Inverse problems for abstract evolution equations with applications in electrodynamics and elasticity

It is common knowledge { mainly based on experience { that parameter identification problems in partial differential equations are ill-posed. Yet, a mathematical sound argumentation is missing, except for some special cases. We present a general theory for inverse problems related to abstract evolution equations which explains not only their local ill-posedness but also provides the Fréchet derivative of the corresponding parameter-to-solution map which is needed, e.g., in Newton-like solvers. Our abstract results are applied to inverse problems related to the following first order hyperbolic systems: Maxwell\u27s equation (electromagnetic scattering in conducting media) and elastic wave equation (seismic imaging)

Proceedings of the International Workshop on Medical Ultrasound Tomography: 1.- 3. Nov. 2017, Speyer, Germany

Ultrasound Tomography is an emerging technology for medical imaging that is quickly approaching its clinical utility. Research groups around the globe are engaged in research spanning from theory to practical applications. The International Workshop on Medical Ultrasound Tomography (1.-3. November 2017, Speyer, Germany) brought together scientists to exchange their knowledge and discuss new ideas and results in order to boost the research in Ultrasound Tomography

Non-Smooth Optimization by Abs-Linearization in Reflexive Function Spaces

Nichtglatte Optimierungsprobleme in reflexiven Banachräumen treten in vielen Anwendungen auf. Häufig wird angenommen, dass alle vorkommenden Nichtdifferenzierbarkeiten durch Lipschitz-stetige Operatoren wie abs, min und max gegeben sind. Bei solchen Problemen kann es sich zum Beispiel um optimale Steuerungsprobleme mit möglicherweise nicht glatten Zielfunktionen handeln, welche durch partielle Differentialgleichungen (PDG) eingeschränkt sind, die ebenfalls nicht glatte Terme enthalten können. Eine effiziente und robuste Lösung erfordert eine Kombination numerischer Simulationen und spezifischer Optimierungsalgorithmen. Lokal Lipschitz-stetige, nichtglatte Nemytzkii-Operatoren, welche direkt in der Problemformulierung auftreten, spielen eine wesentliche Rolle in der Untersuchung der zugrundeliegenden Optimierungsprobleme. In dieser Dissertation werden zwei spezifische Methoden und Algorithmen zur Lösung solcher nichtglatter Optimierungsprobleme in reflexiven Banachräumen vorgestellt und diskutiert. Als erste Lösungsmethode wird in dieser Dissertation die Minimierung von nichtglatten Operatoren in reflexiven Banachräumen mittels sukzessiver quadratischer Überschätzung vorgestellt, SALMIN. Ein neuartiger Optimierungsansatz für Optimierungsprobleme mit nichtglatten elliptischen PDG-Beschränkungen, welcher auf expliziter Strukturausnutzung beruht, stellt die zweite Lösungsmethode dar, SCALi. Das zentrale Merkmal dieser Methoden ist ein geeigneter Umgang mit Nichtglattheiten. Besonderes Augenmerk liegt dabei auf der zugrundeliegenden nichtglatten Struktur des Problems und der effektiven Ausnutzung dieser, um das Optimierungsproblem auf angemessene und effiziente Weise zu lösen.Non-smooth optimization problems in reflexive Banach spaces arise in many applications. Frequently, all non-differentiabilities involved are assumed to be given by Lipschitz-continuous operators such as abs, min and max. For example, such problems can refer to optimal control problems with possibly non-smooth objective functionals constrained by partial differential equations (PDEs) which can also include non-smooth terms. Their efficient as well as robust solution requires numerical simulations combined with specific optimization algorithms. Locally Lipschitz-continuous non-smooth non-linearities described by appropriate Nemytzkii operators which arise directly in the problem formulation play an essential role in the study of the underlying optimization problems. In this dissertation, two specific solution methods and algorithms to solve such non-smooth optimization problems in reflexive Banach spaces are proposed and discussed. The minimization of non-smooth operators in reflexive Banach spaces by means of successive quadratic overestimation is presented as the first solution method, SALMIN. A novel structure exploiting optimization approach for optimization problems with non-smooth elliptic PDE constraints constitutes the second solution method, SCALi. The central feature of these methods is the appropriate handling of non-differentiabilities. Special focus lies on the underlying structure of the problem stemming from the non-smoothness and how it can be effectively exploited to solve the optimization problem in an appropriate and efficient way

New PDE models for imaging problems and applications

Variational methods and Partial Differential Equations (PDEs) have been extensively employed for the mathematical formulation of a myriad of problems describing physical phenomena such as heat propagation, thermodynamic transformations and many more. In imaging, PDEs following variational principles are often considered. In their general form these models combine a regularisation and a data fitting term, balancing one against the other appropriately. Total variation (TV) regularisation is often used due to its edgepreserving and smoothing properties. In this thesis, we focus on the design of TV-based models for several different applications. We start considering PDE models encoding higher-order derivatives to overcome wellknown TV reconstruction drawbacks. Due to their high differential order and nonlinear nature, the computation of the numerical solution of these equations is often challenging. In this thesis, we propose directional splitting techniques and use Newton-type methods that despite these numerical hurdles render reliable and efficient computational schemes. Next, we discuss the problem of choosing the appropriate data fitting term in the case when multiple noise statistics in the data are present due, for instance, to different acquisition and transmission problems. We propose a novel variational model which encodes appropriately and consistently the different noise distributions in this case. Balancing the effect of the regularisation against the data fitting is also crucial. For this sake, we consider a learning approach which estimates the optimal ratio between the two by using training sets of examples via bilevel optimisation. Numerically, we use a combination of SemiSmooth (SSN) and quasi-Newton methods to solve the problem efficiently. Finally, we consider TV-based models in the framework of graphs for image segmentation problems. Here, spectral properties combined with matrix completion techniques are needed to overcome the computational limitations due to the large amount of image data. Further, a semi-supervised technique for the measurement of the segmented region by means of the Hough transform is proposed

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described