Stochastic timeseries analysis in electric power systems and paleo-climate data

In this thesis a data science study of elementary stochastic processes is laid, aided with the development of two numerical software programmes, applied to power-grid frequency studies and Dansgaard--Oeschger events in paleo-climate data. Power-grid frequency is a key measure in power grid studies. It comprises the balance of power in a power grid at any instance. In this thesis an elementary Markovian Langevin-like stochastic process is employed, extending from existent literature, to show the basic elements of power-grid frequency dynamics can be modelled in such manner. Through a data science study of power-grid frequency data, it is shown that fluctuations scale in an inverse square-root relation with their size, alike any other stochastic process, confirming previous theoretical results. A simple Ornstein--Uhlenbeck is offered as a surrogate model for power-grid frequency dynamics, with a versatile input of driving deterministic functions, showing not surprisingly that driven stochastic processes with Gaussian noise do not necessarily show a Gaussian distribution. A study of the correlations between recordings of power-grid frequency in the same power-grid system reveals they are correlated, but a theoretical understanding is yet to be developed. A super-diffusive relaxation of amplitude synchronisation is shown to exist in space in coupled power-grid systems, whereas a linear relation is evidenced for the emergence of phase synchronisation. Two Python software packages are designed, offering the possibility to extract conditional moments for Markovian stochastic processes of any dimension, with a particular application for Markovian jump-diffusion processes for one-dimensional timeseries. Lastly, a study of Dansgaard--Oeschger events in recordings of paleoclimate data under the purview of bivariate Markovian jump-diffusion processes is proposed, augmented by a semi-theoretical study of bivariate stochastic processes, offering an explanation for the discontinuous transitions in these events and showing the existence of deterministic couplings between the recordings of the dust concentration and a proxy for the atmospheric temperature

Learning in adaptive networks: analytical and computational approaches

The dynamics on networks and the dynamics of networks are usually entangled with each other in many highly connected systems, where the former means the evolution of state and the latter means the adaptation of structure. In this thesis, we will study the coupled dynamics through analytical and computational approaches, where the adaptive networks are driven by learning of various complexities. Firstly, we investigate information diffusion on networks through an adaptive voter model, where two opinions are competing for the dominance. Two types of dynamics facilitate the agreement between neighbours: one is pairwise imitation and the other is link rewiring. As the rewiring strength increases, the network of voters will transform from consensus to fragmentation. By exploring various strategies for structure adaptation and state evolution, our results suggest that network configuration is highly influenced by range-based rewiring and biased imitation. In particular, some approximation techniques are proposed to capture the dynamics analytically through moment-closure differential equations. Secondly, we study an evolutionary model under the framework of natural selection. In a structured community made up of cooperators and cheaters (or defectors), a new-born player will adopt a strategy and reorganise its neighbourhood based on social inheritance. Starting from a cooperative population, an invading cheater may spread in the population occasionally leading to the collapse of cooperation. Such a collapse unfolds rapidly with the change of external conditions, bearing the traits of a critical transition. In order to detect the risk of invasions, some indicators based on population composition and network structure are proposed to signal the fragility of communities. Through the analyses of consistency and accuracy, our results suggest possible avenues for detecting the loss of cooperation in evolving networks. Lastly, we incorporate distributed learning into adaptive agents coordination, which emerges as a consequence of rational individual behaviours. A generic framework of work-learn-adapt (WLA) is proposed to foster the success of agents organisation. To gain higher organisation performance, the division of labour is achieved by a series of events of state evolution and structure adaptation. Importantly, agents are able to adjust their states and structures through quantitative information obtained from distributed learning. The adaptive networks driven by explicit learning pave the way for a better understanding of intelligent organisations in real world

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

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