Well-Behavior, Well-Posedness and Nonsmooth Analysis

AMS subject classification: 90C30, 90C33.We survey the relationships between well-posedness and well-behavior. The latter notion means that any critical sequence (xn) of a lower semicontinuous function f on a Banach space is minimizing. Here “critical” means that the remoteness of the subdifferential ∂f(xn) of f at xn (i.e. the distance of 0 to ∂f(xn)) converges to 0. The objective function f is not supposed to be convex or smooth and the subdifferential ∂ is not necessarily the usual Fenchel subdifferential. We are thus led to deal with conditions ensuring that a growth property of the subdifferential (or the derivative) of a function implies a growth property of the function itself. Both qualitative questions and quantitative results are considered

Investigations in Hadamard spaces

Kjo tezë e doktoratës hulumton ndërveprimin midis gjeometrisë dhe analizës konvekse në hapësirat Hadamard. E motivuar nga aplikime të shumta të gjeometrisë CAT(0), puna jonë bazohet në rezultatet e shumë autorëve të mëparshëmnë mbi analizën konvekse dhe gjeometrinë në sensin e Alexandrovit. Hetimet tona u përgjigjen disa pyetjeve në teorinë e hapësirave CAT(0) prej të cilave disa janë parashtruar si probleme të hapura në literaturën e fundit. Teza jonë e doktoratës zhvillohet sipas linjave të mëposhtme: 1. Topologjitë e dobëta në hapësirat Hadamard, 2. Konveksifikimi i bashkësive kompakte, 3. Problemi i pemës mesatare në hapësirat e pemëve filogjenetike, 4. Konvergjenca Mosko në hapësirat Hadamard, 5. Operatorët (plotësisht) jo-ekspansivë dhe aplikimet e tyre në hapësirat Hadamard.Diese Doktorarbeit untersucht das Zusammenspiel zwischen Geometrie und konvexer Analyse in Hadamardräumen. Motiviert durch zahlreiche Anwendungen der CAT(0)-Geometrie baut unsere Arbeit auf den Ergebnissen vieler früherer Autoren in der konvexen Analysis und der Alexandrov-Geometrie auf. Unsere Untersuchungen beantworten mehrere Fragen in der Theorie von CAT(0)-Räumen, von denen einige in der neueren Literatur als offene Probleme gestellt wurden. Zusammengefasst entwickelt sich unsere Dissertation in folgende Richtungen: 1. Schwache Topologien in Hadamard-Räumen, 2. Konvexe Hüllen kompakter Mengen, 3. Mittleres Baumproblem in phylogenetischen Baumräumen, 4. Mosco-Konvergenz in Hadamard-Räumen, 5. Fest nichtexpansive Operatoren und ihre Anwendungen in Hadamard-Räumen.This thesis investigates the interplay between geometry and convex analysis in Hadamard spaces. Motivated by numerous applications of CAT(0) geometry, our work builds upon the results in convex analysis and Alexandrov geometry of many previous authors. Our investigations answer several questions in the theory of CAT(0) spaces some of which were posed as open problems in recent literature. In a nutshell our thesis develops along the following lines: 1. Weak topologies in Hadamard spaces, 2. Convex hulls of compact sets, 3. Mean tree problem in phylogenetic tree spaces, 4. Mosco convergence in Hadamard spaces, 5. Firmly nonexpansive operators and their applications in Hadamard spaces

Applications of Optimal Transportation in the Natural Sciences (online meeting)

Concepts and methods from the mathematical theory of optimal transportation have reached significant importance in various fields of the natural sciences. The view on classical problems from a "transport perspective'' has lead to the development of powerful problem-adapted mathematical tools, and sometimes to a novel geometric understanding of the matter. The natural sciences, in turn, are the most important source of ideas for the further development of the optimal transport theory, and are a driving force for the design of efficient and reliable numerical methods to approximate Wasserstein distances and the like. The presentations and discussions in this workshop have been centered around recent analytical results and numerical methods in the field of optimal transportation that have been motivated by specific applications in statistical physics, quantum mechanics, and chemistry

Numerical Methods in Shape Spaces and Optimal Branching Patterns

The contribution of this thesis is twofold. The main part deals with numerical methods in the context of shape space analysis, where the shape space at hand is considered as a Riemannian manifold. In detail, we apply and extend the time-discrete geodesic calculus (established by Rumpf and Wirth [WBRS11, RW15]) to the space of discrete shells, i.e. triangular meshes with fixed connectivity. The essential building block is a variational time-discretization of geodesic curves, which is based on a local approximation of the squared Riemannian distance on the manifold. On physical shape spaces this approximation can be derived e.g. from a dissimilarity measure. The dissimilarity measure between two shell surfaces can naturally be defined as an elastic deformation energy capturing both membrane and bending distortions. Combined with a non-conforming discretization of a physically sound thin shell model the time-discrete geodesic calculus applied to the space of discrete shells is shown to be suitable to solve important problems in computer graphics and animation. To extend the existing calculus, we introduce a generalized spline functional based on the covariant derivative along a curve in shape space whose minimizers can be considered as Riemannian splines. We establish a corresponding time-discrete functional that fits perfectly into the framework of Rumpf and Wirth, and prove this discretization to be consistent. Several numerical simulations reveal that the optimization of the spline functional—subject to appropriate constraints—can be used to solve the multiple interpolation problem in shape space, e.g. to realize keyframe animation. Based on the spline functional, we further develop a simple regression model which generalizes linear regression to nonlinear shape spaces. Numerical examples based on real data from anatomy and botany show the capability of the model. Finally, we apply the statistical analysis of elastic shape spaces presented by Rumpf and Wirth [RW09, RW11] to the space of discrete shells. To this end, we compute a Fréchet mean within a class of shapes bearing highly nonlinear variations and perform a principal component analysis with respect to the metric induced by the Hessian of an elastic shell energy. The last part of this thesis deals with the optimization of microstructures arising e.g. at austenite-martensite interfaces in shape memory alloys. For a corresponding scalar problem, Kohn and Müller [KM92, KM94] proved existence of a minimizer and provided a lower and an upper bound for the optimal energy. To establish the upper bound, they studied a particular branching pattern generated by mixing two different martensite phases. We perform a finite element simulation based on subdivision surfaces that suggests a topologically different class of branching patterns to represent an optimal microstructure. Based on these observations we derive a novel, low dimensional family of patterns and show—numerically and analytically—that our new branching pattern results in a significantly better upper energy bound

Knots and Links in Three-Dimensional Flows

The closed orbits of three-dimensional flows form knots and links. This book develops the tools - template theory and symbolic dynamics - needed for studying knotted orbits. This theory is applied to the problems of understanding local and global bifurcations, as well as the embedding data of orbits in Morse-smale, Smale, and integrable Hamiltonian flows. The necesssary background theory is sketched; however, some familiarity with low-dimensional topology and differential equations is assumed

Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition

Estimation and control of non-linear and hybrid systems with applications to air-to-air guidance

Issued as Progress report, and Final report, Project no. E-21-67

Transformation-Invariant Analysis of Visual Signals with Parametric Models

The analysis of collections of visual data, e.g., their classification, modeling and clustering, has become a problem of high importance in a variety of applications. Meanwhile, image data captured in uncontrolled environments by arbitrary users is very likely to be exposed to geometric transformations. Therefore, efficient methods are needed for analyzing high-dimensional visual data sets that can cope with geometric transformations of the visual content of interest. In this thesis, we study parametric models for transformation-invariant analysis of geometrically transformed image data, which provide low-dimensional image representations that capture relevant information efficiently. We focus on transformation manifolds, which are image sets created by parametrizable geometric transformations of a reference image model. Transformation manifolds provide a geometric interpretation of several image analysis problems. In particular, image registration corresponds to the computation of the projection of the target image onto the transformation manifold of the reference image. Similarly, in classification, the class label of a query image can be estimated in a transformation-invariant way by comparing its distance to transformation manifolds that represent different image classes. In this thesis, we explore several problems related to the registration, modeling, and classification of images with transformation manifolds. First, we address the problem of sampling transformation manifolds of known parameterization, where we focus on the target applications of image registration and classification in the sampling. We first propose an iterative algorithm for sampling a manifold such that the selected set of samples gives an accurate estimate of the distance of a query image to the manifold. We then extend this method to a classification setting with several transformation manifolds representing different image classes. We develop an algorithm to jointly sample multiple transformation manifolds such that the class label of query images can be estimated accurately by comparing their distances to the class-representative manifold samples. The proposed methods outperform baseline sampling schemes in image registration and classification. Next, we study the problem of learning transformation manifolds that are good models of a given set of geometrically transformed image data. We first learn a representative pattern whose transformation manifold fits well the input images and then generalize the problem to a supervised classification setting, where we jointly learn multiple class-representative pattern transformation manifolds from training images with known class labels. The proposed manifold learning methods exploit the information of the type of the geometric transformation in the data to compute an accurate data model, which is ignored in previous manifold learning algorithms. Finally, we focus on the usage of transformation manifolds in multiscale image registration. We consider two different methods in image registration, namely, the tangent distance method and the minimization of the image intensity difference with gradient descent. We present a multiscale performance analysis of these methods. We derive upper bounds for the alignment errors yielded by the two methods and analyze the variations of these bounds with noise and low-pass filtering, which is useful for gaining an understanding of the performance of these methods in image registration. To the best of our knowledge, these are the first such studies in multiscale registration settings. Geometrically transformed image sets have a particular structure, and classical image analysis methods do not always suit well for the treatment of such data. This thesis is motivated by this observation and proposes new techniques and insights for handling geometric transformations in image analysis and processing