# 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).

Waterloo faculty, students, and staff can contact us or visit the UWSpace guide to learn more about depositing their research.

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### Recent Submissions

Item Degenerate Parabolic Diffusion Equations: Theory and Applications in Climatology(University of Waterloo, 2024-08-27) Vlachos, PanagiotisDiffusion models describe the spread of particles, energy, or other entities within a medium. Perturbations of mechanical systems, random walks(discrete case), and Brownian motion(continuous-time stochastic process) are some classical methods used to model diffusion. Among these, those generated by stochastic processes have been extensively studied by employing the Fokker-Planck equation—a one-dimensional parabolic partial differential equation—to examine these systems by analyzing the probability density function. Given the incomplete theory surrounding degenerate diffusion equations, our objective is to generalize and expand existing results for degenerate diffusion processes by examining cases where weak degeneracy occurs at the boundaries, utilizing a Fokker-Planck-like equation. More precisely, we first address the well-posedness results, which ensure the existence and uniqueness of a solution and are critical for investigating other qualitative properties such as controllability, observability, stabilization, and optimal control. Additionally, we explore the interval of the existence or absence of stationary states, which is fundamental in the analysis of mechanical or physical systems. To this end, we examine sufficient conditions for both the non-existence and existence of stationary points. Furthermore, to verify and illustrate our analytical results, we delve into the Budyko-Sellers model, a climate model, providing results on its well-posedness and addressing the inverse problem of determining the insolation function. Throughout this study, we primarily employ semigroup theory, op- erator and functional analysis, and weighted Sobolev spaces to manage the non-ellipticity of the diffusion and well-posedness of the parabolic equation, while using the theory of Lyapunov functions to ensure the existence of stationary states.Item Systems biology models for cancer immunotherapy(University of Waterloo, 2024-08-19) Cotra, SonjaCancer is a complex disease that continues to affect millions of people around the world every year. With ever-improving science and technology, several forms of treatment have been introduced within the past century and continue to be developed so as to provide increasing chances of survival and comfort to patients. Particularly, the 21st century has seen the blossoming of immunotherapy methods, which exploit the natural immune system's ability to kill tumor cells. Several varieties of immunotherapy exist in order to use all sorts of immune cells, targeting specific antigens expressed on tumors or blocking checkpoints which inhibit necessary immune responses. Unfortunately, there is no perfect immunotherapy that can provide a safe and effective path to remission for every patient. Traditional clinical experimentation, while providing important insight, remains a costly option in increasing our understanding of immunotherapies against cancer. Systems biology methods provide a unique and effective channel for exploring the complex dynamics involved in tumor micro-environments between cancer cells, native immune cells and administered drugs. Resulting insight may be used to inform drug development leading to safe, effective, and personalized therapeutic routines. In this thesis, we start by providing a general overview of cancer biology starting from the cell, and systems biology. We then detail equations and parameters comprising a particular systems biology model for nivolumab, an anti-PD-1 immune checkpoint inhibitor, informed by ex vivo data extracted from patients suffering from head and neck squamous cell carcinoma. We then present results of an examination of sex differences in regards to patient response to nivolumab monotherapy as well as combination therapy with recombinant IL12. Here, the aforementioned model was used alongside basal immune differences between the sexes from the literature to generate virtual cohorts of male and female patients receiving these treatments. Finally, we conclude with a general summary as well as potential future directions involving a similar systems biology model describing cytokine release syndrome as a side-effect of CAR-T cell therapy.Item Analyzing Bacterial Conjugation with Graphical Models: A Model Comparison Approach(University of Waterloo, 2024-08-12) Kendal-Freedman, NatConjugation is a mechanism for horizontal gene transfer that allows microbes to share genetic material with nearby cells. It plays an important role in the spread of antibiotic resistance in bacteria and is used as a tool for genetic engineering. Understanding which factors affect conjugation frequency is an ongoing challenge due to the stochastic nature of cell-cell interactions. In this thesis, we present a proof of concept of a model comparison approach for analyzing experimental data of bacterial conjugation. We develop a Bayesian network structure to model the interactions within a single experimental trial. We model different versions of biological mechanisms by assigning different conditional probability distributions to those structures. Identifying distributions that predict events consistent with the experimental results provides insight into the mechanisms governing conjugation. We compare 12 model variations for each of 6 experimental trials. Our results suggest that individual cell features and contact quality both impact the likelihood of conjugation. We also provide insight into the length of the delays involved in conjugation. These results are consistent when compared across multiple trials and metrics.Item Reactive Tracers in Turbulence Models(University of Waterloo, 2024-07-25) Legare, SierraThis thesis characterizes the nature of inaccuracies in the description of two tracers, one passive and one reactive (with a particular choice of model) within Reynolds Averaged Navier Stokes simulations (RANS) and Large Eddy Simulations (LES). Both tracers are taken to be passive with respect to the fluid dynamics and the reactive tracer is assumed to undergo a growth reaction governed by Fisher's equation. To begin, the Navier-Stokes equations and the partial differential equation governing the reactive tracer evolution are Reynolds averaged and spatially filtered to obtain the governing equations for each of the turbulence models. The procedure is applied to a generalized polynomial reaction function and can be extended to other sufficiently smooth non-polynomial reactions. The Reynolds averaged and filtered reaction equations are analyzed using a simplified, zero dimensional toy model. A one dimensional toy model is used to illustrate how a non-linear reaction term, advection, and diffusion each influence the spectral distribution of a reactive tracer. To consider the effect of Reynolds averaging, an ensemble of 50 two-dimensional direct numerical simulations is run. Within each simulation, the reactive tracer is subjected to mixing induced by a Rayleigh-Taylor instability. A posteriori Reynolds averaging is applied to the ensemble data to evaluate the discrepancies between the mean system and the dynamics of the ensemble members. A two-dimensional toy model with specified velocities is used to illustrate the effect of spatial filtering. Further, the dependence of the sub-filter-scale flux and reaction terms on the cutoff wavenumber of a low-pass filter and the reaction rate is evaluated. To investigate a system with a larger range of spatial scales, a posteriori LES is applied to data from a three-dimensional simulation of a reactive tracer subjected to turbulence induced by a Rayleigh-Taylor instability. Various filter choices are applied and the sub-filter-scale terms are quantified. Given the scope of this thesis, discussions of the findings and their implications for modelling can be found at the end of chapters 5 and 6. This thesis concludes with a broader discussion of the findings and highlights avenues for future work.Item Exponential Stability of Maxwell’s Equations(University of Waterloo, 2024-07-25) Poulin, JosephThe control of electromagnetic systems, which is governed by Maxwell's equations, is of interest for various reasons such as controlling plasma in a nuclear fusion reactor or magnetically trapping antiparticles for observation. Proving that a controlled system is exponentially stable is of particular interest, as exponential stability infers that the system will converge asymptotically to a steady state solution. A tool called the multiplier method is considered, which allows for exponential stability to be demonstrated by defining an auxiliary functional and proving it is bounded by the system energy and the energy's time derivative in a particular way. If an appropriate bound is shown, exponential stability is not only guaranteed, but the exponential function which bounds the L2 norm of the system variables will be fully determined in terms of the systems parameters. Currently, there is ongoing work into generalizing the multiplier method approach for a class of problems known as Port-Hamiltonian systems. This thesis aims to contribute to this work by formulating Maxwell's equations as a Port-Hamiltonian system, and using this formulation as a basis for determining how to choose the auxiliary functional needed in the multiplier method.Item Hopf 2-Algebras: Homotopy Higher Symmetries in Physics(University of Waterloo, 2024-06-26) Chen, HankThe theory of Hopf algebras and quantum groups have led to very rich and interesting developments in both mathematics and physics. In particular, they are known to play crucial roles in the interplay between 3d topological quantum field theories, categorical algebras, and the geometry of embedded links and tangles. Moreover, the semiclassical limits of quantum group Hopf algebras, in particular, are vital for the understanding of integrable systems in statistical mechanics and Poisson-Lie dualities in string theory. The goal of this PhD thesis is to study a higher-dimensional version of these correspondences, based on the very successful categorical ladder proposal: higher-dimensional physics and geometry is described by higher-categorical strutures. This is accomplished with the definition of a {\it higher homotopy Hopf algebra}, which can be understood as a quantization of the homotopy Lie bialgebra symmetries that have recently received attention in various fields of theoretical physics. These higher-homotopy symmetries are part of the study of the recently-popular categorical symmetries, which appear in the condensed matter literature, for instance, in relation to 1-form dipole symmetries in topologically ordered phases. However, here I will provide another physical motivation arising from the gauge theoretic perspective, which is natural in the context of the Green-Schwarz anomaly cancellation mechanism in quantum field theories. In particular, I use this perspective to prove various known structural theorems about Lie 2-bialgebras and their associated 2-graded classical $R$-matrices, as well as to provide a new definition and characterization of the so-called "quadratic 2-Casimir" elements. I will apply these higher homotopy symmetries to study the 4d 2-Chern-Simons topological quantum field theory, and to develop a notion of graded classical integrability for 2+1d bulk-boundary coupled systems. By following the philosophy of deformation quantization and the theory of $A_\infty$-algbera, I then introduce the notion of a "Hopf 2-algebra" explicitly, and prove several of their structural theorems. I will in particular derive a novel definition of a universal quantum 2-$R$-matrix and the higher-Yang-Baxter equations they satisfy. The main result of this thesis is that the 2-representation 2-category of Hopf 2-algebras is cohesively braided monoidal iff it is equipped with a universal 2-$R$-matrix, and that (weak) Hopf 2-algebras admit (weak) Lie 2-bialgebras as semiclassical limits. Finally, an application of this quantization framework will be considered, in which I will explicitly compute the higher representation theory of Drinfel'd double Hopf 2-algebras of finite groups. The corresponding 2-group Dijkgraaf-Witten topological field theories are then constructed directly from these Hopf 2-algebras, and I show that they recover the known 2-categorical characterizations of 4d $\mathbb{Z}_2$ symmetry protected topological phases of matter.Item On Enabling Layer-Parallelism for Graph Neural Networks using IMEX Integration(University of Waterloo, 2024-06-20) Kara, Omer EgeGraph Neural Networks (GNNs) are a type of neural networks designed to perform machine learning tasks with graph data. Recently, there have been several works to train differential equation-inspired GNN architectures, which are suitable for robust training when equipped with a relatively large number of layers. Neural networks with more layers are potentially more expressive. However, the training time increases linearly with the number of layers. Parallel-in-layer training is a method that was developed to overcome the increase in training time of deeper networks and was first applied to training residual networks. In this thesis, we first give an overview of existing works on layer-parallel training and graph neural networks inspired by differential equations. We then discuss issues that are encountered when these graph neural network architectures are trained parallel-in- layer and propose solutions to address these issues. Finally, we present and evaluate experimental results about layer-parallel GNN training using the proposed approach.Item Stability Analysis and Formally Guaranteed Tracking Control of Quadrotors(University of Waterloo, 2024-06-19) Chang, HaochengReach-avoid tasks are among the most common challenges in autonomous aerial vehicle (UAV) applications. Despite the significant progress made in the research of aerial vehicle control during recent decades, the task of efficiently generating feasible trajectories amidst complex surroundings while ensuring formal safety guarantees during trajectory tracking remains an ongoing challenge. In response to this challenge, we propose a comprehensive control framework specifically for quadrotor UAVs reach-avoid tasks with robust formal safety guarantees. Our approach integrates geometric control theory with advanced trajectory generation techniques, enabling the consideration of tracking errors during the trajectory planning phase. Our framework leverages the well-established geometric tracking controller, analyzing its stability to demonstrate the local exponential stability of tracking error dynamics with any positive control gains. Additionally, we derive precise and tight uniform bounds for tracking errors, ensuring guaranteed safety of the system's behavior under certain conditions. In the trajectory generation phase, our approach incorporates these bounds into the planning process, employing sophisticated sampling-based planning algorithms and safe hyper-rectangular set computations to define robust safe tubes within the environment. These safe tubes serve as corridors within which trajectories can be constructed, with piecewise continuous Bezier curves employed to ensure smooth and continuous motion. Furthermore, to enhance the performance and adaptability of our framework, we formulate an optimization problem aimed at determining optimal control gains, thereby enabling the quadrotor UAV to navigate with optimal safety guarantees. To demonstrate the validation of the proposed framework, we conduct comprehensive numerical simulations as well as real experiments, demonstrating its ability to successfully plan and execute reach-avoid maneuvers while maintaining a high degree of safety and precision. Through these simulations, we illustrate the practical effectiveness and versatility of our framework in addressing real-world challenges encountered in UAV navigation and trajectory planning.Item Controller and Observer Designs for Partial Differential-Algebraic Equations(University of Waterloo, 2024-06-19) Alalabi, Ala'Partial differential-algebraic equations (PDAEs) arise in numerous situations, including the coupling between differential-algebraic equations (DAEs) and partial differential equations (PDEs). They also emerge from the coupling of partial differential equations where one of the equations is in equilibrium, as seen in parabolic-elliptic systems. Stabilizing PDAEs and achieving certain performance necessitate sophisticated controller designs. Although there are well-developed controllers for each of PDEs and DAEs, research into controllers for PDAEs remains limited. Discretizing PDAEs to DAEs or reducing PDAE systems to PDEs, when feasible, often results in undesirable outcomes or a loss of the physical meaning of the algebraic constraints. Consequently, this thesis concentrates on the direct design of controllers based on PDAEs, using two control techniques: linear-quadratic and boundary control. The thesis first addresses the stabilization of coupled parabolic-elliptic systems, an important class of PDAEs with wide applications in fields such as biology, incompressible fluid dynamics, and electrochemical processes. Even when the parabolic equation is exponentially stable on its own, the coupling between the two equations can cause instability in the overall system. A backstepping approach is used to derive a boundary control input to stabilize the system, resulting in an explicit expression for the control law in a state feedback form. Since the system state is not always available, exponentially convergent observers are designed to estimate the system state using boundary measurements. The observation error system is shown to be exponentially stable, again by employing a backstepping method. This leads to the design of observer gains in closed form. By integrating these observers with state feedback boundary control, the thesis also tackles the output feedback problem. Next, the thesis considers finite-time linear-quadratic control of PDAEs that are radial with index 0; this corresponds to a nilpotency degree of 1. The well-known results for PDEs are generalized to this class of PDAEs. Here, the existence of a unique minimizing optimal control is established. In addition, a projection is used to derive a system of differential Riccati-like equation coupled with an algebraic equation, yielding the solution of the optimization problem in a feedback form. These equations, and hence the optimal control, can be calculated without constructing the projected PDAE. Lastly, the thesis examines the linear-quadratic (LQ) control problem for linear DAEs of arbitrary index over a finite horizon. Without index reduction or a behavioral approach, it is shown that a certain projection can lead to the derivation of a differential Riccati equation, from which the optimal control is obtained. Numerical simulations are presented to illustrate the theoretical findings for each objective of the thesis.Item Ice/hydrology feedback in the Siple Coast, Antarctica, from two-way coupled modeling(University of Waterloo, 2024-06-18) McArthur, KoiSubglacial hydrological processes have long been understood to play a critical role in ice dynamics (Budd et al., 1979). Consequently, the recent emergence of complex two-dimensional subglacial hydrology models with both inefficient and efficient drainage components has led to two-way coupling of these complex hydrology models to ice flow models (Cook et al., 2022). Such two-way coupled models bring about questions regarding our implementation of friction in ice flow models and allow us to examine feedback mechanisms between the subglacial hydrological system and the ice sheet. This thesis investigates feedback mechanisms between the subglacial hydrological system and the ice sheet, and analyzes our current implementation of friction in ice flow models. This is accomplished through subglacial hydrology, ice flow, and coupled modeling of the Siple Coast of West Antarctica, which has a history of observable hydrology/ice flow feedback. We use the Glacier Drainage System (Werder et al., 2013, GlaDS) model as a subglacial hydrology model, and the Shallow Shelf Approximation (Larour et al., 2012, SSA) with a mass transport model as an ice flow model, both of which are implemented in the Ice-Sheet and Sea-Level Systems Model (Larour et al., 2012, ISSM). We model the steady state subglacial hydrology of the Ross Sea subglacial hydrologic catchment, along with ice flow and two-way coupled ice flow/hydrology from 2010-2100 using an SSP585 surface mass balance forcing scenario. We test three different friction laws – the Budd friction law, the Schoof friction law, and a version of the Schoof friction law that we modify to ensure the sliding regime is representative of the cavitation at the glacier bed. Additionally, we test coupling with variable melt from frictional heating of ice, coupling with subglacial lake geometry altering glacier driving stress, and coupling with a combination of the two. The effective pressure and the modeled sliding regime were found to be largely responsible for the evolution of fast flowing regions of the domain, highlighting the importance of two-way coupled models, which have a cavitation-dependent sliding regime. Feedback mechanisms between the subglacial hydrologic system and the ice sheet were identified, including a negative feedback mechanism that stabilized the basal shear stress and the effective pressure fields when variable melt was available to the subglacial hydrologic system. The inclusion of subglacial lake geometry on the glacier driving stress was found to have a large control on lake depth, with the potential for large speedup events corresponding to the fast filling of subglacial lakes. When all coupling components were active, a negative feedback mechanism between subglacial lake depth, glacier driving stress, and melt water production, which stabilized subglacial lake depth and ice motion was observed. The methods developed in this thesis and the limitations that we discovered for implementing subglacial processes in ice flow models will be highly valuable to the glaciological modeling community moving forwards.Item Multiscale Modelling of Biological Rhythms and Systems(University of Waterloo, 2024-06-17) Abo, Stéphanie Marie ColetteLiving organisms possess the remarkable ability to both respond to rhythms and generate them. In some instances, unintended rhythms arise, leading to undesirable or even hazardous consequences, such as synchronized neuronal firing during epilepsy. However, in other cases, such biological rhythms are beneficial in regulating essential processes across all life forms. From bacteria to humans, rhythms permeate various aspects of life, influencing everything from biochemical reactions to lifestyle habits. Here, our focus is on understanding systems that actively generate rhythms, known as clocks. Clocks, in particular, are systems that not only generate rhythms but also respond to environmental signals. Examining rhythms in isolation, without considering their generation, alteration, or regulation, would provide limited insights into the complexities of biological systems. We study the interaction between biological clocks and physiological processes: sleep, the immune system, metabolism, and environmental perturbations such as fluctuations in photoperiods. We develop mathematical and computational frameworks to investigate rhythms and their influence on biological processes at tissue and system levels. We specifically study cell-cell interactions at the level of the suprachiasmatic neuleus (SCN) in the hypothalamus of the brain, also called the master circadian clock. We investigate how noise at the level of the individual cells affect properties of the ensemble: period, oscillation amplitude, and bifurcation boundaries. Starting from individual dynamics, we derive macroscopic descriptions called mean field limits for interacting cells. Going up in scale, we also study the interactions between the peripheral circadian clock in the lung and the innate immune system during inflammation. At this organ scale, we investigate protein-protein interactions between clock proteins and immune agents, called cytokines. We are interested in the reciprocal modulation between these two systems, especially when the circadian rhythm is disrupted. Finally, we move from organ-level to the whole-body level. We develop multi-organ models of metabolism. These whole-body models integrate exercise and diet. Given the ubiquity of circadian rhythms at all levels of our physiology, these models are intended for the study of the role of external signals, beside neural signals emanating from the SCN, on (re-)synchronizing rhythms in the periphery. The interplay between such signals and metabolic processes plays a role in maintaining homeostasis, while also organizing and timing physiological processes in a proactive rather than reactive manner. This thesis contributes to the development of novel frameworks aimed at understanding multiscale systems, analyzing the relationships between network structure and dynamics, and ultimately deriving candidate mechanisms that can be experimentally verified.Item Collective Dynamics of Large-Scale Spiking Neural Networks by Mean-Field Theory(University of Waterloo, 2024-05-31) Chen, LiangThe brain contains a large number of neurons, each of which typically has thousands of synaptic connections. Its functionality, whether function or dysfunction, depends on the emergent collective dynamics arising from the coordination of these neurons. Rather than focusing on large-scale realistic simulations of individual neurons and their synaptic coupling to understand these macroscopic behaviors, we emphasize the development of mathematically manageable models in terms of macroscopic observable variables. This approach allows us to gain insight into the underlying mechanisms of collective dynamics from a dynamical systems perspective. It is the central idea of this thesis. We analytically reduce large-scale neural networks to low-dimensional mean-field mod- els that account for spike frequency adaptation, time delay between neuron communication, and short-term synaptic plasticity. These mean-field descriptions offer a precise correspondence between the microscopic dynamics of individual neurons and the macroscopic dynamics of the neural network, valid in the limit of infinitely many neurons in the network. Bifurcation analysis of the mean-field systems is capable of predicting net- work transitions between asynchronous and synchronous states, or different patterns of synchronization, such as slow-fast nested collective oscillations. We discuss how these dynamics are closely related to normal brain functions and neurological disorders. We also investigate the influence on these dynamic transitions induced by current heterogeneity, adaptation intensity, and delayed coupling. By integrating a kinetic model of synapses into the neural network, we describe calcium-dependent short-term synaptic plasticity in a relatively simple mathematical form. Through our mean-field modeling approach, we explore the impact of synaptic dynamics on collective behaviors, particularly the effect of muscarinic activation at inhibitory hippocampal synapses. Together, this thesis provides a tractable and reliable tool for model-based inference of neurological mechanisms from the perspective of theoretical neuroscience.Item A geometric investigation of non-regular separation applied to the bi-Helmholtz equation & its connection to symmetry operators(University of Waterloo, 2024-05-23) Jayyusi, BaselThe theory of non-regular separation is examined in its geometric form and applied to the bi-Helmholtz equation in the flat coordinate systems in 2-dimensions. It is shown that the bi-Helmholtz equation does not admit regular separation in any dimensions on any Riemannian manifold. It is demonstrated that the bi-Helmholtz equation admits non-trivial non-regular separation in the Cartesian and polar coordinate systems in R^2 but does not admit non-trivial non-regular separation in the parabolic and elliptic-hyperbolic coordinate systems of R^2. The results are applied to the study of small vibrations of a thin solid circular plate. It is conjectured that the reason as to why non-trivial non-regular separation occurs in the Cartesian and polar coordinate systems is due to the existence of first order symmetries (Killing vectors) in those coordinate systems. Symmetries of the bi-Helmholtz equation are examined in detail giving supporting evidence of the conjecture.Item Space-time Hybridizable Discontinuous Galerkin Method for the Advection-Diffusion Problem(University of Waterloo, 2024-05-22) Wang, YuanIn this thesis, we analyze a space-time hybridizable discontinuous Galerkin (HDG) method for the time-dependent advection-dominated advection-diffusion problem. It is well-known that solutions to these problems may admit sharp boundary and interior layers and that many numerical methods are prone to non-physical oscillations when resolving these solutions. This challenge has prompted the design of many new numerical methods and stabilization mechanisms. Among others, HDG methods prove to be capable of resolving the sharp layers in a robust manner. The design principles of HDG methods consist of discontinuous Galerkin (DG) methods and their strong stability properties, as well as hybridization to reduce the computational cost of the numerical method. The analysis in this work focuses on a space-time formulation of the time-dependent advection-diffusion problem and an HDG discretization in both space and time. This provides a straightforward approach to discretize the problem on a time-dependent domain, with arbitrary higher-order spatial and temporal accuracy. We present an a priori error analysis that provides Peclet-robust error estimates that are also valid on moving meshes. A key intermediate step towards our error estimates is a Peclet-robust inf-sup stability condition. The second contribution of this thesis is an a posteriori error analysis of the space-time HDG method for the time-dependent advection-dominated advection-diffusion problem on fixed domains. This is motivated by the efficiency of combining a posteriori error estimators with adaptive mesh refinement (AMR) to locally refine or coarsen a mesh in the presence of sharp layers. When the solution admits sharp layers, AMR may still lead to optimal rates of convergence in terms of the number of degrees-of-freedom, unlike uniform mesh refinement. In this thesis, we present an a posteriori error estimator for the space-time HDG method with respect to a locally computable norm. We prove its reliability and local efficiency. The proof of reliability is based on a combination of a Peclet-robust coercivity type result and a saturation assumption. In addition, efficiency, which is local both in space and time, is shown using bubble function techniques. The error estimator in this thesis is fully local, hence it is an estimator for local space and time adaptivity in the AMR procedure. Finally, numerical simulations are presented to demonstrate and verify the theory. Both uniform and adaptive refinement strategies are performed on problems which admit boundary and interior layers.Item Discrete total variation in multiple spatial dimensions and its applications(University of Waterloo, 2024-05-13) Smirnov, AlexeyTotal variation plays an important role in the analysis of stability and convergence of numerical solutions for one-dimensional scalar conservation laws. However, extending this approach to two or more spatial dimensions presents a formidable challenge. Existing literature indicates that total variation diminishing solutions for two-dimensional hyperbolic equations are limited to at most first-order accuracy. The presented research contributes to overcoming the challenges associated with extending total variation to higher dimensions, particularly in the context of hyperbolic conservation laws. By addressing the limitations of conventional discrete total variation definitions, we seek answers to critical questions associated with the total variation diminishing property of solutions of scalar conservation laws in multiple spatial dimensions. We adopt a more accurate dual discrete definition of total variation, recently proposed in Condat, L. (2017) Discrete total variation: New definition and minimization, SIAM Journal on Imaging Sciences, 10(3), 1258-1290, for measuring the total variation of grid-based functions. Dual total variation can be computed as a solution to a constrained optimization problem. We propose a set of conditions on the coefficients of a general five-point scheme so that the numerical solution is total variation diminishing in the dual discrete sense and validate that through numerical experiments. Apart from the contributions to the analysis of numerical methods for two-dimensional scalar conservation laws, we develop an algorithm to efficiently compute the dual discrete total variation and develop an imaging method, based on this algorithm. We study its performance in computed tomography image reconstruction and compare it with the state-of-the-art total variation minimization-based imaging methods.Item On Convergence Analysis of Stochastic and Distributed Gradient-Based Algorithms(University of Waterloo, 2024-05-08) Zhang, MengyaoOptimization is a fundamental mathematical discipline focused on finding the best solution from a set of feasible choices. It is vital in various applications, including engineering, economics, data science, and beyond. Stochastic optimization and distributed optimization are crucial paradigms in the optimization field. Stochastic optimization deals with uncertainty and variability in problem parameters, providing a framework for decision-making under probabilistic conditions. On the other hand, distributed optimization tackles large-scale problems by harnessing the collective power of multiple agents or nodes, each with local information and local communication capabilities. This thesis aims to modify and analyze the existing stochastic methods and develop the algorithms and theory to solve the unconstrained distributed optimization problem. For stochastic adaptive gradient-based methods, including RMSprop, Adadelta, Adam, AdaGrad, Nadam, and AMSgrad, which are popular stochastic optimization methods commonly used in machine learning and deep learning, the Chapter 2 provides a concise and rigorous proof of the almost sure convergence guarantees towards a critical point in the context of smooth and non-convex objective functions. To the best of our knowledge, this work offers the first almost sure convergence rates for these stochastic adaptive gradient-based methods. For non-convex objective functions, we show that a weighted average of the squared gradient norms in each aforementioned method achieves a unified convergence rate of $o(1/{t^{\frac{1}{2}-\theta}})$ for all $\theta\in\left(0,\frac{1}{2}\right)$. Moreover, for strongly convex objective functions, the convergence rates for RMSprop and Adadelta can be further improved to $o(1/{t^{1-\theta}})$ for all $\theta\in\left(0,\frac{1}{2}\right)$. {These rates are arbitrarily close to their optimal convergence rates possible.} As a locking-free parallel stochastic gradient descent algorithm, Hogwild! algorithm is commonly used for training large-scale machine learning models. In Chapter 3, we will provide an almost sure convergence rates analysis for Hogwild! algorithm under different assumptions on the loss function. We first prove its almost sure convergence rate on strongly convex function, which matches the optimal convergence rate of the classic stochastic gradient descent (SGD) method to an arbitrarily small factor. For non-convex loss function, a weighted average of the squared gradient, as well as the last iterations of the algorithm converges to zero almost surely. We also provide a last-iterate almost sure convergence rate analysis for this method on general convex smooth functions. Another aspect of the research addresses the convergence rate analysis of the gradient-based distributed optimization algorithms, which have been shown to achieve computational efficiency and rapid convergence while requiring weaker assumptions. We first propose a novel gradient-based proportional-integral (PI) algorithm in Chapter 4, and prove that its convergence rate matches that of the centralized gradient descent method under the strong convexity assumption. We then relax this assumption and discuss the local linear convergence of its virtual state for strictly convex cost functions. In Chapter 5, we propose the powered proportional-integral (PI) algorithm and prove its convergence in finite time under the assumption of strict convexity. Then, we discuss the fixed-time convergence of its virtual state for strongly convex cost functions. Finally, we demonstrate the practicality of the distributed algorithms proposed in this thesis through simulation results.Item Safety-Critical Control for Dynamical Systems under Uncertainties(University of Waterloo, 2024-02-15) Wang, ChuanzhengControl barrier functions (CBFs) and higher-order control barrier functions (HOCBFs) have shown great success in addressing control problems with safety guarantees. These methods usually find the next safe control input by solving an online quadratic programming problem. However, model uncertainty is a big challenge in synthesizing controllers. This may lead to the generation of unsafe control actions, resulting in severe consequences. In this thesis, we discuss safety-critical control problems for systems with different levels of uncertainties. We first study systems modeled by stochastic differential equations (SDEs) driven by Brownian motion. We propose a notion of stochastic control barrier functions (SCBFs) and show that SCBFs can significantly reduce the control efforts, especially in the presence of noise, and can provide a reasonable worst-case safety probability. Based on this less conservative probabilistic estimation for the proposed notion of SCBFs, we further extend the results to handle higher relative degree safety constraints using higher-order SCBFs. We demonstrate that the proposed SCBFs achieve good trade-offs of performance and control efforts, both through theoretical analysis and numerical simulations. Next, we discuss deterministic systems with imperfect information. We focus on higher relative degree safety constraints and HOCBFs to develop a learning framework to deal with such uncertainty. The proposed method learns the derivatives of a HOCBF and we show that for each order, the derivative of the HOCBF can be separated into the nominal derivative of the HOCBF and some remainders. This implies that we can use a neural network to learn the remainders so that we can approximate the real residual dynamics of the HOCBF. Next, we study stochastic systems with unknown diffusion terms. We propose a data-driven method to handle the case where we cannot calculate the generator of the stochastic barrier functions. We provide guarantees that the data-driven method can approximate the It\^{o} derivative of the stochastic control barrier function (SCBF) under partially unknown dynamics using the universal approximation theorem. Finally, we study completely unknown stochastic systems. We extend our assumption into the case where we do not know either the drift or the diffusion term of SDEs. We employ Bayesian inference as a data-driven approach to approximate the system. To be more specific, we utilize Bayesian linear regression along with the central limit theorem to estimate the drift term. Additionally, we employ Bayesian inference to approximate the diffusion term. We also validate our theoretical results using numerical examples in each chapter.Item Spatial and Temporal Discounting in a Social-Climate Model(University of Waterloo, 2024-01-25) Cameron, MackenzieThis thesis analyzes how individuals' devaluation of distant impacts of climate change affects mitigation behaviours and projected climate conditions. To approach this question, spatial and temporal discounting is applied to a coupled social-climate model. This model represents a two-way feedback between human decision-making, social norms, and human behaviour with changes in the climate. This is achieved through coupling an evolutionary game theoretic model of opinion dynamics and a simple Earth System Model. The results showed that shifting from current-looking to future-looking behaviours (preferring lower discounting scenarios) and considering multiple locations and population groups, supports a higher proportion of the population choosing mitigation strategies. This shift produces a pathway to reducing temperature anomalies and carbon dioxide emissions. However, the approach to a better state of the climate is best achieved by targeting both discounting and social behaviours rather than just one or the other. These results highlight the benefits of including human behaviour in climate models and the need for a more multifaceted approach to mitigating the negative effects of climate change.Item Entropy-Stable Positivity-Preserving Schemes for Multiphase Flows(University of Waterloo, 2024-01-22) Simpson, Benjamin JacobHigh-intensity focused ultrasound is a promising non-invasive medical technology that has been successfully used to ablate tumors, as well as in the treatment of other conditions. Researchers believe high-intensity focused ultrasound could see clinical application in other areas such as disruption of the blood brain barrier and sonoporation. However, such advances in medical technology requires fundamental insight into the physics associated with high-intensity focused ultrasound, such as the phenomena known as acoustic cavitation and the collapse of the ensuing bubble cavity. The multiphase description of flow phenomena is an attractive option for modelling such problems as all fluids in the domain are modelled using a single set of governing equations, as opposed to separate systems of equations for each phase and therefore, separate meshes for each fluid. In this thesis, we are interested in studying the bubble collapse problem numerically, to elucidate the physics behind the collapse of acoustically driven bubbles. We seek to develop high-order numerical methods to solve this problem, due to their potential to increase computational efficiency. However, high-order methods typically have stability issues, especially when considering complex physics. For this reason, high-order entropy-stable summation-by-parts schemes are a popular method used to simulate compressible flow equations. These methods offer provable stability through satisfying a discrete entropy inequality, which is used to prove discrete L2 stability. Such stability proofs rely on the fundamental assumption that the densities and volume (or void) fractions of both phases remain positive. However, we seek numerical schemes that can simulate flows where the densities and volume fractions get arbitrarily close to zero and, as such, could become negative as the simulation progresses. To address this problem, we present a novel high-order entropy-stable positivity-preserving scheme to solve the 1-D isentropic Baer-Nunziato model. The key to our proposed scheme is a novel artificial dissipation operator, which has tuneable dissipation coefficients that allow the scheme to have provable nodewise positivity of the densities and volume fractions. This new scheme is constructed by mixing a high-order entropy-conservative scheme with a first-order entropy-stable positivity-preserving scheme to create a high-order entropy-stable positivity-preserving scheme. Numerical results which demonstrate the convergence, positivity, and shock capturing capabilities of the scheme are presented.Item Optimal trajectory calculation using neural networks(University of Waterloo, 2024-01-08) Majumder, SounakOptimal control methods for linear systems have reached a substantial level of maturity, both in terms of conceptual understanding and scalable computational implementation. For non-linear systems, an open-loop feedback control may be calculated using Pontryagin's Maximum Principle. Alternatively, the Hamilton-Jacobi-Bellman (HJB) equation may be used to calculate the optimal control in a state-feedback form. However, it is an established fact that this equation becomes progressively harder to solve as the number of state variables increases. In this thesis, we discuss a Neural Network (NN)-based method [1] to approximate the solution to the HJB equation arising from high-dimensional ODE systems. We leverage the equivalency between the HJB equation and Pontryagin's Principle to generate the training and test datasets and define a physics-based loss function. The NN is then trained using a supervised optimization approach. We also examine an existing toolkit [2] to approximate the optimal control based on a power series expansion of the system around an equilibrium point in an infinite time horizon setting. We examine the possibility of incorporating this toolkit in the NN training procedure at different stages. The proposed methods are applied to three problems: optimal control of a 6 degree-of-freedom rigid body and the stabilization of ODE systems arising from the discretization of a Burgers'-like non-linear PDE and the damped wave equation. References: [1] Tenavi Nakamura-Zimmerer, Qi Gong, and Wei Kang. Adaptive deep learning for high-dimensional Hamilton-Jacobi-Bellman equations. SIAM Journal of Scientific Computing, 43(2):A1221–A1247, 2021. [2] Arthur J. Krener. Nonlinear systems toolbox v.1.0, 1997. MATLAB based toolbox available by request from ajkrener@ucdavis.edu