Applied Mathematics

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This is the collection for the University of Waterloo's Department of Applied Mathematics.

Research outputs are organized by type (eg. Master Thesis, Article, Conference Paper).

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Learning-Based Stability Certification and System Identification of Nonlinear Dynamical Systems
(University of Waterloo, 2025-10-23) Zhou, Ruikun
In recent decades, by taking advantage of the abundance of sensory measurements, learning-based methods have been prevalent and shown their effectiveness in tackling challenging or intractable problems for classical approaches in systems and control. For instance, many systems with complex nonlinearities, high-dimensional state spaces, or unknown dynamics cannot be effectively handled by classical mathematical tools, and computing stability certifications for such systems is often intractable. This thesis aims to construct systematic approaches to perform system identification tasks and learning-based Lyapunov functions for nonlinear dynamical systems, with some extensions to optimal control. The first aspect of this thesis is to develop an efficient method based on a special feedforward neural network structure, an extreme learning machine, to compute stability certificates for nonlinear systems by solving linear PDEs when the dynamics are accessible. Differing from the typical neural network-based approaches that require training on high-performance computing platforms, one only needs to solve a convex optimization problem. On top of that, the proposed method can also be used to efficiently solve the notable HJB equation via policy iteration to obtain optimal control policies for nonlinear systems. The second aspect of this research is to tackle these issues for nonlinear systems with (partially) unknown dynamics. We first show that with two feedforward neural networks, the unknown system and a Lyapunov-based stability certificate can be learned simultaneously. With the help of satisfiability modulo theories (SMT) solvers, the resulting Lyapunov function can be formally verified to provide stability certificates for the unknown nonlinear system. Alternatively, in the past two decades, the Koopman operator and its generator have demonstrated advantages in identifying discrete-time systems and continuous-time systems, respectively, requiring significantly less data while achieving better performance than most existing classical methods. For unknown continuous-time dynamical systems, we propose a novel resolvent operator-based learning framework to learn the Koopman generator, which is a linear operator that describes the infinitesimal evolution of the Koopman operator. The learned generator, thereafter, can be used to identify the vector field of the nonlinear systems. Moreover, with the learned high-accuracy Koopman generator, we can also construct a Lyapunov-based stability certificate for the unknown nonlinear system in the same function space. By formulating the linear PDEs as a linear least squares problem, Lyapunov functions can be computed efficiently. The learned Lyapunov functions can be formally verified using an SMT solver and provide less conservative estimates of the region of attraction, compared to existing methods. Taken together, these contributions provide a coherent pathway that begins with model-based stability certification computation and continues to fully data-driven system identification and thereafter computing Lyapunov-based stability certificates.
Quantum Data Processing Inequalities and their Reverse
(University of Waterloo, 2025-10-02) Natarajan, Shreyas
Any reasonable measure of quantum information must satisfy a data processing inequality, that is, it must not increase under the action of a quantum channel. The same is, therefore, true for measures of distinguishability of quantum states. In this thesis, we study two families of distinguishability measures that are particularly interesting: the Riemannian metric (more precisely, the corresponding semi-norm) and the standard quantum f-divergences (sometimes referred to as just standard f-divergences). However, rather than focusing on the information lost, we ask about the information preserved - namely, a reverse data processing inequality. As is established in this thesis, an exact reverse data processing inequality for all states acted on by a specific channel is not possible for these measures if the output dimension of the quantum channel is no greater than the input dimension (which includes several important channels). Instead, we settle for a reverse data processing inequality on a restricted set of input states, or oftentimes it suffices to only compare the loss of information incurred via two given quantum channels in general. This thesis demonstrates cases of a restricted reverse data processing inequality for these measures and initiates a study of the similarities between the Riemannian metrics and standard quantum f-divergences in this context.
The Impacts of Ocean-Atmosphere Coupling Near Ocean Mesoscale Fronts
(University of Waterloo, 2025-09-23) Braga, Jakob
Understanding the interaction between the atmosphere and ocean is crucial to modeling or predicting the dynamics of both systems. Observational studies and numerical simulations of atmospheric responses to ocean mesoscale fronts have found strong correlation between the sea surface temperature (SST) and near surface wind speeds, which is interpreted as the SST driving the atmosphere at these scales. Furthermore, correlations studies have shown that SST fronts are capable of modifying surface turbulent heat and momentum fluxes which can affect winds, clouds and rainfall, as well as have larger scale effects on storm tracks. These larger scale effects have been observed to feed back onto the ocean and further evolve the SST fronts, thus making the need to understand the fully coupled atmosphere-ocean system more significant. This thesis uses idealized large eddy simulations to investigate the evolution of a coupled atmosphere-ocean model initialized with a filament-like warm SST anomaly in the presence of weak 3 m s⁻¹ winds blowing across the front, as well as no winds, over a period of 24 hours. Uncoupled systems are considered, holding the ocean constant in time for the atmosphere and vice versa, following an analysis of the simulations coupled together to compare. Large scale structures of the potential temperature and zonal velocity are considered alongside the mean and turbulent kinetic energy. To assist in the understanding of dominant factors of both systems a momentum budget and turbulent kinetic energy budget will be analyzed term by term. The coupling is found to have a small impact on the atmosphere over the short time scale, mainly affecting the turbulent kinetic energy strength and distribution, while having a significant impact on the ocean, causing large scale flow changes as well as altering the turbulent kinetic energy distribution.
Modeling the Frequency Response of the Graphene/Electrolyte Interface in Electrochemical Systems
(University of Waterloo, 2025-09-22) Yavarian, Mahdi
The graphene/electrolyte interface plays a central role in applications such as supercapacitors and biosensors. Traditionally modeled as two capacitors in series—the Debye capacitance of the electrolyte and the quantum capacitance of graphene—the interface is predominantly governed by the latter. While prior studies have focused on graphene’s voltage-dependent capacitance, its frequency response remains underexplored theoretically. This thesis develops a rigorous mathematical framework to model and analyze the frequency response of the graphene/electrolyte interface under diverse conditions, including finite-conductivity neutral graphene, charged graphene with infinite conductivity, and systems with room-temperature ionic liquids (RTILs). We first examine a graphene electrode in a dilute electrolyte under small AC voltages. By linearizing and normalizing the Poisson–Nernst–Planck (PNP) equations, we derive analytical impedance expressions for graphene-metal and metal-metal systems, elucidating the role of quantum capacitance across electrolyte concentrations. For finite-sized graphene disk electrodes, we incorporate graphene’s intrinsic conductivity to obtain explicit quantum impedance expressions. The results reveal a transition from Warburg-type behavior at high frequencies to RC-circuit behavior at low frequencies, governed by quantum capacitance and conductivity. The analysis extends to charged graphene electrodes under DC bias, using matched asymptotic expansions in the thin double-layer limit. We derive an analytical impedance expression that highlights the dependence of frequency response on ion concentration and bias voltage. Finally, we explore concentrated electrolytes with RTILs, introducing a new length scale to capture electrostatic correlations and characterizing their impact on low-frequency impedance.
State estimation using machine learning
(University of Waterloo, 2025-09-22) Kaur, Avneet
State estimation refers to determining the states of a dynamical system that evolves under disturbances, based on noisy measurements, partially known or unknown initial condition, and a known system model. JRNs have a structure that mimics that of a dynamical system and are thus attractive for estimator design. We show that a JRN performs better than an EKF and UKF for several examples. We also provide an input-to-state stability analysis of the error dynamics of JRNs. The stability of the error dynamics of several examples is shown. We then extend the Jordan structure to long-short-term memory networks to obtain a JLSTM which, as we show in several examples, is comparatively more robust to changes in initial conditions and noise and performs better than a EKF and PF. It also trains faster than an ELSTM for state estimation when trained to achieve a similar normalized MSE. We also compare a shallow and deep JLSTM and observe that they perform almost similarly in terms of average error across time-steps and MSE but the deep JLSTM takes longer to train due to more layers. We also train a JLSTM with a modified maximum likelihood equivalent loss function(JLSTM-ML). We observe that for Gaussian initial conditions and disturbances, the average error at each time step is best for estimates of JLSTM-ML. It is also the most robust to changes in initial conditions and disturbances in the systems considered. The measures, time taken to train, time taken to test, mean squared error, and average error at each time-step were used for comparison for various networks. We discretized the following systems to use as examples in data generation, training, and testing: mass-spring system, down pendulum, reversed Van der Pol oscillator, Galerkin approximation of Burger's partial differential equation and Kuramoto-Sivashinsky partial differential equation.
Heterogeneity and homophily in coupled behavior-disease dynamics: from model structure to early warnings
(University of Waterloo, 2025-09-19) He, Zitao
Understanding how human behavior and infectious disease dynamics interact is essential for anticipating and mitigating outbreaks. While coupled behavior-disease models have provided valuable insights into the feedback between disease transmission and vaccination behavior, many assume homogeneous populations and neglect the influence of social structure in shaping individual vaccination strategies. Traditional surveillance systems often lack timely data on vaccination behavior, making it difficult to monitor changes in public vaccine sentiment. Moreover, existing statistical methods for detecting early warning signals of critical transitions rely on assumptions that do not always hold in real-world settings. This thesis addresses these limitations by incorporating population heterogeneity and homophily into a coupled behavior-disease model, and by using the resulting simulations to support the training of data-driven models for forecasting outbreak risks from high-frequency social media data. Specifically, we develop a coupled behavior-disease model that distinguishes social media users from non-users, capturing indirect heterogeneity in how individuals access vaccine-related information. The model demonstrates that homophily slows the spread of pro-vaccine strategies, pushing the population closer to tipping points. It also suggests that early vaccine-related online discussions may offer predictive signals of future outbreaks. Building on these findings, we generate synthetic time series with heavy-tailed noise to mimic real-world social media data. These model-generated data are used to train deep learning classifiers, under CNN-LSTM and ResNet architectures, to detect early warning signals in social media data. These classifiers outperform conventional statistical indicators, such as variance and lag-1 autocorrelation, in both sensitivity and specificity. Finally, we extend the modeling framework to a generalized multi-group vaccination game, considering direct heterogeneity in levels of vaccine support. Simulations reveal that homophily contributes to the persistence of opinion polarization in the population, regardless of the presence of diseases. Together, these studies highlight the need to account for heterogeneity in modeling vaccination behavior and that homophily can have various effects depending on the states of the system. We also show that combining mechanistic models and data-driven techniques can help detect emerging risks of disease outbreaks, informing more proactive public health policies.
Nonlinear Fields in Gravitation: Investigations in Black Hole and Cosmological Spacetimes
(University of Waterloo, 2025-09-19) Hull, Brayden
This thesis is devoted to the study of non-linear effects in gravity, with a focus on black hole and cosmological spacetimes. We wish to study the phenomenological consequences that non-linearity can play on these particular spacetimes. These non-linear affects are primarily focused through higher curvature corrections, non-linear electrodynamics, and scaler fields. As we focus on black hole and cosmological space time we split this thesis into three parts, with Part I serving as a review of classical gravity, and alternative theories studied within. Part II is dedicated to the discussion and study of black holes within these theories. We study various regimes with both single and multiple non-linear effects present. We study the thermodynamics of black holes in higher curvature and higher dimensional black holes, where higher curvature corrections serve as the source of non-linear effects. We show novel phase behaviour for a certain black hole class we refer to as exotic black holes. We show a further result of these exotic black holes is that they can posses negative mass in spacetimes which are pseudo-de Sitter spaces -- asymptotically possessing positive curvature, but not constant. We investigate lower dimensional black hole solutions in general relativity coupled to nonlinear electrodynamics, and show how they all follow similar solution structure. We also look at lower dimensional black holes non-minimally coupled to non-linear scalar fields with linear and non-linear electrodynamics. We show that all black hole solutions sourced via a radial electric field are characterized by the solution from the general relativity case. Part III focuses on the study of cosmology. We look at cosmological constraints on the novel 4-dimensional Einstein-Gauss-Bonnet Gravity. This is done through construction of a complete set of equations of motion for 1st order scalar perturbations, and makes use of ACTpol cosmological data. We also look to determine whether this theory of gravity can resolve cosmological fine tuning problems such as the horizon and flatness problems. We then discuss constraints of quantum energy conditions on FLRW spacetimes in general relativity. How a variable equations of state parameter will be constrained, and its connection with the recent DESI results. We also look at how a bouncing universe through a non-minimally coupled non-linear massive scalar can also be constrained.
Harvesting entanglement from quantum fields: from theory to proposed superconducting implementations
(University of Waterloo, 2025-08-26) Teixido-Bonfill, Adam
Entanglement harvesting is a relativistic quantum information protocol through which initially uncorrelated particle detectors become entangled by locally interacting with a quantum field. This process allows one to extract and repurpose the entanglement naturally present in quantum fields, even from spacelike-separated regions. Entanglement harvesting has not been realized in the lab, but multiple experimental platforms to realize it have been proposed, among which superconducting circuits stand out due to their controllability and ability to implement strong interactions of detectors with 1+1 dimensional quantum fields. In this thesis, we start by investigating entanglement harvesting using particle detectors coupled to a massless scalar field through its derivative, a coupling that captures important features of light-matter interaction and is naturally realized in superconducting circuits. We show that detectors in causal contact can still harvest genuine entanglement from the field, with harvested entanglement peaking when the detectors are fully light-connected. Additionally, we find that communication and harvesting contributions to the detectors’ entanglement can interfere both constructively and destructively. Surprisingly, this implies that the presence of entanglement in the field can sometimes inhibit, rather than enhance, the entangling of the detectors. We then broaden the analysis to more general entanglement harvesting protocols involving detectors with arbitrary number of energy levels and a general class of couplings to the field. Furthermore, we study the longitudinal (diagonal) and transversal (off-diagonal) components of the detector-field interaction. We show that at leading order, entanglement harvesting is dominated by the component transversal to the detectors’ initial state. Through an explicit qubit model, we further illustrate how increasing the strength of longitudinal coupling can suppress harvested entanglement via higher-order effects. Finally, motivated by the prospect of experimental realizations, we introduce a variable-gap detector model that bridges the gap between idealized Unruh-DeWitt particle detectors and existing implementations in superconducting circuits. Using parameters tailored to potential experimental setups, we investigate entanglement harvesting in both spacelike-separated and causally connected scenarios. We find that, while variations in the energy gap reduce the ability to harvest entanglement in spacelike scenarios, detectors in causal contact detectors can still become entangled through their interaction with the field. Notably, our analysis shows that (due to the derivative coupling nature of the model) even for causally connected detectors, there are setups where entanglement primarily originates from the field's correlations. This demonstrates the potential for genuine entanglement harvesting in the lab and opens the door to near-future entanglement harvesting experiments in superconducting circuits.
Additivity of the Quantum and Classical Capacities of Quantum Channels
(University of Waterloo, 2025-08-06) Kazachek, Alexander
Quantum channels enable communication through the transmission of quantum states. Quantum Shannon theory investigates these channels, aiming to characterize their capacity for information transmission under various conditions. While this characterization is well-established for classical communication channels, quantum channels exhibit significantly more complex and mathematically intricate behavior, making a complete understanding elusive. A key challenge is the phenomenon of non-additivity, where combining quantum channels can enhance information flow by leveraging quantum effects. In this work, we focus on two types of non-additivity: those of classical capacity and quantum capacity. We present new constructive counterexamples demonstrating the non-additivity of the minimum output p-Renyi entropy for p>2. These examples achieve non-additivity at lower values of p than previously known constructions of the same dimension. We also show that several plausible generalizations of antisymmetric spaces -- such as through alternative symmetries or higher tensor powers -- cannot produce non-additivity using current techniques. Additionally, we advance the study of resonant multilevel amplitude damping channels. We analytically derive their degradability regions, previously inferred using a heuristic assumption supported by numerical evidence, and formulate conjectures on their capacity based on our own numerical evidence. Specifically, we conjecture that their coherent information is optimized on diagonal states and that they are always weakly additive. However, we find that coherent information activation is possible, as strong non-additivity arises in certain regions when combined with erasure channels.
Mathematical modeling of whole-body electrolyte homeostasis
(University of Waterloo, 2025-06-24) Stadt, Melissa
Electrolyte balance is crucial for many physiological processes, including cellular signaling, muscle contractions, membrane potentials, hormonal secretion, and bone structure. Disruptions to electrolyte balance, arising from disease, diet, or drugs can have severe health consequences, such as muscle weakness, bone fragility, and life-threatening cardiac arrythmias. Therefore, a comprehensive understanding of these regulatory systems and how they may be disrupted is important for developing effective preventative and therapeutic strategies. Mathematical modeling provides a powerful tool for investigating these systems through simulations and analysis. In this thesis, we present the development and analysis of mathematical models focused on the regulation of key electrolytes, potassium and calcium. For potassium homeostasis, we first developed a detailed, whole-body model incorporating known regulatory mechanisms. We conducted model simulations to quantify the individual contributions of these regulatory mechanisms on long-term potassium balance and responses to a meal. Additionally, we conducted sensitivity analyses to understand how parameter variations impact potassium levels in the extracellular and intracellular fluid. Furthermore, we integrated recent experimental data on renal adaptations to high potassium intake to analyze these findings from a whole-body perspective. For calcium homeostasis, we developed mathematical models representing a male, female, late pregnant, and lactating rat to quantify sex-specific differences and maternal adaptations in calcium regulation. These models synthesized literature data to identify key mechanisms that enable females to meet the high calcium demands of pregnancy and lactation. Finally, we developed an integrated model that represents the renin-angiotensin system, calcium regulation, and bone remodeling to investigate the impact of estrogen deficiency in post-menopausal women and common antihypertensive treatments on bone density and calcium regulation. The research provided in this thesis contributes frameworks for understanding electrolyte homeostasis and predicting the impacts of physiological changes and pharmacological interventions on electrolyte and bone homeostasis.
Multi-scale Modelling of Neurosteroid-mediated Seizure Trajectories in Childhood Absence Epilepsy
(University of Waterloo, 2025-06-20) Ahmed, Maliha
Childhood absence epilepsy (CAE) is a pediatric generalized epilepsy disorder characterized by brief episodes of impaired consciousness and distinctive 2.5--5 Hz spike-wave discharges (SWDs) on electroencephalography. With a well-established genetic aetiology, this condition tends to resolve spontaneously during adolescence in most cases. While several mechanisms have been proposed for remission, understanding remains insufficient to guide early intervention practices. In this thesis, we first utilize a conductance-based thalamocortical network model that exhibits characteristic SWDs, to demonstrate that allopregnanolone---a progesterone metabolite known to enhance GABAa receptor-mediated inhibition---has an ameliorating effect on SWDs. To investigate the divergence between this finding and clinical observations, we developed an enhanced thalamocortical model that incorporates a layered cortical structure to explore regional cortical heterogeneity and frontocortical connectivity as potential resistance factors to ALLO-mediated recovery. Our results suggest that non-resolving CAE may be due not only to increased frontocortical connectivity but also to the composition of cell types within the network. Specifically, a higher proportion of bursting-type cells may prevent the therapeutic effects of allopregnanolone. We extended our investigation to examine whether these findings apply to CAE caused by different genetic mechanisms, particularly mutations in sodium channel genes by modelling their effects at the individual neuron level. Furthermore, we examined the degree to which these alterations lead to network-level pathological activity, as well as the influence of ALLO on these genetically distinct networks. Our results demonstrate that ALLO facilitates recovery from SWDs regardless of the underlying mutation type. However, enhanced frontocortical connectivity prevents recovery in some mutation types, particularly when mutation effects are severe. Altogether, the multi-scale computational framework developed in this thesis demonstrates that CAE remission is determined by complex interactions between hormonal influences, genetic factors, and network connectivity patterns. The results suggest that certain genetic mutations may predispose individuals toward either remission or non-remission, which can be further modulated by connectivity profiles. In particular, enhanced frontocortical connectivity appears to be a significant factor in resistance to hormone-mediated remission. Additionally, this thesis develops techniques for analyzing transitions between distinct dynamical states in neural systems, incorporates genetic and hormonal factors into conductance-based models, and provides a computational framework to identify key parameters governing epileptic activity. These approaches not only advance our understanding of CAE specifically, but offer generalizable insights into the mathematical modelling of neurological conditions characterized by spontaneous shifts in brain dynamics.
Phase Model Analysis of the Effect of Acetylcholine on the Neural Synchrony in Hippocampal Networks
(University of Waterloo, 2025-06-20) Manoj, Megha
Neural assemblies—transiently coordinated groups of neurons—are observed in the hippocampus and are thought to underlie the encoding and consolidation of episodic memories. Acetylcholine (ACh), a key neuromodulator, plays a critical role in learning and memory and has been implicated in neurodegenerative disorders involving hippocampal dysfunction. A well-supported hypothesis suggests that high levels of ACh during active exploration and rapid eye movement (REM) sleep promote memory encoding, while low levels during quiet waking and slow-wave sleep (SWS) support memory consolidation. In this study, we examine the bidirectional role of ACh in modulating neural assembly formation through its effect on neural synchrony in the CA3 region of the hippocampus. We construct a computational model of a network of excitatory pyramidal neurons, each equipped with a slow, voltage-dependent, non-inactivating potassium current (M-current), which is downregulated in the presence of ACh. Neural assemblies are modelled mathematically as cluster solutions—special types of phase-locked states. Using a phase model reduction of a pair of weakly coupled neurons, we analyze the existence and stability of cluster solutions that may emerge in larger networks equipped with all-to-all globally homogeneous, symmetric distance-dependent and nearest-neighbours coupling architectures. Our results suggest that ACh shapes assembly formation by modulating network dynamics in CA3. Under low ACh conditions, the network tends to fully synchronize, whereas high ACh levels enable the emergence of multiple stable cluster states, allowing for distinct patterns of activity associated with memory encoding. These findings propose a mechanism by which ACh regulates transitions in hippocampal network states, supporting distinct stages of memory formation.
Kinetic Energy Spectra, Backscatter, and Subgrid Parameterization Analysis in Radiative-Convective Equilibrium
(University of Waterloo, 2025-06-16) Lai, Kwan Tsaan
This thesis explores how energy is distributed and transferred across scales in convective-permitting radiative-convective equilibrium (RCE) simulations and how these processes can be more accurately represented in numerical models through improved subgrid parameterizations. Aggregation steepens the horizontal kinetic energy spectra by enhancing the large-scale energy, which results in horizontal kinetic energy spectra in both the upper troposphere and lower stratosphere that are close to the mesoscale -5/3 spectrum. In the upper troposphere, spectral energy budget analysis indicates that this is the result of the balance between buoyancy flux and vertical energy flux, rather than a classic direct energy cascade. In the lower stratosphere, there is inverse energy transfer, which may be explained by wave-mean-flow-interaction. Subfilter energy transfer analysis is performed on an idealized RCE simulation by filtering 1-km high-resolution simulation to a horizontal scale of 4 km. The net subfilter energy transfer rate is dissipative in the upper troposphere and backscattering in the lower stratosphere, which are consistent with the direction of energy transfer in the nonlinear transfer energy flux. The stochastic backscatter TKE scheme, a stochastic backscatter-allowing subgrid turbulence scheme created by adding a zero-mean stochastic forcing to the eddy viscosity of the TKE scheme, is proposed and tested on idealized RCE simulations. The stochastic backscatter TKE scheme improves the subgrid local energy transfer when compared to a common stochastic backscatter scheme and the standard TKE scheme without backscatter. Despite the fact that backscatter is still weaker than dissipation in the lower stratosphere in the stochastic backscatter TKE simulations, the kinetic energy spectra are closer to the -5/3 spectrum when compared to the standard TKE simulation. This study advances our understanding of the interscale distribution and transfer of energy in RCE, and the introduction of the stochastic backscatter TKE scheme provides a more realistic representation of dissipation and backscatter by matching the distribution of subfilter energy transfer rate from a high-resolution simulation.
Quickest Change Detection in Nonlinear Hidden Markov Models Using a Generalized CUSUM Procedure
(University of Waterloo, 2025-05-27) Li, Dongchang
Fault diagnosis in modern aircraft engines is crucial for monitoring due to the rising need for high performance and safety. Early detection of system changes within a controllable range can prevent significant breakdowns. It's essential to track jet engine states and detect real-time dynamic mode shifts. This thesis explores change detection theories and procedures to handle dynamic instabilities in jet engines. We use the Moore-Greitzer equation to model flow and pressure changes in axial-flow compressors, specifically focusing on a reduced planar system. The research addresses QCD problems in nonlinear hidden Markov models, using pressure rise coefficients as observational data. The standard QCD scheme doesn't apply, so we employ the generalized CUSUM procedure, where the post-change distribution depends on an unknown change point, despite its non-recursive nature. We adopt standard filtering theory to approximate log-likelihood ratios with particle methods. To manage computational costs, we adjust the CUSUM-like procedure with an assumption of immediate change to enable recursion. This research focuses on how changes in the system excite shifts from a steady state to a new equilibrium or periodic oscillations. We assess the performance of generalized CUSUMs with particle filters through random simulations in surge modes of the Moore–Greitzer model with external forcing. Observations reveal similarities and differences between generalized CUSUMs and a special CUSUM that assumes immediate change, influenced by phase errors and the relationship of the steady state to the limit cycle. Signal noise mitigates the phase effects of the limit cycle. This research addresses change detection in the Moore-Greitzer PDE model, where disturbances in axial flow at the compressor inlet are combined with a modified ODE system. We simulate the PDE system by obtaining time series from a finite-dimensional Moore-Greitzer system using the Fourier spectral method. Employing proper orthogonal decomposition (POD), we reduce model dimensions while maintaining fidelity with fewer basis functions. The reduced model is validated by reconstructing the PDE system with $N$ POD modes capturing about 95% energy of the full model in $L^2$ inner product and $H^1$ Sobolev spaces. We examine generalized CUSUM statistics to detect dynamic changes, using POD bases and particle filters, with simulations showing the effectiveness of CUSUM statistics using {\em in situ} and {\em a priori} POD modes. Validated reduced models implement the generalized CUSUM for stall detection in stochastic systems, showing that the generalized CUSUM statistics are more effective and robust than recursive CUSUM-like procedures.
A data-centric view of LQR algorithms in continuous time
(University of Waterloo, 2025-05-21) Song, Christopher
Many control theorists are interested in how measurements of input and state trajectories can determine the properties of a control system, in lieu of the differential or difference equation models that usually play this role. In situations of practical interest, such models may be wholly or partially unknown, while input and state data are readily available. In the discrete- and continuous-time linear quadratic regulator (LQR) settings, this area of research has produced a proliferation of algorithms that use input and state data to solve the LQR problem, the system identification problem, or both. In different algorithms, these data and the requirements imposed on them take different forms that are not directly comparable, making it difficult to assess the relative efficiencies with which different algorithms makes use of data. In [30], the authors show that the LQR and system identification problems are essentially equivalent in the discrete-time setting. In this thesis, we extend this result to the continuous-time setting, showing that, assuming input and state data are collected on intervals, every algorithm that solves the LQR problem requires at least as much data as system identification. From this, we show that the map from the data to the optimal gain defined by these algorithms is continuous, establishing a connection between interval data and sampled data algorithms. The possibility of using sampled data in place of interval data leads to a weaker convergence criterion on sampled data approximations, and a natural connection with numerical integration. We do some numerical experiments that show the critical importance of choosing when to make input and state measurements, and emphasize the possibility of doing so without knowledge of the system or its optimal gain.
Minimum Energy Estimation of Nonlinear Dynamical Systems in Continuous and Discrete Time
(University of Waterloo, 2025-05-20) Tres Vilanova, Albert
In many applications, accurately estimating a system's state is crucial for effective control. This thesis gives an in-depth introduction to the minimum energy estimator (MEE), an optimal estimator that is based on the minimum energy cost function. Although less widely used than the extended Kalman filter or the unscented Kalman filter, the MEE offers a rigorous optimality guarantee. Using tools from optimal control theory - such as the Pontryagin maximum principle and the Hamilton-Jacobi-Bellman (HJB) equation - we derive both continuous and discrete time formulations of the MEE. Conditions for the stability and converge of the MEE are reviewed. We also demonstrate that the MEE coincides with the Kalman filter in the case of linear dynamics. The derivation leads to an estimator equation for the MEE that requires solving an HJB equation, prompting the introduction of various numerical methods to address this challenge. To further explore the relationship between continuous and discrete time filtering, simulations are conducted to compare the accuracy of the discrete time Kalman filter and a "hybrid" Kalman filter, where the continuous time filter is discretized in the update step.
Reduced-Order Modeling and Data Assimilation of the El Niño–Southern Oscillation
(University of Waterloo, 2025-05-16) Aydogdu, Yusuf
Simulations of complex fluid dynamics problems or climate models take weeks to complete even when run parallel in state-of-the-art supercomputers. Given computational resource constraints and the need for adaptable simulation settings, cost-efficient and accurate algorithms are essential. In this thesis, we explore stable, efficient, and accurate methodologies when applied to the El Niño–Southern Oscillation (ENSO), which integrates coupled atmosphere, ocean, and sea surface temperature (SST) mechanisms in the equatorial Pacific. ENSO is one of the most influential and complex climate phenomena, affecting weather patterns across the globe. ENSO consists of irregular oscillations between warm (El Niño) and cold (La Niña) phases in the Pacific Ocean, significantly impacting global weather patterns. Due to ENSO's inherent complexity and uncertainties, it is particularly suited for stochastic modeling. By modeling these uncertainties, stochastic simulations offer a more accurate representation of ENSO's variability, including its irregular periods and amplitudes. We first study the effects of stochastic perturbations on ENSO dynamics and introduce novel modeling and numerical schemes based on the Wiener Chaos Expansion (WCE). The key idea behind WCE is the explicit discretization of white noise through Fourier expansion. We also compare these methods with Monte Carlo (MC) simulations. Our findings demonstrate that the simulation of the linear stochastic ENSO model driven by the Ornstein-Uhlenbeck process using WCE requires far less computational resources and gives more accurate results compared to MC ensembles. This part of the thesis provides an alternative efficient approach for simulations of stochastic climate models and quantification of statistical moments,i.e, mean and variance. In the next stage of this research, we explore a reduced-order modeling (ROM) framework based on the POD-Galerkin method when applied to a nonlinear ENSO model. POD-Galerkin reduced order modeling aims to reduce the computational complexity and present high-dimensional problems~(usually PDEs) with reduced-order equations (ODEs). POD modes are optimal in capturing the system’s dominant features, making it particularly effective for reducing the dimensionality of systems governed by PDEs. By capturing the full-order ENSO (PDE) model with only four modes and four reduced-order equations, we achieve a substantial reduction in computational complexity without significant loss of accuracy. Due to the special properties of the model, we introduce a novel approach using different POD bases, but the same time coefficients for all model components. Moreover, we employ machine learning methods to explore different ROM and model discovery techniques in this part. The final part of this thesis focuses on the data assimilation of the nonlinear stochastic ENSO model, which forms the core results of this research. We first demonstrate the validity of POD-Galerkin reduced order modeling for the stochastic ENSO driven by the Ornstein-Uhlenbeck process. We project SPDEs onto POD modes derived from the deterministic model and introduce the reduced-order stochastic equations (SDEs). After setting up the filtering framework, we combine these equations and artificial observations in the Pacific ocean, based on realistic experiments, to estimate the ENSO-related SST anomalies. We employ particle filters and test the efficiency using different number of particles and ensembles. From novel ENSO modeling to uncertainty quantification, from reduced order modeling to nonlinear filtering, this thesis provides a promising approach for accurate and efficient predictions of ENSO-related climate variables.
Measurements of quantum fields with particle detectors
(University of Waterloo, 2025-05-14) Polo Gomez, Jose
This thesis aims to provide a working measurement theory for quantum fields built upon measurements using localized non-relativistic quantum systems, widely known as particle detectors. In Chapters 3 to 5, we focus on using detector-based schemes to carefully formulate a measurement theory that is compatible with relativity. To do so, we provide a rigorous analysis of the causal behaviour of the induced non-selective channels, as well as an update rule that is consistent with relativistic causality. In the process, we establish a body of results, including a characterization of localized causal channels in real scalar QFT, and the formulation of a formalism that allows a consistent treatment of non-relativistic multipartite systems in relativistic setups. In Chapters 6 to 8, we focus on verifying that the measurement theory formulated in the first part is a working measurement theory, meaning that it can actually be used in practice to measure specific targeted features of the quantum field. To ensure this, we introduce a measurement strategy where particle detectors always undergo the same measurement protocol—regardless of the quantity or feature of interest—and then are subjected to a tomography process. The resulting data is fed into a trained neural network that is capable of inferring the targeted quantity or feature from the detector's readings. This strategy has the potential to be applicable to measure any quantity of the field that is accessible through local measurements. More tangentially, we also introduce a method beyond perturbation theory that uses trains of delta couplings to efficiently approximate the final state of a detector undergoing a continuous interaction. Additionally, we apply the detector-based measurement theory to analyze the effect of measurements on the protocol of entanglement harvesting.
Cosmic Frontiers: From the Origins to Gravitational Wave Astronomy
(University of Waterloo, 2025-05-14) Dehghanizadeh, Amirhossein
In this thesis, we present different aspects of cosmology, focusing on the early universe, large scale structure formation, and gravitational wave astrophysics. We begin by introducing cosmological perturbation theory as a framework to describe the origin of the universe, emphasizing its observational implications. We discuss early universe models and, motivated by the avoidance of singularities, we propose bouncing cosmologies. In particular, we discuss the Cuscuton model; which modifies gravity to generate a bounce without introducing new degrees of freedom, provides a smooth transition between contraction and expansion, and generates scale-invariant entropy perturbations. We analyze scalar perturbations and demonstrate that the model remains stable and weakly coupled. Additionally, we compute three-point functions, showing negligible non Gaussianities on observable scales, with potential signals arising from the conversion of isocurvature to curvature perturbations. Building on this foundation, we investigate the formation of large scale structures and study bias parameters that describe the statistical distribution of dark matter halos and galaxies. We extend this analysis to gravitational wave (GW) sources, introducing the GW bias parameter. We propose a methodology to link binary black hole (BBH) mergers to their host galaxies using phenomenological GW host-galaxy probability functions. These functions are constructed based on observed astrophysical properties of galaxies, including stellar mass, star formation rate, and metallicity. We calculate the GW bias using the angular power spectrum for photometric surveys and the 3D power spectrum for spectroscopic surveys. We find that the GW bias depends on the parameters of the host-galaxy probability functions, which are linked to the formation channels of BBHs. Future measurements of the GW bias could provide insights into the astrophysical processes governing GW sources and their relation to large-scale structure. Lastly, we emphasize that cosmology may be entering a golden era, driven by a wealth of upcoming experiments, including the next generation of Cosmic Microwave Background (CMB) telescopes, galaxy surveys, and GW detectors. These experiments have the potential to provide hints for modifications to ΛCDM cosmology, the nature of dark matter and dark energy, black hole physics, and possible deviations from general relativity, paving the way for new theoretical and observational breakthroughs.
Interaction of electromagnetic fields with two-dimensional materials at the interface between dielectric media
(University of Waterloo, 2025-05-08) Hansen, Asbjorn
This thesis investigates the interaction of electromagnetic plane waves with two-dimensional (2D) materials approximated by infinitesimal sheets positioned at interfaces between non-magnetic, isotropic, lossless dielectrics. On their own, 2D materials offer a host of useful properties, including high carrier mobility, dopant-free tuning through the application of external electromagnetic fields or stacking, as well as significant anisotropy. When combined with the excitation of plasmon modes to enhance light-matter interactions, the range of potential applications broadens greatly, including, for example, high-speed optical computation and enhanced light trapping in solar cells. In this thesis, a new derivation of the boundary condition for out-of-plane response is presented for a 2D material immersed in an infinite homogeneous dielectric. To account for the influence of distinct dielectric surrounding media, Fresnel coefficients are derived using a vacuum gap method. This approach yields novel dispersion relations for surface plasmons that incorporate the out-of-plane response of 2D materials. In the presence of distinct surrounding dielectrics, the out-of-plane mode hybridizes with the in-plane longitudinal plasmon mode. The hybridization is destroyed when the surrounding materials are made identical. Additionally, expressions for the conservation of energy and momentum are derived in terms of the Fresnel field amplitudes for the 2D materials described by singular current sheets. The vacuum gap method is also applied to solve the conservation equations in the presence of dielectric media, as this avoids controversial notions of electromagnetic momentum in media. The electromagnetic stress tensor derived through the vacuum gap method satisfies conservation of linear momentum, thus confirming that the forces on a 2D material can be self-consistently separated from those on surrounding dielectrics. This theoretical framework is applied to both graphene and phosphorene. For graphene, the reflectance, transmittance, and absorption characteristics are compared with and without considering the out-of-plane response. The analysis confirms that the influence of the out-of-plane component is minimal for plasmon modes at low frequencies as is often assumed, but becomes significant at high frequencies. Forces calculated around frequencies and wavenumbers defined by the plasmon dispersion relation showed significant amplification. However, the new non-monotonic dispersion profile of the hybridized plasmon mode introduces additional resonant and antiresonant behaviour in the forces. To emphasize the in-plane anisotropy of phosphorene, only in-plane response was considered, with a focus on forces in the plasmonic regime at low frequencies. Based on numerical examples, predictions are made for possible future experimental studies of both plasmon modes in the presence of the out-of-plane response and forces on 2D materials for possible optomechanical and sensing applications.

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