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- Institut für Mathematik (124) (entfernen)
We establish elements of a new approach to ellipticity and parametrices within operator algebras on manifolds with higher singularities, only based on some general axiomatic requirements on parameter-dependent operators in suitable scales of spaes. The idea is to model an iterative process with new generations of parameter-dependent operator theories, together with new scales of spaces that satisfy analogous requirements as the original ones, now on a corresponding higher level. The "full" calculus involves two separate theories, one near the tip of the corner and another one at the conical exit to infinity. However, concerning the conical exit to infinity, we establish here a new concrete calculus of edge-degenerate operators which can be iterated to higher singularities.
Data assimilation has been an active area of research in recent years, owing to its wide utility. At the core of data assimilation are filtering, prediction, and smoothing procedures. Filtering entails incorporation of measurements' information into the model to gain more insight into a given state governed by a noisy state space model. Most natural laws are governed by time-continuous nonlinear models. For the most part, the knowledge available about a model is incomplete; and hence uncertainties are approximated by means of probabilities. Time-continuous filtering, therefore, holds promise for wider usefulness, for it offers a means of combining noisy measurements with imperfect model to provide more insight on a given state.
The solution to time-continuous nonlinear Gaussian filtering problem is provided for by the Kushner-Stratonovich equation. Unfortunately, the Kushner-Stratonovich equation lacks a closed-form solution. Moreover, the numerical approximations based on Taylor expansion above third order are fraught with computational complications. For this reason, numerical methods based on Monte Carlo methods have been resorted to. Chief among these methods are sequential Monte-Carlo methods (or particle filters), for they allow for online assimilation of data. Particle filters are not without challenges: they suffer from particle degeneracy, sample impoverishment, and computational costs arising from resampling.
The goal of this thesis is to:— i) Review the derivation of Kushner-Stratonovich equation from first principles and its extant numerical approximation methods, ii) Study the feedback particle filters as a way of avoiding resampling in particle filters, iii) Study joint state and parameter estimation in time-continuous settings, iv) Apply the notions studied to linear hyperbolic stochastic differential equations.
The interconnection between Itô integrals and stochastic partial differential equations and those of Stratonovich is introduced in anticipation of feedback particle filters. With these ideas and motivated by the variants of ensemble Kalman-Bucy filters founded on the structure of the innovation process, a feedback particle filter with randomly perturbed innovation is proposed. Moreover, feedback particle filters based on coupling of prediction and analysis measures are proposed. They register a better performance than the bootstrap particle filter at lower ensemble sizes.
We study joint state and parameter estimation, both by means of extended state spaces and by use of dual filters. Feedback particle filters seem to perform well in both cases. Finally, we apply joint state and parameter estimation in the advection and wave equation, whose velocity is spatially varying. Two methods are employed: Metropolis Hastings with filter likelihood and a dual filter comprising of Kalman-Bucy filter and ensemble Kalman-Bucy filter. The former performs better than the latter.
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. We consider the Seiberg-Witten moduli space on a simply connected Käuhler 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 Kähler surface is a Kähler metric.
Change points in time series are perceived as heterogeneities in the statistical or dynamical characteristics of the observations. Unraveling such transitions yields essential information for the understanding of the observed system’s intrinsic evolution and potential external influences. A precise detection of multiple changes is therefore of great importance for various research disciplines, such as environmental sciences, bioinformatics and economics. The primary purpose of the detection approach introduced in this thesis is the investigation of transitions underlying direct or indirect climate observations. In order to develop a diagnostic approach capable to capture such a variety of natural processes, the generic statistical features in terms of central tendency and dispersion are employed in the light of Bayesian inversion. In contrast to established Bayesian approaches to multiple changes, the generic approach proposed in this thesis is not formulated in the framework of specialized partition models of high dimensionality requiring prior specification, but as a robust kernel-based approach of low dimensionality employing least informative prior distributions.
First of all, a local Bayesian inversion approach is developed to robustly infer on the location and the generic patterns of a single transition. The analysis of synthetic time series comprising changes of different observational evidence, data loss and outliers validates the performance, consistency and sensitivity of the inference algorithm. To systematically investigate time series for multiple changes, the Bayesian inversion is extended to a kernel-based inference approach. By introducing basic kernel measures, the weighted kernel inference results are composed into a proxy probability to a posterior distribution of multiple transitions. The detection approach is applied to environmental time series from the Nile river in Aswan and the weather station Tuscaloosa, Alabama comprising documented changes. The method’s performance confirms the approach as a powerful diagnostic tool to decipher multiple changes underlying direct climate observations.
Finally, the kernel-based Bayesian inference approach is used to investigate a set of complex terrigenous dust records interpreted as climate indicators of the African region of the Plio-Pleistocene period. A detailed inference unravels multiple transitions underlying the indirect climate observations, that are interpreted as conjoint changes. The identified conjoint changes coincide with established global climate events. In particular, the two-step transition associated to the establishment of the modern Walker-Circulation contributes to the current discussion about the influence of paleoclimate changes on the environmental conditions in tropical and subtropical Africa at around two million years ago.
Microsaccades
(2015)
The first thing we do upon waking is open our eyes. Rotating them in our eye sockets, we scan our surroundings and collect the information into a picture in our head. Eye movements can be split into saccades and fixational eye movements, which occur when we attempt to fixate our gaze. The latter consists of microsaccades, drift and tremor. Before we even lift our eye lids, eye movements – such as saccades and microsaccades that let the eyes jump from one to another position – have partially been prepared in the brain stem. Saccades and microsaccades are often assumed to be generated by the same mechanisms. But how saccades and microsaccades can be classified according to shape has not yet been reported in a statistical manner. Research has put more effort into the investigations of microsaccades’ properties and generation only since the last decade. Consequently, we are only beginning to understand the dynamic processes governing microsaccadic eye movements. Within this thesis, the dynamics governing the generation of microsaccades is assessed and the development of a model for the underlying processes. Eye movement trajectories from different experiments are used, recorded with a video-based eye tracking technique, and a novel method is proposed for the scale-invariant detection of saccades (events of large amplitude) and microsaccades (events of small amplitude). Using a time-frequency approach, the method is examined with different experiments and validated against simulated data. A shape model is suggested that allows for a simple estimation of saccade- and microsaccade related properties. For sequences of microsaccades, in this thesis a time-dynamic Markov model is proposed, with a memory horizon that changes over time and which can best describe sequences of microsaccades.