Mathematical Sciences - Doctoral Theses

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https://hdl.handle.net/10468/859

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Rate-induced tipping to metastable Zombie fires
(University College Cork, 2023) O'Sullivan, Eoin; Mulchrone, Kieran F.; Wieczorek, Sebastian; ATSR Ltd.
Surface wildfires are generally believed to be the cause of so-called Zombie fires observed in peatlands, that disappear from the surface, smoulder underground during the winter, and ''come back to life" in the spring. Here, we propose rate-induced tipping (R-tipping) to a subsurface hot metastable state in bioactive peat soils as a main cause of Zombie fires. Our hypothesis is based on a conceptual soil-carbon model subjected to realistic changes in weather and climate patterns, including global warming scenarios and summer heatwaves. Mathematically speaking, R-tipping to the hot metastable state is a nonautonomous instability, due to crossing an elusive quasithreshold, in a multiple-timescale dynamical system. The instability is {\em reversible}, in the sense that the system eventually returns to its base state. To explain this instability, we provide a framework that combines a special compactification technique with concepts from geometric singular perturbation theory. This framework allows us to reduce a reversible R-tipping problem due to crossing a quasithreshold to a heteroclinic orbit problem in a singular limit. Thus, we identify generic cases of such R-tipping via: (i) unfolding of a codimension-two heteroclinic folded saddle-node type-I singularity for global warming, and (ii) analysis of a codimension-one saddle-to-saddle hetroclinic orbit for summer heatwaves, which in turn reveal new types of excitability quasithresholds.
Nonlinear dynamics & stochastic processes in cybersecurity applications
(University College Cork, 2023) Ryan, Pierce; Amann, Andreas; Healy, Sorcha; Irish Research Council for Science, Engineering and Technology; McAfee
The Internet is an extremely complex system which has a significant impact on the world we live in. In this thesis, we formalise Internet-based problems as mathematical models to better understand their dynamics. Modelling these problems requires dynamical features such as time delay, periodic forcing, switching and stochasticity. We study several dynamical systems which employ a combination of these features from Internet applications, including targeted ransomware, data networks, and signal processing. We also study a climate science system which shares features with the signal processing system and exhibits similar dynamics. Stochasticity is found to be critical in the modelling of the negotiations involved in targeted ransomware, while time delay is a crucial feature in the modelling of data networks. The signal processing and climate science systems give rise to extremely rich dynamics, which we are able to study analytically due to the presence of switching. This yields further insights into related smooth systems.
Theory and applications of multifunctional reservoir computers
(University College Cork, 2023) Flynn, Andrew; Amann, Andreas; Tsachouridis, Vassilios A.; Irish Research Council
In the pursuit of developing artificially intelligent systems there is much to be gained from dually integrating further physiological features of biological neural networks and knowledge of dynamical systems into machine learning environments. In this Thesis such a two-armed approach is employed in order to translate 'multifunctionality' from biological to artificial neural networks via the reservoir computing machine learning paradigm. Multifunctionality describes the ability of a single neural network that exploits a form of multistability to perform a multitude of mutually exclusive tasks. The dynamics of multifunctional RCs are assessed across several tasks and from this many new application areas are explored which include, data-driven modelling of multistability, generating chaotic itinerancy, and reconstructing dynamical transitions present in the epileptic brain.
Metaheuristics and machine learning for joint stratification and sample allocation in survey design
(University College Cork, 2022-01) O'Luing, Mervyn; Prestwich, Steve; Tarim, Armagan; European Regional Development Fund; Science Foundation Ireland
In this thesis, we propose a number of metaheuristics and machine learning techniques to solve the joint stratification and sample allocation problem. Finding the optimal solution to this problem is hard when the sampling frame is large, and the evaluation algorithm is computationally burdensome. To advance the research in this area, we explore and evaluate different algorithmic methods of modelling and solving this problem. Firstly, we propose a new genetic algorithm approach using "grouping" genetic operators instead of traditional operators. Experiments show a significant improvement in solution quality for similar computational effort. Next, we combine the capability of a simulated annealing algorithm to escape from local minima with delta evaluation to exploit the similarity between consecutive solutions and thereby reduce evaluation time. Comparisons with two recent algorithms show the simulated annealing algorithm attaining comparable solution qualities in less computation time. Then, we consider the combination of the k-means and clustering algorithms with a hill climbing algorithm in stages and report the solution costs, evaluation times and training times. The multi-stage combinations generally compare well with recent algorithms, and provide the survey designer with a greater choice of algorithms to choose from. Finally, we combine the explorative properties of an estimation of distribution algorithm (EDA) to model the probabilities of an atomic stratum belonging to different strata with the exploitative search properties of a simulated annealing algorithm to create a hybrid estimation of distribution algorithm (HEDA). Results of comparisons with the best solution qualities from our earlier experiments show that the HEDA finds better solution qualities, but requires a longer total execution time than alternative approaches we considered.
Improved statistical quantitation of dynamic PET scans
(University College Cork, 2023-03-15) Gu, Fengyun; O'Sullivan, Finbarr; Huang, Jian; University College Cork; Science Foundation Ireland; National Cancer Institute USA; Swedish Cancer Foundation; IEEE Foundation
Positron emission tomography (PET) scanning is an important diagnostic imaging technique used in the management of cancer patients and in medical research. It plays a key role in a variety of tasks related to diagnosis, therapy planning, prognosis and treatment monitoring by injecting a radiotracer to characterize the specific biologic process (e.g., tumor metabolism, proliferation or blood flow). The standardized uptake value (SUV) obtained at a single time point is widely employed in clinical practice. However, well beyond this simple static uptake measure, more detailed metabolic information may be recovered from dynamic PET scanning with multiple time frames. Assuming a tracer’s interaction with the tissue is linear and time-invariant, the tissue time course can be expressed as a convolution between arterial input function (AIF) and the tissue impulse response/residue function. Kinetic analysis is concerned with the estimation of residue and associated physiological parameters such as flow, flux and volume of distribution. Some traditional methods including Patlak and compartmental modeling are well-established with a given form of residue function (constant or mixture of exponentials), but they are not flexible to represent data in which in-vivo biochemistry is not clear, especially for the whole-body imaging on the long axial field of view (LAFOV) PET systems. The main goals of this thesis are to develop novel statistical approaches for improving and evaluating parametric imaging extracted from dynamic PET scans. The non-parametric residue mapping (NPRM) procedure has been constructed by a fully automatic process incorporating data-adaptive segmentation, non-parametric residue analysis of segment data and voxel-level kinetic mapping scheme. Based on this approach, the benefits of pooling data in multiple injection PET scans are investigated. Spatial and temporal patterns of residuals recovered by model diagnostics exhibit a non-Gaussian structure, which defines a bootstrap data generation process (DGP) in the image domain. The proposed bootstrap method has been used to assess the uncertainty (standard errors) in kinetic information and more complex regional summaries. We also examine its potential to improve the mean square error (MSE) characteristics of kinetic maps generated from either compartmental modeling or NPRM approach by averaging results from individual bootstrap samples. Dynamic breast cancer studies on the early, recent and latest LAFOV PET scanners are presented to illustrate these techniques. The performance of above models and schemes has been evaluated in a series of one and two-dimensional numerical simulations. Both direct filtered backprojection (FBP) and iterative maximum likelihood (ML) reconstructions are considered. The proposed NPRM approach has some important features like the flexibility for diverse tissue environments and consideration of delays for different parts, which make it promising to be applied to the emerging total-body PET imaging. The developed image-domain bootstrap provides a practical way to quantify the uncertainties of biomarkers. This mechanism has the potential to further support clinical decision-making and enhance personalized medicine.

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