Colin MacDonald

Associate Professor

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Graduate Student Supervision

Doctoral Student Supervision

Dissertations completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest dissertations.

An asymptotic and numerical analysis of localized solutions to some linear and nonlinear pattern formation problems with heterogeneities (2021)

Reaction-diffusion (RD) systems are indispensable models tounderstand pattern formation. To reproduce sophisticatedpattern-forming behaviors that resemble observed biologicalpatterns, we can increase the complexity of a RD model byintroducing spatial heterogeneity. In the first part of thisthesis, we consider two-component RD systems in thesingular limit of a large diffusivity ratio for which localized spotpatterns occur. We will use a hybrid asymptotic-numerical approachto analyze the existence, linear stability, and dynamics oflocalized spot patterns in the presence of certain spatialheterogeneities. More specifically, for the Schnakenberg model, we willinvestigate the effect on spot patterns that occur for distincttypes of localized spatial heterogeneity for which spots are eitherrepelled from or attracted to, respectively. For the Klausmeier model, we study the effect of water advection on the dynamics and steady-state behavior of localized patches or ``spots'' of vegetation. Moreover, we study the effect of a slowly varying rainfall rate on spot dynamics, and the induced bifurcation with delays. In many previous studies of spot instabilities, it has been observedfrom numerical PDE simulations that a linear shape-deforminginstability of a localized spot is the trigger of a nonlinearspot-replication event in the absence of heterogeneity. To provide a theoretical basis for theseobservations, we derive an amplitude equation from a weaklynonlinear analysis which confirms that a peanut-shaped instability ofa spot is subcritical. In the second part of this thesis, we study mean first passage time(MFPT) for a Brownian particle to be captured by small circulartraps in a 2-D confining domain. Our focus is to understand how thedeviations from a radially symmetric domain, which represents a domainheterogeneity in a general sense, alters the optimal spatialconfiguration of a collection of small circular traps that minimizesthe average MFPT. In this direction, we develop a numerical methodand perform asymptotic analysis to approximate the MFPT for general2-D domains. In particular, by deriving a new explicit analyticalformula for the Neumann Green's function, we demonstrate the fullpower of these tools for an elliptical domains of arbitrary aspectratio.

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Topics in the asymptotic analysis of narrow escape and quorum-sensing behavior for PDE models with biological applications (2021)

In this thesis, we develop novel numerical and analytical techniques for calculating the MFPT for a Brownian particle to be captured by either small stationary or mobile traps inside a bounded 2-D domain. Of particular interest is identifying the optimal arrangements of small traps that minimize the average MFPT. Although the MFPT and the associated optimal trap arrangement problem havebeen well-studied for disk-shaped domains, there are very few analytical or numerical results available for general star-shaped domains or thin domains with large aspect ratio. We develop an embedded numerical method for both stationary and periodic mobile trap problems, based on the Closest Point Method (CPM), to perform MFPT simulations on various confining 2-D domains. Optimal trap arrangements are identified numerically through either a refined discrete samplingapproach or a particle-swarm optimization procedure. To confirm some of the numerical findings, novel perturbation approaches are developed to approximate the average MFPT and identify optimal trap configurations for a class of near-disk confining domains or an arbitrary thin domain of large aspect ratio.We also analyze cell-bulk coupled ODE-PDE models for describing the communication between localized spatially segregated dynamically active signaling compartments or cells, coupled through a passive extracellular bulk diffusion field. In a 2-D bounded domain, where the cells are small disks of a common radius ε
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Pattern formation on curved surfaces (2019)

Patterns emerge in various growing biological organisms, like conifer embryos, often from an homogeneous preceding state. The embryo tip, initially hemispherical shaped gradually flattens as cotyledons arranged in a roughly regular pattern emerge. A common way to model these patterns is through reaction-diffusion systems of partial differential equations. This thesis relays results obtained studying such systems and provides various new results about pattern formation on curved surfaces via diffusion driven bifurcations.We first describe situations where, for certain critical values of domain or differential equation parameters, the unpatterned solution is unstable to two different linear normal modes. We use centre manifold and normal form reductions to analyze the existence and stability of pure and mixed modes of nonlinear patterned solutions of the reaction-diffusion system, for parameters near two cases of critical values. In one case, the system reduces to a well known example of mode interaction. In the other case, the mode interaction is new, due to very small quadratic terms in the normal form.We then perform a reduction of a nonautonomous Brusselator reaction-diffusion system of partial differential equations on a spherical cap with time dependent curvature using an asymptotic series expansion on the centre manifold reduction. Parameter values are chosen such that the change in curvature would cross critical values which would change the stability of the patternless solution in the constant domain case. The non-isotropic nature of the domain evolution insert a small patterned component to the previously patternless state, which we call the `quasi-patternless solution'. The evolving domain functions and quasi-patternless solutions are derived as well as a method to obtain this nonautonomous normal form. The obtained reduction solutions are then compared to numerical solutions.Finally we provide an adaptation of the closet point method to evolving domains. We perform several convergence analysis experiments of the heat equation on test surfaces and obtain quadratic convergence. Then, a few reaction diffusion simulations are shown on evolving surfaces, some featuring non-isotropic evolution. The previously shown application of the Brusselator system on an evolving spherical cap are compared to a centre manifold reduction and also show quadratic convergence.

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Master's Student Supervision

Theses completed in 2010 or later are listed below. Please note that there is a 6-12 month delay to add the latest theses.

Microtubules on curved surfaces (2022)

The self-organization of ordered cortical microtubule arrays plays an important role in the development of plant cells. This is observed to emerge from a combination various factors such as microtubule-microtubule interactions, nucleation, and localization of microtubule-associated proteins. Distilling this process into the interaction of one-dimensional bodies on the two-dimensional cortex, quantitative models have been proposed to emulate array formation. These have assisted in understanding the importance of each aspect in addition to identifying potential avenues of experimentation. Until recently, the direct mechanical influence of cell geometry on the constrained microtubule trajectories have been largely ignored in computational models. Modelling microtubules as thin elastic rods constrained on a surface, it has been found that microtubules shapes may differ significantly from the previously assumed geodesics. Restricting to cylinders, we formulate a model incorporating the mechanics of curvature-induced deflection and examine its implications. First, we introduce a minimal anchoring process necessary to describe the geometry of individual microtubules as they grow and become fixed to the cortex. Curvature-induced deflection is governed by the extent to which microtubules can explore the surface curvature, as prescribed by the segment lengths between anchoring. This, in turn, can be modulated by the anchoring kinetics proposed. We implement this model as an event-driven simulation in Python, accommodating for the curvilinear shapes prescribed by the anchoring process. In doing so, ambiguities between differing models are identified. Although these are not resolved here, these provide a possible explanation to the reported conflicting results among differing models. Based on some preliminary simulations using our model, we find that the curvature mechanics provides strong influence for longitudinal alignment with realistic anchoring rates. In particular, catastrophe-inducing edges are unable to overcome this influence to reproduce the observed transverse arrays during the cell elongation phase. This simulation provides the opportunity for further exploration into mechanical influences on array formation and their regulation by the anchoring process.

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A fast solver for the pseudo-two-dimensional model of lithium-ion batteries (2021)

The pseudo-two-dimensional (P2D) model is a complex mathematical model that can capture the electrochemical process in Li-ion batteries. However, simulation of the model also brings a heavy computational burden. Many simplifications to the model have been introduced in the literature to reduce the complexity. We present methods for fast computation of two models: an asymptotically scaled P2D model and the full P2D model. The techniques developed in this work can be used when simplifications are not accurate enough. By rearranging the calculations, we reduce the complexity of the linear algebra problem. We also employ automatic differentiation, using the open source package JAX, for robustness, which also allows easy implementation of changes to coefficient expressions. The method alleviates the computational bottleneck in P2D models without compromising accuracy.

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Closest point methods with polyharmonic spline radial basis functions and local refinement (2021)

Closest point methods are a class of embedding methods that have been used to solve partial differential equations on surfaces with the closest point representation of the surface. Recently, several studies replaced the standard Cartesian grid methods in the original Closest point methods with radial basis function generated finite differences. This reduces the computational cost and allows scattered and unstructured grids as well as locally refined uniform grids. This thesis uses the polyharmonic spline function as the radial basis function in the combined method which is different from the usual choice of Gaussian or multiquadric to avoid the shape parameter.We first perform convergence tests of the combined method. In all cases, the radial basis function closest point method uses fewer points in the embedding space while achieving a similar accuracy and convergence rate as the original closest point method.We then focus on solving partial differential equation problems with irregular grids that match features of the surface or the solution. These include using more points near high curvature regions or using more points near fine scale solution features. This can reduce the computational cost compared to using a uniform fine grid over the entire surface. Lastly, we provide an adaptive version of the combined method that is able to solve partial differential equation problems on surfaces when either or both of the surface features and problem features are changing in time.

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Point cloud classification by nearest samples of random rays (2021)

Point clouds serve as a common type of data representation for general geometric objects and abstract data structures in arbitrary dimensions. With the accessibility of reliable and efficient 3D scanning technology as well as the pervasiveness of data science, point clouds widely appear in various research areas and applications, especially in geometric processing, computer vision, and robotics, making robust point cloud processing algorithms an urgent need. However, due to the inherently irregular structure of point clouds, the remarkable success of many traditional algorithms cannot be duplicated directly on point cloud processing. Besides, semantic understanding of underlying shapes, which is essential in many applications in computer vision, requires interpretation of refined information, yet how to properly extract useful geometric information from point clouds is still an open problem in this field.To this end, in this thesis, we focus on point cloud classification, one of the fundamental point cloud processing tasks, and present a novel framework, RaySense$+$RayNN, to conquer this challenge. First, we use RaySense, an innovative subsampling strategy, to compute the signature of a target point cloud by finding the nearest neighbors of points on a set of randomly generated rays. Using rays, we incorporate computational structure into the point cloud thereby different strategies and algorithms can be implemented. We then propose a convolutional neural network, RayNN, that operates on the RaySense signature to perform point cloud classification.By analysis and experiments, we reveal the RaySense signature is not merely a subsample of the original point clouds, it also preserves permutation invariance of the input points and extracts certain statistical as well as geometric information, independent of the choice of ray set. We show that even with the simple architecture, our framework can achieve comparable performance against the well-tuned state-of-the-art algorithms on the benchmark dataset. Furthermore, we explore the robustness of our framework by exposing it to a variety of data corruptions, indicating its capacity for practical applications. Lastly, we conclude with a discussion on potential applications and directions for future research.

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