrate advantages of the non-parametric approach versus complex parametric models and LRT.

Eine einfache Darstellung der Grundgleichungen für 'Große Deformationen' und Herleitung eines geeigneten Fehlerschätzers für die adaptive FEM.

Dissertation ueber die effektive Vorkonditionierung linearer Gleichungssysteme resultierend aus der Diskretisierung eines elliptischen Randwertproblems 2. Ordnung mittels der Methode der Finiten Elementen. Als Vorkonditionierer werden multi-level artige Vorkonditionierer (BPX, Multi-grid, Wavelets) benutzt.

We establish multiresolution norm equivalences in weighted spaces <i>L²_w</i>((0,1)) with possibly singular weight functions <i>w(x)</i>&ge;0 in (0,1). Our analysis exploits the locality of the biorthogonal wavelet basis and its dual basis functions. The discrete norms are sums of wavelet coefficients which are weighted with respect to the collocated weight function <i>w(x)</i> within each scale. Since norm equivalences for Sobolev norms are by now well-known, our result can also be applied to weighted Sobolev norms. We apply our theory to the problem of preconditioning <i>p</i>-Version FEM and wavelet discretizations of degenerate elliptic problems.

In this paper, finite element discretizations of the degenerated operator -&omega;²(y) u_xx-&omega;²(x)u_yy=g in the unit square are investigated, where the weight function satisfies &omega;(&xi;)=&xi;^&alpha; with &alpha; &ge; 0. We propose two multi-level methods in order to solve the resulting system of linear algebraic equations. The first method is a multi-grid algorithm with line-smoother. A proof of the smoothing property is given. The second method is a BPX-like preconditioner which we call MTS-BPX preconditioner. We show that the upper eigenvalue bound of the MTS-BPX preconditioned system matrix grows proportionally to the level number.