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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Aging affects circadian clock and metabolism and modulates timing of medication
(Elsevier, 2021-04) Sadria, Mehrshad; Layton, Anita T.
Aging is associated with impairments in the circadian rhythms, and with energy deregulation that affects multiple metabolic pathways. The goal of this study is to unravel the complex interactions among aging, metabolism, and the circadian clock. We seek to identify key factors that inform the liver circadian clock of cellular energy status and to reveal the mechanisms by which variations in food intake may disrupt the clock. To address these questions, we develop a comprehensive mathematical model that represents the circadian pathway in the mouse liver, together with the insulin/IGF-1 pathway, mTORC1, AMPK, NAD+, and the NAD+ -consuming factor SIRT1. The model is age-specific and can simulate the liver of a young mouse or an aged mouse. Simulation results suggest that the reduced NAD+ and SIRT1 bioavailability may explain the shortened circadian period in aged rodents. Importantly, the model identifies the dosing schedules for maximizing the efficacy of anti-aging medications.
Algebraic Multigrid for Markov Chains and Tensor Decomposition
(University of Waterloo, 2013-01-23T19:50:12Z) Miller, Killian
The majority of this thesis is concerned with the development of efficient and robust numerical methods based on adaptive algebraic multigrid to compute the stationary distribution of Markov chains. It is shown that classical algebraic multigrid techniques can be applied in an exact interpolation scheme framework to compute the stationary distribution of irreducible, homogeneous Markov chains. A quantitative analysis shows that algebraically smooth multiplicative error is locally constant along strong connections in a scaled system operator, which suggests that classical algebraic multigrid coarsening and interpolation can be applied to the class of nonsymmetric irreducible singular M-matrices with zero column sums. Acceleration schemes based on fine-level iterant recombination, and over-correction of the coarse-grid correction are developed to improve the rate of convergence and scalability of simple adaptive aggregation multigrid methods for Markov chains. Numerical tests over a wide range of challenging nonsymmetric test problems demonstrate the effectiveness of the proposed multilevel method and the acceleration schemes. This thesis also investigates the application of adaptive algebraic multigrid techniques for computing the canonical decomposition of higher-order tensors. The canonical decomposition is formulated as a least squares optimization problem, for which local minimizers are computed by solving the first-order optimality equations. The proposed multilevel method consists of two phases: an adaptive setup phase that uses a multiplicative correction scheme in conjunction with bootstrap algebraic multigrid interpolation to build the necessary operators on each level, and a solve phase that uses additive correction cycles based on the full approximation scheme to efficiently obtain an accurate solution. The alternating least squares method, which is a standard one-level iterative method for computing the canonical decomposition, is used as the relaxation scheme. Numerical tests show that for certain test problems arising from the discretization of high-dimensional partial differential equations on regular lattices the proposed multilevel method significantly outperforms the standard alternating least squares method when a high level of accuracy is required.
Algorithms and Models for Tensors and Networks with Applications in Data Science
(University of Waterloo, 2016-01-19) Winlaw, Manda
Big data plays an increasingly central role in many areas of research including optimization and network modeling. We consider problems applicable to large datasets within these two branches of research. We begin by presenting a nonlinearly preconditioned nonlinear conjugate gradient (PNCG) algorithm to increase the convergence speed of iterative unconstrained optimization methods. We provide a concise overview of several PNCG variants and their properties and obtain a new convergence result for one of the PNCG variants under suitable conditions. We then use the PNCG algorithm to solve two different problems: computing the rank-R canonical tensor decomposition and finding the solution to a latent factor model where latent factor models are often used as important building blocks in many practical recommendation systems. For both problems, the alternating least squares (ALS) algorithm is typically used to find a solution and as such we consider it as a nonlinear preconditioner. Note that the ALS algorithm can be viewed as a nonlinear preconditioner for the NCG algorithm or alternatively, NCG can be viewed as an acceleration process for ALS. We demonstrate numerically that the convergence acceleration mechanism in PNCG often leads to important pay-offs for difficult tensor decomposition problems, with convergence that is significantly faster and more robust than for the stand-alone NCG or ALS algorithms. As well, we show numerically that the PNCG algorithm requires many fewer iterations and less time to reach desired ranking accuracies than stand-alone ALS in solving latent factor models. We next turn to problems within the field of network or graph modeling. A network is a collection of points joined together by lines and networks are used in a broad variety of fields to represent connections between objects. Many large real-world networks share similar properties which has garnered considerable interest in developing models that can replicate these properties. We begin our discussion of graph models by closely examining the Chung-Lu model. The Chung-Lu model is a very simple model where by design the expected degree sequence of a graph generated by the model is equal to a user-supplied degree sequence. We explore what happens both theoretically and numerically when simple changes are made to the model and when the model assumptions are violated. As well, we consider an algorithm used to generate instances of the Chung-Lu model that is designed to be faster than the traditional algorithm but find that it only generates instances of an approximate Chung-Lu model. We explore the properties of this approximate model under a variety of conditions and examine how different the expected degree sequence is from the user-supplied degree sequence. We also explore several ways of improving this approximate model to reduce the approximation error in the expected degree sequence and note that when the assumptions of the original model are violated this error remains very large. We next design a new graph generator to match the community structure found in real-world networks as measured using the clustering coefficient and assortativity coefficient. Our graph generator uses information generated from a clustering algorithm run on the original network to build a synthetic network. Using several real-world networks, we test our algorithm numerically by creating a synthetic network and then comparing the properties to the real network properties as well as to the properties of another popular graph generator, BTER, developed by Seshadhri, Kolda and Pinar. Our graph generator does well at preserving the clustering coefficient and typically outperforms BTER in matching the assortativity coefficient, particularly when the assortativity coefficient is negative.
Almost-Sure Stability of a Noisy Autoparametric Vibration Absorber
(University of Waterloo, 2022-12-13) Oluyemi, Momoiyioluwa
This thesis investigates the almost-sure stability of the single mode solution of a two degree-of-freedom, noisy, nonlinear autoparametric system. While only the first mode is forced in such a system, the nonlinear coupling often transfers energy to the second mode. Equations of motion of autoparametric systems model the dynamics of a number of structural and mechanical systems, such as, a randomly excited and initially deformed shallow arch, a suspended elastic cable driven by planar excitation, or a vessel subject to longitudinal wave action. To keep things as simple as possible, we consider a very simple system, namely, a type of autoparametric vibration absorber with randomly excited base - a pendulum attached to a mass-spring oscillator. Under the assumption of small random perturbations and small damping, the maximal Lyapunov exponent which determines the almost-sure stability of the single mode solution is calculated. Putting the maximal Lyapunov exponent to zero provides the second-order approximation of the almost-sure stability boundary in terms of the excitation intensity and the dissipation coefficients. A plot of this stability boundary reveals several trends of practical importance to engineering applications.
Analysis of a pressure-robust hybridized discontinuous Galerkin method for the stationary Navier-Stokes equations
(Springer, 2019-08-30) Kirk, Keegan L.A.; Rhebergen, Sander
We present well-posedness and an a priori error analysis of the hybridized discontinuous Galerkin method for the stationary form of the Navier-Stokes problem proposed in (J Sci Comput, 76(3):1484{ 1501, 2018). This scheme was shown to result in an approximate velocity eld that is pointwise divergence-free and divergence-conforming. As a consequence we show that the velocity error estimate is independent of the pressure. Furthermore, we show that estimates for both the velocity and pressure are optimal. Numerical examples demonstrate pressure-robustness and optimality of the scheme.
Analysis of Asymptotic Solutions for Cusp Problems in Capillarity
(University of Waterloo, 2007-09-27T18:13:11Z) Aoki, Yasunori
The capillary surface $u(x,y)$ near a cusp region satisfies the boundary value problem: \begin{eqnarray} \nabla \cdot \frac{\nabla u}{\sqrt{1+\left|\nabla u \right|^2}}&=&\kappa u \qquad \textrm{in }\left\{(x,y): 0
Analysis of Light-Matter Systems
(University of Waterloo, 2022-01-18) El Mandouh, Mohamed
In this thesis we introduce the simplest model of a two–level system coupled to a single mode of an optical cavity, called the Jaynes-Cummings model. This model is then extended to an ensemble of identical two-level systems and is studied in more detail, also known as the Tavis-Cummings model. This model is intractable, but we show that by a clever but simple choice of basis one can reduce the dimensionality of the Tavis-Cumming system. We then demonstrate the effectiveness of this reduction by calculating interesting statistics of the system, and simulating large ensembles of two–level systems which were not practical before. Finally, we examine some dynamics of the Tavis-Cummings model in the presence of photon losses, and introduce a method for population transfer by modulating TLS-cavity interaction strength.
Analysis of Plasmons Sustained on the Surface of Graphene
(University of Waterloo, 2014-08-11) Lyon, Keenan
This thesis is broken into two parts, both dealing with the role of two-dimensional graphene in electronic and optical applications. The first section develops a phenomenological relationship for the polarizability of the graphene sheet using a hybrid semi-classical and QFT-derived (Quantum Field Theory) model for different energy regimes. Fits are made and our results are compared to data from two distinct experimental setups. The effects of contamination and rippling of the sheet are considered. The second section shows a phenomenological model for the rough surfaces of graphene and its underlying substrate for a sheet grown on a conducting material. Three different perturbative mathematical models are then explored to justify the shift in the plasmon frequency and the energy loss dispersion due to roughness, using input from experimental roughness data. The models are compared and corrected to include physical effects like crumpling.
An Analysis of Stockwell Transforms, with Applications to Image Processing
(University of Waterloo, 2014-04-30) Ladan, John
Time-frequency analysis is a powerful tool for signal analysis and processing. The Fourier transform and wavelet transforms are used extensively as is the Short-Time Fourier Transform (or Gabor transform). In 1996 the Stockwell transform was introduced to maintain the phase of the Fourier transform, while also providing the progressive resolution of the wavelet transform. The discrete orthonormal Stockwell transform is a more efficient, less redundant transform with the same properties. There has been little work on mathematical properties of the Stockwell transform, particularly how it behaves under operations such as translation and modulation. Previous results do discuss a resolution of the identity, as well as some of the function spaces that may be associated with it [2]. We extend the resolution of the identity results, and behaviour under translation, modulation, convolution and differentiation. boundedness and continuity properties are also developed, but the function spaces associated with the transform are unrelated to the focus of this thesis. There has been some work on image processing using the Stockwell transform and discrete orthonormal Stockwell transform. The tests were quite preliminary. In this thesis, we explore some of the mathematics of the Stockwell transform, examining properties, and applying it to various continuous examples. The discrete orthonormal Stockwell transform is compared directly with Newland’s harmonic wavelet transform, and we extend the definition to include varitions, as well as develop the discrete cosine based Stockwell transform. All of these discrete transforms are tested against current methods for image compression.
Analyzing Bacterial Conjugation with Graphical Models: A Model Comparison Approach
(University of Waterloo, 2024-08-12) Kendal-Freedman, Nat
Conjugation 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.
Angles, Majorization, Wielandt Inequality and Applications
(University of Waterloo, 2013-05-29T19:14:03Z) Lin, Minghua
In this thesis we revisit two classical definitions of angle in an inner product space: real-part angle and Hermitian angle. Special attention is paid to Krein’s inequality and its analogue. Some applications are given, leading to a simple proof of a basic lemma for a trace inequality of unitary matrices and also its extension. A brief survey on recent results of angles between subspaces is presented. This naturally brings us to the world of majorization. After introducing the notion of majorization, we present some classical as well as recent results on eigenvalue majorization. Several new norm inequalities are derived by making use of a powerful decomposition lemma for positive semidefinite matrices. We also consider coneigenvalue majorization. Some discussion on the possible generalization of the majorization bounds for Ritz values is presented. We then turn to a basic notion in convex analysis, the Legendre-Fenchel conjugate. The convexity of a function is important in finding the explicit expression of the transform for certain functions. A sufficient convexity condition is given for the product of positive definite quadratic forms. When the number of quadratic forms is two, the condition is also necessary. The condition is in terms of the condition number of the underlying matrices. The key lemma in our derivation is found to have some connection with the generalized Wielandt inequality. A new inequality between angles in inner product spaces is formulated and proved. This leads directly to a concise statement and proof of the generalized Wielandt inequality, including a simple description of all cases of equality. As a consequence, several recent results in matrix analysis and inner product spaces are improved.
An Anisotropic Subgrid-Scale Parameterization for Large-Eddy Simulations of Stratified Turbulence
(AMS, 2020-10-01) Khani, Sina; Waite, Michael L
Subgrid-scale (SGS) parameterizations in atmosphere and ocean models are often defined independently in the horizontal and vertical directions because the grid spacing is not the same in these directions (anisotropic grids). In this paper, we introduce a new anisotropic SGS model in large-eddy simulations (LES) of stratified turbulence based on hor izontal filtering of the equations of motion. Unlike the common horizontal SGS parameterizations in atmosphere and ocean models, the vertical derivatives of the horizontal SGS fluxes are included in our anisotropic SGS scheme, and therefore the horizontal and vertical SGS dissipation mechanisms are not disconnected in the newly developed model. Our model is tested with two vertical grid spacings and various horizontal resolutions, where the horizontal grid spacing is comparatively larger than that in the vertical. Our anisotropic LES model can successfully reproduce the results of direct numerical simulations, while the computational cost is significantly reduced in the LES. We suggest the new anisotropic SGS model as an alternative to current SGS parameterizations in atmosphere and ocean models, in which the schemes for horizontal and vertical scales are often decoupled. The new SGS scheme may improve the dissipative performance of atmosphere and ocean models without adding any backscatter or other energizing terms at small horizontal scales.
Annular Capillary Surfaces: Properties and Approximation Techniques
(University of Waterloo, 2007-09-17T14:43:53Z) Gordon, James
The capillary surface formed within a symmetric annular tube is analyzed. Assuming identical contact angles along each boundary, we consider surfaces u(x,y) that satisfy the capillary problem on an annular region. Several qualitative properties of u are determined and in particular, the behaviour of u is examined in the limiting cases of the annular domain approaching a disk as well as a thin ring. The iterative method of Siegel is also applied to the boundary value problem and convergence is demonstrated under conditions which include a contact angle of zero. Moreover, some geometries still yield interleaving iterates, allowing for upper and lower bounds to be placed on the boundary values of u. However, the interleaving properties no longer hold universally and for other geometries, another more complex behaviour is described. Finally, a numerical method is designed to approximate the iterative scheme.
An Antibiotic Protocol To Minimize Emergence Of Drug-Resistant Tuberculosis
(Elsevier, 2014-04-15) de Espindola, Aquino L.; Girardi, Daniel; Penna, T. J. P.; Bauch, Chris T.; Troca Cabella, Brenno C.; Martinez, Alexandre Souto
A within-host model of the spread of tuberculosis is proposed here where the emergence of drug resistance and bacterial dormancy are simultaneously combined. We consider both sensitive and resistant strains of tuberculosis pathogens as well as a dormant state of these bacteria. The dynamics of the within-host system is modeled by a set of coupled differential equations which are numerically solved to find a relation between the within-host bacterial populations and the host health states. The values of the parameters were taken from the current literature when available; a sensitivity analysis was performed for the others. Antibiotic treatment for standard, intermittent and oscillating intermittent protocols is analyzed for different conditions. Our results suggest that the oscillating protocol is the most effective one, that would imply a lower treatment cost.
Application of Mixture Theory to solid tumors and normal pressure hydrocephalus
(University of Waterloo, 2014-01-15) Burazin, Andrijana
In this thesis, the theory of poroelasticity, namely the Mixture Theory version -- a homogenized, macroscopic scale approach used to describe fluid flow through a porous medium -- is employed in three separate cases pertaining to a biological phenomenon. The first investigation explores the behavior of interstitial fluid pressure (IFP) in solid tumors. Thus, in Chapter 2, a Mixture Theory based approach is developed to describe the evolution of the IFP from that in a healthy interstitium to the elevated levels in cancerous tumors. Attention is focused on angiogenesis, a tightly regulated process in healthy tissue that provides all necessary nutrients through the creation of new blood vessels. Once this process becomes unruly within a tumor, angiogenesis gives rise to an abnormal vasculature by forming convoluted and leaky blood vessels. Thus, the primary focus of the model is on the capillary filtration coefficient and vascular density as they increase in time, which in turn elevates the tumor IFP. Later, the Mixture Theory model is extended to simulate the effects of vascular normalization, where the cancer therapy not only prunes blood vessels, but reverts the chaotic vasculature to a somewhat normal state, thereby temporarily lowering the tumor IFP. In Chapter 3, the validity of an assumption that was made in order to facilitate the mathematical calculations is investigated. In addition to all of the Mixture Theory assumptions, it is assumed that the pore pressure p is proportional to the tissue dilatation e. This assumption is examined to determine how appropriate and accurate it is, by using a heat type equation without the presence of sources and sinks under the assumption of a spherical geometry. The results obtained under the proportionality of p and e, are compared with the results obtained without this assumption. A substantial difference is found, which suggests that great care must be exercised in assuming the proportionality of p and e. The last application is reported in Chapter 4 and it investigates the pathogenesis of normal pressure hydrocephalus. In a normal brain, cerebrospinal fluid (CSF) is created by the choroid plexus, circulates around the brain and the spinal cord without any impediment, and then is absorbed at various sites. However, normal pressure hydrocephalus occurs when there is an imbalance between the production and absorption of CSF in the brain that causes the impaired clearance of CSF and the enlargement of ventricles; however, the ventricular pressure in this case is frequently measured to be normal. Thus, a mathematical model using Mixture Theory is formulated to analyze a possible explanation of this brain condition. Levine (1999) proposed the hypothesis that CSF seeps from the ventricular space into the brain parenchyma and is efficiently absorbed in the bloodstream. To test this hypothesis, Levine used the consolidation theory version of poroelasticity theory, with the addition of Starling's law to account for the absorption of CSF in the brain parenchyma at steady state. However, the Mixture Theory model does not agree with the results obtained by Levine (1999) which leads one to conclude that the pathogenesis of normal pressure hydrocephalus remains unknown. To conclude the thesis, all three applications of Mixture Theory are discussed and the importance and contribution of this work is highlighted. In addition, possible future directions are indicated based on the findings of this thesis.
Application of repetitive control to the lateral motion in a roll-to-roll web system
(University of Waterloo, 2012-04-10T16:45:28Z) Jin, Zhao
In a roll-to-roll web system lateral motion of a web caused by disturbances, which are often periodic, results in poor product quality. To reduce the effect of such disturbances, two control strategies are applied. First, the internal model principle is used to reject a sinusoidal disturbance. Second, repetitive control theory is used to reject a general periodic disturbance. We provide the synthesis procedure for both strategies, and demonstrate its use in several simulation studies on a five-roller web system. The simulation results show that the effect of disturbances, either sinusoidal or triangular, on lateral motion are significantly reduced with the internal model controller or the modified repetitive controller.
Application of the LANS-alpha Model to Gravity Currents
(University of Waterloo, 2019-10-21) Zhao, Yukun
This thesis aims to investigate how HERCULES performs when running a lock-exchange gravity current case. The LANS-alpha model with stratification is also tested as a subgrid model in HERCULES using the same gravity current case. Gravity currents have been studied using both direct numerical simulation (DNS) and large eddy simulation (LES). On the other hand, the LANS-alpha model has only been applied to several test cases which mainly focus on isotropic turbulence and wall-bounded unstratified flows. We begin by reviewing the characteristics of the turbulent structures in the gravity currents and the motivation to use the LANS-alpha model. This is followed by the implementation of the model in HERCULES with both grid-dependent alpha and flow-dependent alpha. For the numerical study, a gravity current is generated using a lock release in a horizontal channel. With a fine grid, the front location and the three-dimensionality of the gravity current can be resolved accurately using HERCULES. When the grid resolution is coarse, the LANS-alpha model can improve the results considerably using grid-dependent alpha with both subgrid terms. The flow-dependent alpha requires modification in its definition as the grid-dependent alpha outperforms it in resolving the front location and the small-scale, three-dimensional structures.
Applications of Deep Learning to Differential Equation Models in Oncology
(University of Waterloo, 2023-07-25) Meaney, Cameron
The integration of quantitative tools in biology and medicine has led to many groundbreaking advances in recent history, with many more promising discoveries on the horizon. Conventional mathematical models, particularly differential equation-based models, have had great success in various biological applications, including modelling bacterial growth, disease propagation, and tumour spread. However, these approaches can be somewhat limited due to their reliance on known parameter values, initial conditions, and boundary conditions, which can dull their applicability. Furthermore, their forms are directly tied to mechanistic phenomena, making these models highly explainable, but also requiring a comprehensive understanding of the underlying dynamics before modelling the system. On the other hand, machine learning models typically require less prior knowledge of the system but require a significant amount of data for training. Although machine learning models can be more flexible, they tend to be black boxes, making them difficult to interpret. Hybrid models, which combine conventional and machine learning approaches, have the potential to achieve the best of both worlds. These models can provide explainable outcomes while relying on minimal assumptions or data. An example of this is physics-informed neural networks, a novel deep learning approach that incorporates information from partial differential equations into the optimization of a neural network. This hybrid approach offers significant potential in various contexts where differential equation models are known, but data is scarce or challenging to work with. Precision oncology is one such field. This thesis employs hybrid conventional/machine learning models to address problems in cancer medicine, specifically aiming to advance personalized medicine approaches. It contains three projects. In the first, a hybrid approach is used to make patient-specific characterizations of brain tumours using medical imaging data. In the second project, a hybrid approach is employed to create subject-specific projections of drug-carrying cancer nanoparticle accumulation and intratumoral interstitial fluid pressure. In the final project, a hybrid approach is utilized to optimize radiation therapy scheduling for tumours with heterogeneous cell populations and cancer stem cells. Overall, this thesis showcases several examples of how quantitative tools, particularly those involving both conventional and machine learning approaches, can be employed to tackle challenges in oncology. It further supports the notion that the continued integration of quantitative tools in medicine is a key strategy in addressing problems and open questions in healthcare.
Assessing The Pandemic Potential Of Mers-Cov
(Elsevier, 2013-08-24) Bauch, Chris T.; Oraby, Tamer
No Abstract Available
Backscatter in stratified turbulence
(Elsevier, 2016-11) Khani, Sina; Waite, Michael L
In this paper, kinetic and potential energy transfers around a spectral test fil ter scale in direct numerical simulations of decaying stratified turbulence are studied in both physical and spectral domains. It is shown that while the domain-averaged effective subgrid scale energy transfer in physical space is a net downscale cascade, it is actually a combination of large values of downscale and upscale transfer, i.e. forward- and backscatter, in which the forward scatter is slightly dominant. Our results suggest that spectral backscatter in stratified turbulence depends on the buoyancy Reynolds number Reb and the filtering scale ∆test. When the test filter scale ∆test is around the dissipation scale Ld, transfer spectra show spectral backscatter from sub-filter to intermediate scales, as reported elsewhere. However, we find that this spectral backscatter is due to viscous effects at vertical scales around the test filter. It is also shown that there is a non-local energy transfer from scales larger than the buoyancy scale Lb to small scales.The effective turbulent Prandtl number spectra demonstrate that the assumption P rt ≈ 1 is reasonable for the local energy transfer.

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