Effiziente numerische Methoden für fraktionale Differentialgleichungen und ihr analystischer Hintergrund

Efficient numerical methods for fractional differential equations and their theoretical background are presented. A historical review introduces and motivates the field of fractional calculus. Analytical results on classical calculus as well as special functions and integral transforms are repeated for completeness. Known analytical results on non-integer order differentiation and integrations are presented and corrected and extended where needed. On those results several numerical methods for the solution of fractional differential equations are based. These methods are described and compared to each other in detail. Special attention is paid to the question of applicability of higher oder methods and in connection the practical implementation of such methods is analyzed. Different ways of improvements of the presented numerical methods are given. Numerical calculations confirm the results which were deduced theoretically. Moreover, some of the presented methods are generalized to deal with partial differential equations of fractional order. Finally a problem of physics/chemistry is presented and some of the presented numerical methods are applied.Effiziente numerische Methoden für fraktionale Differentialgleichungen und ihr theoretischer Hintergrund werden betrachtet. Ein historischer Rückblick liefert eine Einführung und Motivation in das Gebiet der fraktionalen Differential- und Integralrechnung. Analytische Ergebnisse der klassischen Differential- und Integralrechnung, sowie spezielle Funtktionen und Integraltransformationen werden zur Vollständigkeit wiederholt. Bekannte analytische Ergebnisse nicht-ganzzahliger Differentiation und Integration werden dargelegt und berichtigt und erweitert falls nötig. Auf diesen Ergebnissen beruhen mehrere numerische Methoden für die Lösung fraktionaler Differentialgleichungen. Diese Methoden werden detailliert beschrieben und untereinander verglichen. Besonderer Wert wird auf die Frage nach der Anwendbarkeit der Methoden höherer Ordnung gelegt und in diesem Zusammenhang die praktische Implementierung solcher Methoden untersucht. Verschiedene Möglichkeiten zur Verbesserung der beschriebenen Methoden werden vorgestellt. Numerische Berechnungen bestätigen die theoretisch hergeleiteten Ergebnisse. Des Weiteren werden einige der vorgestellten Methoden verallgemeinert, um auf partielle Differentialgleichungen fraktionaler Ordnung angewendet werden zu können. Letzlich wird ein Problem aus der Physik/Chemie vorgestellt und einige der dargestellten numerischen Methoden darauf angewendet

Statistical computation with kernels

Modern statistical inference has seen a tremendous increase in the size and complexity of models and datasets. As such, it has become reliant on advanced com- putational tools for implementation. A first canonical problem in this area is the numerical approximation of integrals of complex and expensive functions. Numerical integration is required for a variety of tasks, including prediction, model comparison and model choice. A second canonical problem is that of statistical inference for models with intractable likelihoods. These include models with intractable normal- isation constants, or models which are so complex that their likelihood cannot be evaluated, but from which data can be generated. Examples include large graphical models, as well as many models in imaging or spatial statistics. This thesis proposes to tackle these two problems using tools from the kernel methods and Bayesian non-parametrics literature. First, we analyse a well-known algorithm for numerical integration called Bayesian quadrature, and provide consis- tency and contraction rates. The algorithm is then assessed on a variety of statistical inference problems, and extended in several directions in order to reduce its compu- tational requirements. We then demonstrate how the combination of reproducing kernels with Stein’s method can lead to computational tools which can be used with unnormalised densities, including numerical integration and approximation of probability measures. We conclude by studying two minimum distance estimators derived from kernel-based statistical divergences which can be used for unnormalised and generative models. In each instance, the tractability provided by reproducing kernels and their properties allows us to provide easily-implementable algorithms whose theoretical foundations can be studied in depth

A theoretical and methodological framework for machine learning in survival analysis: Enabling transparent and accessible predictive modelling on right-censored time-to-event data

Survival analysis is an important field of Statistics concerned with mak- ing time-to-event predictions with ‘censored’ data. Machine learning, specifically supervised learning, is the field of Statistics concerned with using state-of-the-art algorithms in order to make predictions on unseen data. This thesis looks at unifying these two fields as current research into the two is still disjoint, with ‘classical survival’ on one side and su- pervised learning (primarily classification and regression) on the other. This PhD aims to improve the quality of machine learning research in survival analysis by focusing on transparency, accessibility, and predic- tive performance in model building and evaluation. This is achieved by examining historic and current proposals and implementations for models and measures (both classical and machine learning) in survival analysis and making novel contributions. In particular this includes: i) a survey of survival models including a crit- ical and technical survey of almost all supervised learning model classes currently utilised in survival, as well as novel adaptations; ii) a survey of evaluation measures for survival models, including key definitions, proofs and theorems for survival scoring rules that had previously been missing from the literature; iii) introduction and formalisation of composition and reduction in survival analysis, with a view on increasing transparency of modelling strategies and improving predictive performance; iv) imple- mentation of several R software packages, in particular mlr3proba for machine learning in survival analysis; and v) the first large-scale bench- mark experiment on right-censored time-to-event data with 24 survival models and 66 datasets. Survival analysis has many important applications in medical statistics, engineering and finance, and as such requires the same level of rigour as other machine learning fields such as regression and classification; this thesis aims to make this clear by describing a framework from prediction and evaluation to implementation

MATLAB

This excellent book represents the final part of three-volumes regarding MATLAB-based applications in almost every branch of science. The book consists of 19 excellent, insightful articles and the readers will find the results very useful to their work. In particular, the book consists of three parts, the first one is devoted to mathematical methods in the applied sciences by using MATLAB, the second is devoted to MATLAB applications of general interest and the third one discusses MATLAB for educational purposes. This collection of high quality articles, refers to a large range of professional fields and can be used for science as well as for various educational purposes