1,127 research outputs found

    Unstable Periodic Orbits: a language to interpret the complexity of chaotic systems

    Get PDF
    Unstable periodic orbits (UPOs), exact periodic solutions of the evolution equation, offer a very powerful framework for studying chaotic dynamical systems, as they allow one to dissect their dynamical structure. UPOs can be considered the skeleton of chaotic dynamics, its essential building blocks. In fact, it is possible to prove that in a chaotic system, UPOs are dense in the attractor, meaning that it is always possible to find a UPO arbitrarily near any chaotic trajectory. We can thus think of the chaotic trajectory as being approximated by different UPOs as it evolves in time, jumping from one UPO to another as a result of their instability. In this thesis we provide a contribution towards the use of UPOs as a tool to understand and distill the dynamical structure of chaotic dynamical systems. We will focus on two models, characterised by different properties, the Lorenz-63 and Lorenz-96 model. The process of approximation of a chaotic trajectory in terms of UPOs will play a central role in our investigation. In fact, we will use this tool to explore the properties of the attractor of the system under the lens of its UPOs. In the first part of the thesis we consider the Lorenz-63 model with the classic parameters’ value. We investigate how a chaotic trajectory can be approximated using a complete set of UPOs up to symbolic dynamics’ period 14. At each instant in time, we rank the UPOs according to their proximity to the position of the orbit in the phase space. We study this process from two different perspectives. First, we find that longer period UPOs overwhelmingly provide the best local approximation to the trajectory. Second, we construct a finite-state Markov chain by studying the scattering of the trajectory between the neighbourhood of the various UPOs. Each UPO and its neighbourhood are taken as a possible state of the system. Through the analysis of the subdominant eigenvectors of the corresponding stochastic matrix we provide a different interpretation of the mixing processes occurring in the system by taking advantage of the concept of quasi-invariant sets. In the second part of the thesis we provide an extensive numerical investigation of the variability of the dynamical properties across the attractor of the much studied Lorenz ’96 dynamical system. By combining the Lyapunov analysis of the tangent space with the study of the shadowing of the chaotic trajectory performed by a very large set of unstable periodic orbits, we show that the observed variability in the number of unstable dimensions, which shows a serious breakdown of hyperbolicity, is associated with the presence of a substantial number of finite-time Lyapunov exponents that fluctuate about zero also when very long averaging times are considered

    Geometric Data Analysis: Advancements of the Statistical Methodology and Applications

    Get PDF
    Data analysis has become fundamental to our society and comes in multiple facets and approaches. Nevertheless, in research and applications, the focus was primarily on data from Euclidean vector spaces. Consequently, the majority of methods that are applied today are not suited for more general data types. Driven by needs from fields like image processing, (medical) shape analysis, and network analysis, more and more attention has recently been given to data from non-Euclidean spaces–particularly (curved) manifolds. It has led to the field of geometric data analysis whose methods explicitly take the structure (for example, the topology and geometry) of the underlying space into account. This thesis contributes to the methodology of geometric data analysis by generalizing several fundamental notions from multivariate statistics to manifolds. We thereby focus on two different viewpoints. First, we use Riemannian structures to derive a novel regression scheme for general manifolds that relies on splines of generalized Bézier curves. It can accurately model non-geodesic relationships, for example, time-dependent trends with saturation effects or cyclic trends. Since Bézier curves can be evaluated with the constructive de Casteljau algorithm, working with data from manifolds of high dimensions (for example, a hundred thousand or more) is feasible. Relying on the regression, we further develop a hierarchical statistical model for an adequate analysis of longitudinal data in manifolds, and a method to control for confounding variables. We secondly focus on data that is not only manifold- but even Lie group-valued, which is frequently the case in applications. We can only achieve this by endowing the group with an affine connection structure that is generally not Riemannian. Utilizing it, we derive generalizations of several well-known dissimilarity measures between data distributions that can be used for various tasks, including hypothesis testing. Invariance under data translations is proven, and a connection to continuous distributions is given for one measure. A further central contribution of this thesis is that it shows use cases for all notions in real-world applications, particularly in problems from shape analysis in medical imaging and archaeology. We can replicate or further quantify several known findings for shape changes of the femur and the right hippocampus under osteoarthritis and Alzheimer's, respectively. Furthermore, in an archaeological application, we obtain new insights into the construction principles of ancient sundials. Last but not least, we use the geometric structure underlying human brain connectomes to predict cognitive scores. Utilizing a sample selection procedure, we obtain state-of-the-art results

    Implementation of a real time Hough transform using FPGA technology

    Get PDF
    This thesis is concerned with the modelling, design and implementation of efficient architectures for performing the Hough Transform (HT) on mega-pixel resolution real-time images using Field Programmable Gate Array (FPGA) technology. Although the HT has been around for many years and a number of algorithms have been developed it still remains a significant bottleneck in many image processing applications. Even though, the basic idea of the HT is to locate curves in an image that can be parameterized: e.g. straight lines, polynomials or circles, in a suitable parameter space, the research presented in this thesis will focus only on location of straight lines on binary images. The HT algorithm uses an accumulator array (accumulator bins) to detect the existence of a straight line on an image. As the image needs to be binarized, a novel generic synchronization circuit for windowing operations was designed to perform edge detection. An edge detection method of special interest, the canny method, is used and the design and implementation of it in hardware is achieved in this thesis. As each image pixel can be implemented independently, parallel processing can be performed. However, the main disadvantage of the HT is the large storage and computational requirements. This thesis presents new and state-of-the-art hardware implementations for the minimization of the computational cost, using the Hybrid-Logarithmic Number System (Hybrid-LNS) for calculating the HT for fixed bit-width architectures. It is shown that using the Hybrid-LNS the computational cost is minimized, while the precision of the HT algorithm is maintained. Advances in FPGA technology now make it possible to implement functions as the HT in reconfigurable fabrics. Methods for storing large arrays on FPGA’s are presented, where data from a 1024 x 1024 pixel camera at a rate of up to 25 frames per second are processed

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

    Get PDF
    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Shedding Light on Hearing in Coma: Investigating the Applicability of Functional Near-infrared Spectroscopy for Assessing Auditory Function and Aiding Prognosis in Patients with Acute Disorders of Consciousness

    Get PDF
    There is a critical need for a bedside neuroimaging tool to aid in the prediction of functional recovery outcomes for patients with acute disorders of consciousness (DoC) in the early days following severe brain injury. Current neurobehavioral examinations and prognosis tools have limitations in predicting good outcomes, leading to potential mistreatment or premature withdrawal of life support. Functional near-infrared spectroscopy (fNIRS) is a viable candidate for such purposes due to its portability and cost-effectiveness. Auditory processing, viewed as a multi-level and multifaceted brain function, could provide a sensitive and specific marker of residual cognitive function in unresponsive patients. This study aimed to investigate the effectiveness of fNIRS for hierarchical assessment of auditory function and evaluate its applicability for predicting recovery outcomes in acute DoC. The capability of fNIRS for such an application was demonstrated by validating it against fMRI in a healthy population and cross-validating it in an entirely unresponsive patient with cognitive-motor dissociation. An innovative fNIRS-focused method was developed to quantify patients’ auditory function, and a data-driven method was explored to improve the sensitivity and specificity of auditory scores. Using these analytical tools, a direct association was found between auditory function and recovery outcome in a small patient cohort. Based on the study’s findings, the crucial role of methodological considerations in the use of fNIRS was discussed, and specific modifications in the stimulus and optical montage designs were suggested to enhance the method’s reliability

    Electromagnetic scattering from thin tubular objects and an application in electromagnetic chirality

    Get PDF
    Asymptotic perturbation formulas characterize the effective behavior of waves as the volume of the scattering object tends to zero. In this work, wave propagation is described by time-harmonic Maxwell\u27s equations in free space and the corresponding scattering objects are thin tubular objects that feature a different electric permittivity and a different magnetic permeability than their surrounding medium. For this setting, we derive an asymptotic representation of the scattered electric field away from the thin tubular object and use the corresponding leading order term in a shape identification problem and for designing highly electromagnetically chiral objects. In inverse problems, the leading order term may be used to find the center curve of a thin wire that is supposed to emit a scattered field, which is reasonably close to a given measured field. For the optimal design of electromagnetically chiral structures, the representation formula provides an explicit formula for the leading order term of an asymptotic far field operator expansion. A chirality measure, usually requiring the far field operator, will now map aforementioned leading order term to a value between 00 and 11 dependent on the level of electromagnetic chirality of the thin tubular scatterer. This approximation greatly simplifies the challenge to maximize the chirality measure with respect to thin tubular objects. The fact that neither the evaluation of the leading order term nor the calculation of corresponding derivatives require a Maxwell system to be solved implies that the shape optimization scheme is highly efficient compared to shape optimization algorithms that use e.g. domain derivatives. In the visible range, the metallic nanowires obtained by our optimization scheme attain high values of electromagnetic chirality and even exceed those attained by traditional metallic helices

    Self-generated turbulent reconnection

    Get PDF

    Convex Optimization for Machine Learning

    Get PDF
    This book covers an introduction to convex optimization, one of the powerful and tractable optimization problems that can be efficiently solved on a computer. The goal of the book is to help develop a sense of what convex optimization is, and how it can be used in a widening array of practical contexts with a particular emphasis on machine learning. The first part of the book covers core concepts of convex sets, convex functions, and related basic definitions that serve understanding convex optimization and its corresponding models. The second part deals with one very useful theory, called duality, which enables us to: (1) gain algorithmic insights; and (2) obtain an approximate solution to non-convex optimization problems which are often difficult to solve. The last part focuses on modern applications in machine learning and deep learning. A defining feature of this book is that it succinctly relates the “story” of how convex optimization plays a role, via historical examples and trending machine learning applications. Another key feature is that it includes programming implementation of a variety of machine learning algorithms inspired by optimization fundamentals, together with a brief tutorial of the used programming tools. The implementation is based on Python, CVXPY, and TensorFlow. This book does not follow a traditional textbook-style organization, but is streamlined via a series of lecture notes that are intimately related, centered around coherent themes and concepts. It serves as a textbook mainly for a senior-level undergraduate course, yet is also suitable for a first-year graduate course. Readers benefit from having a good background in linear algebra, some exposure to probability, and basic familiarity with Python

    Adaptive swarm optimisation assisted surrogate model for pipeline leak detection and characterisation.

    Get PDF
    Pipelines are often subject to leakage due to ageing, corrosion and weld defects. It is difficult to avoid pipeline leakage as the sources of leaks are diverse. Various pipeline leakage detection methods, including fibre optic, pressure point analysis and numerical modelling, have been proposed during the last decades. One major issue of these methods is distinguishing the leak signal without giving false alarms. Considering that the data obtained by these traditional methods are digital in nature, the machine learning model has been adopted to improve the accuracy of pipeline leakage detection. However, most of these methods rely on a large training dataset for accurate training models. It is difficult to obtain experimental data for accurate model training. Some of the reasons include the huge cost of an experimental setup for data collection to cover all possible scenarios, poor accessibility to the remote pipeline, and labour-intensive experiments. Moreover, datasets constructed from data acquired in laboratory or field tests are usually imbalanced, as leakage data samples are generated from artificial leaks. Computational fluid dynamics (CFD) offers the benefits of providing detailed and accurate pipeline leakage modelling, which may be difficult to obtain experimentally or with the aid of analytical approach. However, CFD simulation is typically time-consuming and computationally expensive, limiting its pertinence in real-time applications. In order to alleviate the high computational cost of CFD modelling, this study proposed a novel data sampling optimisation algorithm, called Adaptive Particle Swarm Optimisation Assisted Surrogate Model (PSOASM), to systematically select simulation scenarios for simulation in an adaptive and optimised manner. The algorithm was designed to place a new sample in a poorly sampled region or regions in parameter space of parametrised leakage scenarios, which the uniform sampling methods may easily miss. This was achieved using two criteria: population density of the training dataset and model prediction fitness value. The model prediction fitness value was used to enhance the global exploration capability of the surrogate model, while the population density of training data samples is beneficial to the local accuracy of the surrogate model. The proposed PSOASM was compared with four conventional sequential sampling approaches and tested on six commonly used benchmark functions in the literature. Different machine learning algorithms are explored with the developed model. The effect of the initial sample size on surrogate model performance was evaluated. Next, pipeline leakage detection analysis - with much emphasis on a multiphase flow system - was investigated in order to find the flow field parameters that provide pertinent indicators in pipeline leakage detection and characterisation. Plausible leak scenarios which may occur in the field were performed for the gas-liquid pipeline using a three-dimensional RANS CFD model. The perturbation of the pertinent flow field indicators for different leak scenarios is reported, which is expected to help in improving the understanding of multiphase flow behaviour induced by leaks. The results of the simulations were validated against the latest experimental and numerical data reported in the literature. The proposed surrogate model was later applied to pipeline leak detection and characterisation. The CFD modelling results showed that fluid flow parameters are pertinent indicators in pipeline leak detection. It was observed that upstream pipeline pressure could serve as a critical indicator for detecting leakage, even if the leak size is small. In contrast, the downstream flow rate is a dominant leakage indicator if the flow rate monitoring is chosen for leak detection. The results also reveal that when two leaks of different sizes co-occur in a single pipe, detecting the small leak becomes difficult if its size is below 25% of the large leak size. However, in the event of a double leak with equal dimensions, the leak closer to the pipe upstream is easier to detect. The results from all the analyses demonstrate the PSOASM algorithm's superiority over the well-known sequential sampling schemes employed for evaluation. The test results show that the PSOASM algorithm can be applied for pipeline leak detection with limited training datasets and provides a general framework for improving computational efficiency using adaptive surrogate modelling in various real-life applications

    Modelling, Dimensioning and Optimization of 5G Communication Networks, Resources and Services

    Get PDF
    This reprint aims to collect state-of-the-art research contributions that address challenges in the emerging 5G networks design, dimensioning and optimization. Designing, dimensioning and optimization of communication networks resources and services have been an inseparable part of telecom network development. The latter must convey a large volume of traffic, providing service to traffic streams with highly differentiated requirements in terms of bit-rate and service time, required quality of service and quality of experience parameters. Such a communication infrastructure presents many important challenges, such as the study of necessary multi-layer cooperation, new protocols, performance evaluation of different network parts, low layer network design, network management and security issues, and new technologies in general, which will be discussed in this book
    • …
    corecore