Technical Program

Stoichiometric principles allow the construction of robust mechanistic, predictive, and empirically testable models via rigorous chemical and physical laws. Experimental and fundamental evidence motivates the application of this microscopic approach to understand macroscopic phenomena. I will introduce some stoichiometric models and their novel dynamics that resolve some biological paradoxes and lead to new insights. The application in methane biogenesis illustrates how this approach contributes to achieving the ambitious goal of carbon neutrality, essential for mitigating global climate change.

This talk will consist of two main parts. The first part will serve as an introduction to Summation by Parts discretizations and their advantageous mathematical properties, including provable stability, discrete conservation, geometric flexibility, and equivalent interpretations under both finite difference and discontinuous spectral element frameworks. The second part will explore the applications of these schemes to stationary curvilinear grids. We compare various approaches of calculating the resulting geometric terms, including grid metric terms, metric Jacobians, and boundary normals. Using both structured and unstructured discretizations of the 3D Euler equations, we then discuss the impact these various approaches have on the accuracy, discrete conservation, and dual consistency of the resulting schemes. Theoretical arguments are verified through numerical experiments involving superconverging wall-boundary functionals.

Summation-by-parts (SBP) operators are widely used in computational fluid dynamics for approximating partial differential equations. In conjunction with proper boundary procedures, they allow for simple stability proofs mimicking the analogous proof for the continuous problem. In this talk, we consider the injection method for imposing Dirichlet boundary conditions strongly. That is, we require the solution at the boundary nodes to satisfy the boundary data exactly. Specifically, we extend the results of [1] where this boundary procedure, together with the $(2,1)$ finite-difference SBP operator, was proven to yield a stable scheme for the compressible Navier-Stokes equations. Herein, we show that the strong imposition of Dirichlet boundary condition leads to stable schemes when used in combination with different kinds of SBP operators, including high-order operators, operators defined on unstructured grids and tensor-product operators. We consider the linear advection equation as a model problem to introduce the ideas, and show that the methodology also leads to entropy estimates for the compressible Navier-Stokes equations. [1]: A. Gjesteland and M. Svärd. Entropy stability for the compressible Navier-Stokes equations with strong imposition of the no-slip boundary condition. Journal of Computational Physics, 470, 111572, 2022.

While nonlinear stability of discretizations of the Euler and Navier-Stokes equations has been realized by satisfying the discrete entropy inequality using the summation-by-parts (SBP) framework, improving efficiency of these discretizations remains an active area of research. The use of two-point fluxes in entropy-stable discretizations presents computational efficiency challenges, particularly for high-order operators on simplices, which typically produce dense matrix operators. Simplicial elements lend themselves well to automatic mesh generation around complex geometries, suggesting significant potential for the development of robust and versatile fluid dynamics solvers. Conversely, tensor-product operators on quadrilateral and hexahedral elements produce matrix operators that are sparse and substantially more accurate than multidimensional SBP operators---if only there is a way to combine the two! In this presentation, we propose a promising method to do just that. We present simplicial elements with tensor-product structure, referred to as tensor-split-simplex (TSS) SBP operators, on triangles and tetrahedra that are constructed by fitting $d+1$ tensor-product operators into the simplex, where $d$ is the spatial dimension. The TSS operators are different from existing methods involving splitting simplicial meshes into tensor-product operators in that they do not have repeated degrees of freedom along the interior edges. Several numerical studies involving both linear and nonlinear problems are presented to illustrate the efficiency gains afforded by the TSS operators.

We present a new approach to the construction of provably entropy-stable spatial discretizations of arbitrary order for nonlinear hyperbolic systems of conservation laws on curved triangular and tetrahedral unstructured grids. In order to obtain efficient low-storage algorithms at high polynomial degrees, the proposed methods employ sparse tensor-product summation-by-parts operators in collapsed coordinates within a weight-adjusted flux-differencing formulation using a Proriol-Koornwinder-Dubiner modal polynomial basis. Exploiting sum factorization in tensor-product operator evaluation and exploiting sparsity in flux differencing then enables the explicit evaluation of the time derivative in $O(p^{d+1})$ floating-point operations, where $p$ is the polynomial degree of the discretization and $d$ is the number of spatial dimensions. Particularly for higher polynomial degrees, the proposed approach significantly reduces the number of required entropy-conservative two-point flux evaluations relative to entropy-stable formulations of comparable accuracy based on (non-tensor-product) multidimensional summation-by-parts operators, and exhibits the same provable conservation, free-stream preservation, and nonlinear stability properties provided by such existing schemes. In addition to a description and analysis of the algorithmic aspects of the proposed methods, we demonstrate their accuracy and robustness in numerical experiments involving the solution of the compressible Euler equations in two and three dimensions.

Temporal difference (TD) learning with linear function approximation is one of the earliest methods in reinforcement learning and the basis of many modern methods. We revisit the analysis of TD learning through a new lens and show that TD may be viewed as a modification of gradient descent. This leads not only to a better explanation of what TD does but also improved convergence guarantees. In particular, we are able to show that TD learning does well in the limit as the discount factor approaches one. We will also discuss applications to the analysis of actor-critic methods.

In chemical engineering applications such as safety verification of batch reactors, some problems require global optimization of dynamic systems that are modeled as systems of ordinary differential equations (ODEs). Typical methods for deterministic global optimization compute essential lower bounds by minimizing convex relaxations of the original problem. Hence, global dynamic optimization warrants convex relaxations of parametric ODE solutions that are computationally inexpensive, yet are also tight relaxations. In this presentation, we detail several of our recent advances in constructing effective convex relaxations for ODE solutions. Specifically, we explain how to incorporate known relaxations of the ODE right-hand side into effective ODE solution relaxations, how to compute subgradients of these cheaply and accurately, and how to use constructions from derivative-free optimization to improve efficiency. Implications are discussed, and numerical examples are presented, based on proof-of-concept implementations in Julia.

In controlling complex dynamical systems, one often seeks to simultaneously optimize several competing objectives subject to constraints. Linear scalarization is a common approach for solving such multi-objective optimization problems. This approach involves formulating a single cost function made up of the weighted sum of the individual objectives. For each specific vector of weights, the resultant solution is Pareto optimal with respect to the original problem. However, determining the influence of different weight vectors on system behaviour can be complex. To address this, we discuss how to compute a finite set of weight vectors that offer a comprehensive representation of potential system behaviours. Specifically, we determine a set of weight vectors such that any other weight vector has a corresponding set element whose resultant solution is close to the optimal solution. One limitation of linear scalarization is its lack of Pareto-completeness; there are Pareto-optimal solutions that are not the solution of the scalarized optimization problem for any vector of weights. To overcome this, we introduce an alternate form of scalarization based on a weighted maximum of objectives, which achieves Pareto-completeness. We demonstrate the application of this scalarization in controlling complex systems in both continuous and discrete spaces. We provide examples in robot motion planning showing the utility of these methods in learning user preferences on robot behaviour and in identifying trade-offs between objectives that are not attainable with linear scalarization.

In this talk, I will present flexible-step Model Predictive Control (MPC), which is based on the novel notion of generalized discrete-time control Lyapunov functions. For the case of linear control systems, we develop a systematic method for constructing such generalized discrete-time control Lyapunov functions. The main consequence of this construction is that, when combined with flexible-step MPC, we can effectively stabilize switched control systems, for which no quadratic common (classical) Lyapunov function exists.

We present a distributed delay differential equations system modeling the in-host evolution of an HIV infection, describing the interactions between uninfected CD4-T cells, infected cells, virus particles and the immune response. The asymptotic behavior of the solutions is entirely characterized by the delay, denoted $\tau$, representing the time before an infected cell produces virus particles. We show that for a sufficiently large value of $\tau$ above an explicitly determined threshold $\tau_1$ , the infection dies out since the disease-free steady-state is asymptotically stable. Then, for a delay below this threshold, the infection persists, the disease-free equilibrium being unstable. In this case, the endemic steady-state and a chronic infection stage exchange stability after another threshold $\tau_2$ is crossed. Numerical simulations supporting the analytical conclusions are presented. [ Joint work with Marc-Antoine Trahan (U de Montréal) ]

Fads are as common in mathematics as in all ares of life, and topics such as spurious solutions, exponential attractors, and differential algebraic equations, have blazed into prominence at different times in recent decades before fading into the background like a burned out star. But, as we approach the 50th anniversary of the publication of the Mackey-Glass equation in 1977, the topic of delay differential equations and in particular their applications in mathematical biology and physiology remains as fertile and active as ever. I will use this talk to discuss what (little) I know and think about why this is, illustrated by some recent and not so recent results. Along the way we will discuss the Mackey-Glass equation, the Burns-Tannock cell cycle, periodic hematological diseases, state-dependent, distributed and threshold delays, and the dynamics, analysis, implementation and tractability of such models.

Recent field experiments on vertebrates showed that mere presence of a predator would cause a dramatic change of prey demography. Fear of predators increases the survival probability of prey but leads to a cost of prey reproduction. Based on the experimental findings, we propose a predator-prey model with the cost of fear and adaptive avoidance of predators. Mathematical analyses show that the fear effect can interplay with maturation delay between juvenile prey and adult prey in determining the long term population dynamics. A positive equilibrium may lose stability with an intermediate value of delay and regain stability if the delay is large. Numerical simulations show that both strong adaptation of adult prey and the large cost of fear have destabilizing effect while large population of predators has a stabilizing effect on the predator-prey interactions. Numerical simulations also imply that adult prey demonstrate stronger anti-predator behaviours if the population of predators is larger and show weaker anti-predator behaviours if the cost of fear is larger.

One of the methods for implementing machine learning tasks such as time series prediction and classification is the Reservoir Computing (RC) scheme. Reservoir computing, a brain-inspired computational model, utilizes a densely interconnected network. This approach enables effective training for time series prediction leveraging the network's high-dimensional properties and complex dynamics. It has been demonstrated that this scheme can separate chaotic signals effectively when a mixture of them is fed as the input to the reservoir [1]. Furthermore, a novel approach employing high-dimensional systems, such as delay differential equations, has been introduced for machine learning tasks, which can be implemented both numerically and experimentally [2]. It is widely known that in this class of dynamical equations, presence of time delays can induce intricate and complex dynamics. We have examined the role of multiple delays in enhancing the prediction accuracy of time delay reservoir computing, utilizing an electro-optic time delay system. Previously, it has been demonstrated that the incorporation of multiple delays that are uniformly distributed, enables the suppression of chaotic dynamics by widening the distribution of delays. Specifically, in first-order nonlinear delay differential equations, having only a few delays that are uniformly distributed, increasing the spacing between delays can lead to the emergence of limit cycle behaviors and stable fixed-point regimes [3]. In the context of time delay reservoir computing, our findings reveal that multiple delays enhance the linear memory capacity (the ability to recall past input data) of time-delay reservoir computing, leading to an improvement in the overall efficacy of the system. We present results for time delay reservoir computing with various architectures, focusing on the classification and simultaneous prediction of signals when the input is the mixture of signals. \[\] [1] - S. Krishnagopal, M. Girvan, E. Ott, and B. R. Hunt, "Separation of chaotic signals by reservoir computing," Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 30, no. 2, p. 023123, 2020. \[\] [2] L. Appeltant, M. C. Soriano, G. Van Der Sande, J. Danckaert, S. Massar, J. Dambre, B. Schrauwen, C. R. Mirasso, and I. Fischer, "Information processing using a single dynamical node as complex system," Nat. Commun., vol. 2, no. 1, p. 468, 2011. \[\] [3] S. K. Tavakoli and A. Longtin, "Multi-delay complexity collapse," Phys. Rev. Research, vol. 2, p. 033485, Sep 2020.

This paper provides a simple framework to analyze the design of contingent capital for depository institutions. We consider a debt portfolio composed of straight debt and contingent convertible (coco) bonds. Coco bonds are automatically written down or converted into equity upon meeting predetermined capitalization ratio thresholds, thus providing capital loss absorption mechanisms. The quality of the design of coco bonds (parameterized by write-down/conversion factors and trigger levels) is gauged with default probabilities at various horizons, shareholder’s incentive for risk-shifting, and the present value of bankruptcy costs.

Nonsmooth payoff functions are common in financial contracts and pose difficulties in obtaining high-order solutions of the contract prices. In this work, we consider parabolic PDEs with initial conditions involving various types of nonsmoothness. We apply a fourth-order finite difference (FD) discretization on a uniform grid in space, and BDF4 time stepping initialized with two steps of an explicit third-order Runge-Kutta (RK3) method and one step of BDF3. Using Fourier analysis on the discrete system, we prove that the low-order errors generated by RK3 for nonsmooth data in the high-frequency domain get damped away by BDF steps, which prevents spurious oscillations in the solution; while low-order errors in the low-frequency domain come from the low-order initial condition discretization. To achieve globally fourth-order convergence, we apply fourth-order smoothing to the initial conditions, and provide explicit formulas of the discretization. By combining the initial condition smoothing with the proposed time-stepping scheme, we mathematically prove and numerically verify that fourth-order convergence is obtained. The analysis is easily generalizable to higher order methods. Numerical examples on the model PDE and various option pricing problems are also given to demonstrate the fourth-order convergence of our method.

This paper develops a dynamic joint model of the implied volatility (IV) surface and its underlying asset which is tractable and seamless to estimate. It combines an asymptotically well-behaved, parametric IV surface representation with a two-component variance, and non-Gaussian asymmetric GARCH specification for the underlying asset returns. Estimated on S&P 500 index return and option data for the 1996-2020 period, the model captures the IV surface movements well and uses them to obtain an improved fit on index returns. It also proves to be an effective risk management tool, producing reliable Value-at-Risk estimates for straddle and strangle positions, and accurate forecasts of the VIX distribution.

Calibrating option pricing models to market data often presents significant numerical challenges, especially when the object of interest (the option price, i.e., a conditional expectation) relies on simulating from the model itself. We proposed a novel calibration method that relies on, essentially, one simulation only and apply this to the problem of calibrating option pricing models with time varying volatility. A distinct advantage of our approach is that the actual option pricing, which requires simulation, can be done "off-line", leaving only a simple optimization to be solved "on-line" in real time. This separation simplifies the calibration procedure enhancing its applicability.

Disease dynamics can be controlled by top-down shutdown regulations, but are also affected by spontaneous behaviour change within the population. These changes in behaviour can occur as a result of opinion dynamics, i.e., shifting opinions following ``conversations'' between individuals, where the infuence of prophylactic voices is affected by disease prevalence. Measuring opinion dynamics is challenging, however, and it is unclear to what extent opinion dynamics change the outcome of the disease dynamics. We present a comparative modelling study of the COVID-19 Omicron wave in the Yukon Territory and the Southern Interior of BC. Vaccination rates were similar in the two regions, but our analysis of the data suggest that different opinion dynamics may have been at play.

During the COVID-19 pandemic, governments implemented non-pharmaceutical interventions (NPI) including self-isolation and testing requirements, to reduce the spread of SARS-CoV-2. These interventions were implemented as strategies, termed elimination, suppression and mitigation, that differ in the strictness and types of NPIs implemented, the conditions for NPI implementation and relaxation, and the resulting epidemiology. Through implementation of NPIs, these strategies reduce infection spread, but they also disrupt the local economy and impact the health care system. Regional factors and disease characteristics determine which strategy is the most cost effective in a jurisdiction. Most existing mathematical models cannot compare elimination, suppression and mitigation strategies, since elimination reduces infection prevalence to zero, and many epidemiological models are systems of ordinary differential equations that cannot achieve this property. We modified the general SIRS model to study the impact of regional factors on the cost-effectiveness of different strategies. We show that the rate that infected travelers spread infections to the local community (the importation rate), and the fraction of infected people that are hospitalized are important factors determining which strategy is optimal. We found that elimination is the best strategy for a small jurisdiction with a relatively high fraction of the population expected to be hospitalized if infected, and a low importation rate. We also found that when a very low fraction of infected individuals become hospitalized due to severe infections, mitigation is the optimal choice. Regions with limited modeling capacity may rely on modeling intended for other jurisdictions, or may make decisions that differ from recommendations made for other regions. Our work aims to support regions with limited modeling capacity by developing models that consider a range of local characteristics, and can recommend either of three pandemic response strategies.

Urban mobility simulations use agent based modelling to estimate the daily activities of individuals within a community. The output from these simulations can be used to generate a detailed synthetic social network that carries information about the duration and venue of all contact events within a city on a given day, as well as the key demographic information (age, occupation, etc) of the participants in each event. We have been working with The Black Arcs, a Fredericton area technology company, to incorporate synthetic social networks generated by their software into network based models of COVID-19 and related diseases. Our goal is to make quantitative predictions about the impact of various public health interventions on disease spread. For example, we are interested in questions like: Is it more effective to shut down schools or retail businesses to control disease spread? What is the effect of different testing strategies and delays on hospitalizations? And, which policies do the best job of protecting vulnerable demographic groups?

\[\text{BACKGROUND}\] In 2020, Defence Research and Development Canada (DRDC) developed a suite of online decision-support tools to help the Canadian Armed Forces (CAF) mitigate the risk of coronavirus disease 2019 (COVID-19) outbreaks. These tools were underpinned by a COVID-19 prevalence model that leveraged case data published openly by public health agencies. For the first one and a half years, we used a compartmented epidemiological model to estimate prevalence. Then, in early 2022, we transitioned to a method developed by Chiu and Ndeffo-Mbah that also considers test positivity. At the request of the Canadian Forces Health Services, we started applying the same prevalence model to other respiratory illnesses, such as influenza, in early 2023 for Canadian provinces and territories. \[\text{METHOD}\] The model by Chiu and Ndeffo-Mbah is based on the fact that the case and test-positivity rates respectively provide lower and upper bounds to prevalence. The prevalence estimate is obtained by taking a weighted geometric mean of the two bounds. The main challenge in applying this method is to determine the appropriate averaging weights, as their value is affected by the testing regime. \[\text{RESULTS}\] While there was a high test volume during part of the COVID-19 pandemic, it was eventually scaled back to a level similar to other respiratory viruses. In this presentation, we will discuss our efforts in determining appropriate averaging weights for use in our prevalence estimates. \[\text{CONCLUSION}\] The DRDC COVID-19 prevalence map provided the CAF and Department of National Defence with critical information for the determination of workplace posture and force health protection measures during the pandemic. Our renewed map with coverage for other respiratory viruses can continue to support medical advisors as they help navigate the annual respiratory illness season year after year.

The talk provides an overview of Early Warning / Situational Awareness tools for public health, particularly those that leverage public data, and those that detect and track disease outbreaks. The principles behind the Canadian GPHIN system will be highlighted. The (increasing) role of AI will be explored, both for current systems and emerging ones. Some forecasting for future game changers will be presented.

With aging population and healthcare worker shortage sending shock waves throughout the healthcare systems in Canada, there is an urgent need to develop and deploy AI-powered healthcare solutions to assist and support our healthcare workers. While AI for healthcare solutions have made tremendous progress in recent years, translating these gains to improve quality of service (QoS) for real-world healthcare systems has been extremely slow despite a widespread interest both from researchers, hospitals, policy-makers. This is primarily because our healthcare systems are not "Artificial Intelligence (AI) Ready". In this talk, I will first outline various ways AI-powered solutions can revolutionize primary healthcare solutions based on my research in liver transplantation and burn surgical candidacy forecasting. Second, via an example of AI-based surgical skill assessment in radical prostatectomy, I will highlight how AI can play a key role in tooling and training our healthcare workforce in highly skilled tasks. Next, I will delve into the data aspect of machine learning for trustworthy and fair AI modeling in healthcare systems and hospitals, followed by a discussion about the current challenges faced by community hospitals in adoption of AI solutions via my current collaboration with Grand River Hospital. I will then discuss some lessons learned from the COVID-19 pandemic, and its impact on policy and healthcare misinformation. I will conclude with a call to bring together hospitals, industry experts, academic researchers, and policy-makers to develop a unified healthcare strategy for building AI ready healthcare systems. This will lead to faster adoption of research results, data transparency initiatives for patients, and provide a blueprint for AI-centric data organization in the health systems in Canada and abroad.

Drug discovery is a discipline defined by failure. The cost of developing a new medicine is roughly $2.6 billion and it requires upwards of 12 years of sustained work. However, AI methods are uniquely suited to accelerate the drug discovery and development process, due to their capabilities of exploring truly vast chemical spaces for ideal molecules that satisfy multiple required properties. In this talk, we will explore the utility and limitations of discriminative and generative AI methods applied to novel drug discovery tasks. We will focus on two therapeutic domains: antibiotic discovery and neuro-oncology applications. Our goal is to more thoroughly understand how to optimally leverage AI to increase the rate of drug discovery and development, as well as decrease the associated costs.

The COVID-19 pandemic caused by SARS-CoV-2 has disproportionately impacted older (aged 50+) people living with HIV (PLWH). Not all PLWH restore their immunity to pre-HIV-infection levels after receiving combination antiretroviral therapy (cART). Further, HIV-mediated immunodeficiency may result in impaired immunological responses to common COVID-19 vaccines, where this effect may be exacerbated in older individuals due to the natural process of immunosenescence. The ability to classify PLWH as immune responders or immune-non-responders based immunological features would be valuable towards treatment plans in an endemic SARS-CoV-2 environment. Recent modelling and clinical work has revealed substantial differences in vaccine-elicited T-cell responses between PLWH and an HIV- age-matched control [1]. In this talk I will discuss our preliminary results of using machine learning classification techniques in order to classify the SARS-CoV-2 vaccine-elicited immunological footprint of PLWH and HIV-negative age-matched individuals. [1] V. A. Matveev et al., “Immunogenicity of COVID-19 vaccines and their effect on HIV reservoir in older people with HIV,” iScience, vol. 26, no. 10, 2023.

The progression and burden of disease outbreaks, and biological invasions more generally, is complicated by heterogeneities in the host population and its environment. There are two sides to this heterogeneity: the risk of infection or transmission, and the cost of disease. For the particular case of SARS-CoV-2 and CoViD, transmission risk is raised or lowered by host behaviour and by host immune history. Familiar aspects of the former being masking and isolation, and of the latter being the time lapsed since vaccination or past infections. Host behaviour and immune history also affects disease risk as does age (through immunosenescence) and various comorbidities. The main objective of this talk is to present preliminary results from simple compartmental and individual-based models designed to predict population-level disease burden and immune landscapes from host behaviour and within-host virus and immune dynamics.

Previous work has failed to fit classic SEIR epidemic models satisfactorily to the prevalence data and the attack rate (proportion of children infected during the outbreak) of the famous English boarding school 1978 influenza A/H1N1 outbreak in the children’s pandemic. Thus it is still an open question that whether a biologically plausible model can fit the prevalence data and the attack rate correctly. We used an intentionally very flexible and overfitted discrete-time model to learn the epidemiological features from the data which led us to the design of a susceptible (S) - exposed (E) - infectious (I) - confined to bed (B) - convalescent (C) - recovered (R) model with the feature of time delay in E and I compartments and multistage (dummy subgroups) in B and C compartments. We simultaneously fitted the reported B and C prevalence curves as well as the attack rate. The model is based on delay differential equations and dummy subgroups to reproduced the non-exponential residence times, which was crucial for good fits. The estimates of the generation time and the basic reproductive number appear to be biologically reasonable. Our work not only provided an answer to this open question, but also demonstrated an approach to constructive model-generation.

In structured models, e.g., age-structured or risk-structured, where there exist heterogeneity in sub-populations’ susceptibility, activity level or transmission risk, these sub-populations may exhibit different R_0’s and thus different capabilities to drive the overall transmission. Furthermore, if there exist more than one possible transmission routes, R_0 can be quantified for each corresponding route, and thereby identifying the dominant transmission path. We will use pertussis and monkeypox as examples to illustrate the ideas.

Bond percolation methods can be used to model disease transmission on complex networks and accommodate social heterogeneity while keeping tractability. Here we review the seminal works on this field by Newman (2002, 2003, 2010), and Miller, Slim & Volz (2011) and present a more clear and systematic discussion about the theoretical background, assumptions, derivation and development of the percolation method. We also present a new R package based on these results that take epidemic and network parameters as input and generates estimates of the epidemic trajectory and final size. Such theoretical framework and calculation tools allow us to apply the edge-based percolation model to solving real world public health emergencies. With syphilis rates continue rising in Ontario at an alarming rate, an ongoing project is collaborating with KFL&A public health to model Syphilis transmissions within the underserved high-risk community from data. The analysis and prediction of the model could provide scientific evidence to optimize implementation strategy based on community structure, thus help public health professionals to better response to the urgent crisis.

The spectral proper orthogonal decomposition (SPOD) is a relatively new way to analyze the spatio-temporal characteristics of a flow. The dominant SPOD modes can be used for prediction, control, design, optimization, and reduced-order model construction. We performed an analysis of the flow fields of a bubbling fluidized bed using SPOD and found there is no singular dominant frequency linked to the highest energy levels; rather, there is a spectrum of frequencies. These frequencies are consistent with the natural frequency of the bed. The SPOD analysis also reveals the presence of multiple co-existing spatio-temporal patterns in the particle volume fraction and velocity fields. This analysis can be used to guide pulsation strategies to enhance mixing quality and consequently heat and mass transfer properties of the bed.

Burger's vortex is one of the few analytically known, three-dimensional, vortical solutions to the Navier-Stokes equation. It relies on vortex stretching, thought to be one of the sustaining mechanisms of turbulence. Remarkably, this solution has been shown to be linearly stable. At the same time, there are theoretical indications that families of less symmetric equilibria exist near Burgers’ solution. We explore the phase space around Burgers flow by direct numerical simulations of finite-sized perturbations for low and intermediate Reynolds numbers, relying on the finite element code OOMPH-LIB for time-stepping as well as for the computation and continuation of equilibria and their spectra.

The importance of the formation of singularities in the 3D Euler and NS equations beginning with a smooth divergence-free velocity field cannot be overstated since it would indicate that the governing equations are not well posed and the issue is of great interest to engineering communities because of the possible connection with the onset of turbulence. In 2014, Luo and Hou presented numerical evidence to show that a certain swirling rotationally symmetric flow of an incompressible inviscid fluid in an axially periodic cylinder of finite radius leads to finite time blow-up in the vorticity $\omega$. Numerical methods used by Luo and Hou (2014), Barkley (2020) and Kolluru et al. (2022) used a vorticity-stream function formulation and pseudospectral or B-spline based Galerkin methods in combination with a Runge-Kutta method. In this talk, we use a Fourier-Chebyshev spectral collocation method and compare both explicit and implicit adaptive Runge-Kutta methods in an investigation of finite time blow-up of the vorticity in the swirling flow proposed by Luo and Hou.

We discuss solutions of the Euler equations of fluid mechanics in three (and higher) dimensions which describe colliding pairs of vortex tubes. We provide rigorous upper and lower bounds for growth rates of the vorticity which generalize and improve on recent estimates of Choi and Jeong. This is joint work with Evan Miller and Tai-Peng Tsai.

We consider enstrophy dissipation in two-dimensional (2D) Navier-Stokes flows and focus on how this quantity behaves in the limit of vanishing viscosity. After recalling a number of a priori estimates providing lower and upper bounds on this quantity, we state an optimization problem aimed at probing the sharpness of these estimates as functions of viscosity. More precisely, solutions of this problem are the initial conditions with fixed palinstrophy and possessing the property that the resulting 2D Navier-Stokes flows locally maximize the enstrophy dissipation over a given time window. This problem is solved numerically with an adjoint-based gradient ascent method and solutions obtained for a broad range of viscosities and lengths of the time window reveal the presence of multiple branches of local maximizers, each associated with a distinct mechanism for the amplification of palinstrophy. The dependence of the maximum enstrophy dissipation on viscosity is shown to be in quantitative agreement with the estimate due to Ciampa, Crippa & Spirito (2021), demonstrating the sharpness of this bound. [Joint work with Pritpal Matharu and Tsuyoshi Yoneda.]

Recent advances in the study of the mechanics of the transition layer between a Darcy porous layer and a Navier-Stokes’ channel employ a Brinkman’s porous layer with variable permeability, sandwiched between the two flow regiments. Flow in the Brinkman’s layer is governed by the well-known Brinkman’s equation which has been shown to give rise to Airy’s differential equation. Exact solutions of the flow in the transition layer are then given in terms Airy’s functions. This approach continues to be implemented in the study of flow through and over porous layers, and has provided an impetus to the transition zone approach, which initiated a number of novel ideas that include: 1) Introducing and reviving the implementation of classical integral functions in the porous media literature (as witnessed by the use of Airy’s differential equation and the Airy functions in providing an analytical solution to the flow in the transition zone). 2) Introducing new integral functions to facilitate solution to the inhomogeneous Airy’s differential equation. This, in turn, facilitates analysis of other related problems governed by the inhomogeneous Airy’s equation. 3) Initiating non-traditional models of permeability variations in porous media. These complement classical models that use elementary mathematical functions and have served the subject matter well. The use of special functions in advancing the topic represents a new generation of models the computations of which is no longer a formidable task. In the current work, modeling aspects of flow through and over porous layers with an embedded transition layer are discussed in order to provide alternative variable permeability models that reduce Brinkman’s equation in the transition zone to equations that complement Airy’s equation. In particular, the interest is to model the flow using generalized Airy’s equation and Weber’s inhomogeneous differential equation, and to provide efficient algorithms for their computations.

We present a study of the two-dimensional water wave problem consisting of a density-stratified fluid composed of two immiscible layers separated by a sharp interface. A goal is to describe the interaction between long, larger amplitude, nonlinear waves on the interface and modulated, smaller amplitude, free wave packets on the surface when the lower fluid is infinitely deep. In the first part, starting from the Hamiltonian formulation of this problem and using techniques from Hamiltonian transformation theory, we describe the resonant interaction of the waves by a system of equations where the internal wave solves a high-order Benjamin-Ono equation coupled to a linear Schroedinger equation for the time evolution of the wave envelope of the free surface. In the second part, we establish a local well-posedness result for the BO-Schroedinger system in the physical regime where the densities of the two fluid layers are close. Neglecting the higher-order coupling terms, we perform a gauge transformation to eliminate the higher-order non-linear terms and reformulate our BO equation, from which our proof follows by a fixed-point argument. This is a joint work with A. Kairzhan and C. Sulem.

Wind energy is rapidly evolving to be the most reliable renewable energy source. Wind farms, particularly those situated over complex terrain and harsh weather conditions, generally present multi-scale challenges of atmospheric turbulence. To the best of our knowledge, past studies have not systematically evaluated the velocity gradient dynamics of atmospheric turbulence for utility-scale wind farms in complex terrain. In the real atmosphere, a role of turbulence is to transport kinetic energy from the free atmosphere to the atmospheric boundary layer, known as entrainment. However, complex terrain itself can quickly generate turbulence within the atmospheric boundary layer. In this study, we consider wavelet-filtered velocity gradient dynamics to encapsulate and model the impact of atmospheric turbulence in wind energy production, where the vortex stretching is regarded as the principal mechanism of energy cascade. Idealized case studies that simulated the impact of complex terrain and turbulence entrainment indicate that the land needed by wind farms in complex terrain may be five times less than that required by offshore wind farms with sustained wind conditions. The study has utilized wind tunnel experiments, field measurements, and the deep learning approach to validate the role of velocity gradient dynamics wind farm design and control. Overall, the study used wavelet-coherence to show that complex terrain is most unlikely to cause fluttering on wind turbine.

Heterogeneity is the norm in biology. The brain is no different: Neuronal cell types are myriad, reflected through their cellular morphology, type, excitability, connectivity motifs, and ion channel distributions. While this biophysical variety adds richness to neural system dynamics, it poses a challenge in maintaining the stability and consistency of brain function over time, known as resilience. To better understand the relationship between excitability heterogeneity (cell-to-cell variability in excitability within a population of neurons) and resilience, we analyzed both analytically and numerically a nonlinear sparse neural network with balanced excitatory and inhibitory connections evolving over long time scales. Homogeneous networks demonstrated increases in excitability, and strong firing rate correlations—signs of instability—in response to slowly varying modulatory fluctuations. Excitability heterogeneity tuned network stability in a context-dependent way by restraining responses to modulatory challenges and limiting firing rate correlations, while enriching dynamics during states of low modulatory drive. Excitability heterogeneity was found to implement a homeostatic control mechanism enhancing network resilience to changes in population size, connection probability, strength and variability of synaptic weights, by quenching the volatility (i.e., its susceptibility to critical transitions) of its dynamics. Together, these results highlight the fundamental role played by cell-to-cell diversity in the robustness of brain function in the face of change.

The expressivity of a neural network where all weights are initialized randomly and only constant inputs (biases) are learned is not well-studied and of interest in two domains. In neuroscience, the contribution of inputs from upstream regions–versus local plasticity–to learning in neural circuits, such as the motor cortex, is poorly understood. In artificial intelligence (AI), recent empirical work has shown that fine-tuning biases alone can yield efficient multi-task learning. However, both fields lack a thorough understanding of the limits of input-only learning. Here, we give theoretical and numerical evidence that a wide class of functions and finite trajectories from many dynamical systems can be well approximated by randomly initialized networks where only biases are optimized. These results extend our collective knowledge of neural networks, providing guidance for future methods in AI and for models of circuit-level plasticity in the brain.

We introduce an approach to image segmentation through spatiotemporal dynamics in a recurrent neural network (RNN). We use a specific RNN, where each node’s state is represented by a complex number. This complex-valued RNN (cv-RNN) generates intricate spatiotemporal patterns that can effectively group structures within a scene. By obtaining an exact solution to the network’s dynamics, we can “open the box” and elucidate the underlying mechanism for how this network performs its computation. This analytical approach then motivates a simple but powerful algorithm that segments objects ranging from simple geometric shapes to complex entities in natural scenes. We find that image segmentation can be accomplished by a single set of recurrent weights obtained from training on simple images and then applied to naturalistic scenes, demonstrating the potential generalizability of the algorithm. Taken together, our results provide a new avenue for machine learning that leverages exact solutions to create algorithms that can be understood through precise mathematical expressions. These results also provide potential insights into computations with spatiotemporal dynamics in the biological neural circuits of the visual system.

We consider a neural field model for a brain network which is a network of Wilson-Cowan nodes with homeostatic adjustment of the inhibitory coupling strength and time delayed, excitatory coupling. Without time delay, the system has been show to exhibit rich dynamics including oscillations, mixed-mode oscillations, and chaos. We explore how synchronization of the nodes depends on both the connectivity structure of the network and the attributes of the distribution of time delays in the connections between nodes. We show that Hopf bifurcations induced by the excitatory coupling, the connectivity structure and the delay lead to different patterns of phase-locked oscillations, either synchronized or desynchronized. Finally, we study how the mean and variance of the distribution affect the results.

Currently, a growing interest is seen in developing solvers that couple high-fidelity and higher-order spatial discretization schemes with higher-order time stepping methods for various time-dependent fluid and plasma models. These problems are famously known to be difficult to solve, requiring either very short time steps for explicit schemes, or expensive implicit time-stepping schemes with certain stability properties to be used. One of the most powerful choices is the family of implicit Runge-Kutta (IRK) methods. However, these are multi-stage schemes, often producing a very large and nonsymmetric system of equations that needs to be solved at each time step. Consequently, there have been several recent efforts to develop efficient and robust solvers for these systems. We have accomplished this by using a Newton-Krylov-multigrid approach that applies a multigrid preconditioner monolithically, preserving the system couplings, and uses Newton's method for linearization wherever necessary. We show robustness of our solver on the single-fluid viscoresistive magnetohydrodynamics (MHD) model, along with the (Navier-)Stokes and Maxwell's equations. For all these, we couple IRK with higher-order (mixed) finite-element (FEM) spatial discretizations. In the Navier-Stokes problem, we further explore achieving higher-order approximations by using nonconforming mixed FEM spaces with added penalty terms for stability. For the Maxwell problem, we focus on the more difficult E-B form needed to model plasma flows and overcome the difficulty of using FEM on curved domains by using an elasticity solve on each level in the non-nested hierarchy of meshes to improve mesh quality for the multigrid method.

Mapping, in a multi-physics context, is the problem of transferring data from the source to target side of a geometric interface of two physics solvers. Accurate and robust mapping technology is critical to any multi-physics coupling engine. In this presentation we develop the conditions for mapping algorithms to preserve the overall order of accuracy of the discretizations of the participating physics solvers, for both volume and surface mapping, and including the case of non-conformal surfaces. The presentation also explores the use of radial basis functions for mapping and investigates several numerical pathologies encountered when using radial basis functions, as well as solutions to these problems.

The exact solution of the governing equations of physical phenomena often enforces auxiliary admissibility criteria, e.g., conservation of entropy or conservation of energies. Often referred to as invariants, these auxiliary admissibility criteria are typically not enforced via conventional discretization techniques. Ultimately, this leads to non-physical solutions, loss of accuracy, and loss of stability. Over the past several decades, significant attention has been placed on low-order entropy or kinetic energy conservative spatial discretizations. These frameworks then have been extended to high-order discretizations, including discontinuous Galerkin (DG) and flux reconstruction (FR). For example, see the work of Chan (2018). However, less attention has been given to the role of time integration in conserving invariants. Recently, Ketcheson (2019) demonstrated relaxation Runge-Kutta (RRK) schemes can be used to enforce energy conservation, subsequently extended to entropy conservation by Ranocha et al. (2020). These introduce a time step relaxation parameter to enforce global conservation. In contrast, Calvo et al. (2006) showed that entropy conservation can be obtained without relaxing the step size by using embedded RK methods to define a more arbitrary projection direction. However, with either RRK or the method of Calvo et al. stability issues may arise and the order of accuracy may decrease. Here, we indicate the constraints in defining the search direction to project the RK prediction to the conservative manifold, and then we present a unified projection framework that behaves superiorly in terms of stability and accuracy. We demonstrate invariant conservation for a range of example applications, and propose that the novel technique is a suitable replacement for the RRK and method of Calvo et al., alleviating the limitations of both previously proposed approaches.

One major drawback of classical spectral methods is their inability to handle irregularly shaped domains, which is why they have only been used sparingly in many engineering problems. Spectral element methods can handle complex geometry, but they are more difficult to implement. There have been many attempts to use classical spectral methods for elliptic PDEs in irregular domains, but only a few studies on spectral collocation methods for time-dependent PDEs in complex geometries. Here, we propose a numerical method to approximate the solution of PDEs in irregular domains using space-time spectral collocation methods. The main idea here is to embed the irregular domain in a regular one and apply Fourier extension. We use Fourier and Chebyshev spectral methods to solve PDEs on regular geometries. Here we assume that the non-homogeneous term is only known in the physical domain and perform a periodic extension to the extended domain. We implemented Boyd's technique for this Fourier extension and observed super-algebraic convergence only. Then we apply the method introduced by Huybrechs to extend the non-homogenous term of the PDE from the physical domain to the regular domain with exponential accuracy and have solved steady and unsteady PDEs.

This study focuses on numerically solving the sub-diffusive Gierer-Meinhardt model with controlled precision. We start by defining and explaining basic concepts of reaction diffusion, highlighting the main differences between normal and anomalous diffusion. Sub-diffusion is modelled at continuum level by fractional derivatives replacing regular ones in PDEs. Therefore an important part of our study involves fractional calculus, which provides a solid framework for describing sub-diffusive processes. We then delve into the well-known Gierer-Meinhardt model, a reaction-diffusion system used to describe pattern formation in biological systems. Leveraging the matched asymptotic expansion technique, which is applicable due to the asymptotic smallness of certain parameters in the system, we transform the differential Gierer-Meinhardt model into a differential algebraic system. This differential algebraic system contains a fractional operator denoted $\mathcal{D}^{\gamma}_{t}$, which involves the integral of a complicated function impossible to determine analytically. This operator depends on multiple parameters, and the number of subdivisions needed for numerical computation varies significantly with these parameters and the desired precision. To address this challenge, we have developed a program capable of precalculating the required number of subdivisions before computation, thus saving significant computation time. All elements in place, we use these tools to study the dynamics of the obtained solutions.

Event-triggered control provides an efficient way to update feedback control signal at a sequence of discrete-time moments, and these discrete times are determined by a certain event that occurs only when the measurement of system states violates a predesigned threshold. The advantage of this control paradigm is that it improves the efficiency of control implementations while still guaranteeing the desired performance level of the control system. In this talk, we present an event-triggered control method for the stabilization of linear time-delay systems. Based on two Halanay-type inequalities, the global asymptotic stability of the event-triggered control system can be guaranteed, and a lower bound of the inter-event times, the intervals between successive control updates, can be derived to ensure the practical implementation of the proposed event-triggering condition.

This talk discusses the leader–follower synchronization problem of complex-valued networked multi-agent systems with time-delay. A hybrid protocol including a continuous-time protocol based on the interaction topology of the follower agents and a pinning delayed impulsive control protocol is proposed. Several delay-dependent leader–follower synchronization criteria are established where various sizes of delays are taken inti account. Particularly, our result shows that leader–follower synchronization of delayed complex-valued multi-agent systems can be achieved even if the proposed hybrid protocol is being subject to relatively large impulse delays. Numerical examples are provided to illustrate the effectiveness of the theoretical results.

Nonsmoothness can arise in difference equations in several ways: because they are a suitable modeling framework for applications that exhibit a mixture of continuous and discrete behavior; because of numerical discretization schemes or analysis techniques applied to nonsmooth continuous-time systems; and because of mathematical techniques that purposefully introduce nonsmoothness to iterative schemes. However, the presence of such nonsmoothness necessitates the development of new tools of analysis since conventional approaches typically require smoothness of the system. After reviewing generalized derivatives theory, we highlight a new sensitivity theory for nonsmooth difference equations that characterizes first-order generalized derivative information of solutions. The theory, which is applicable to both explicit and implicit difference equations, is computationally relevant and it recovers classical theory when participating functions are smooth. We also show how this new theory can be used to analyze the dependence of fixed points and N-cycles on parameters and solve optimal control problems.

We formulate the problem of fake news detection using distributed fact checkers (agents) with limited and unknown trust. We consider the stream of news/statements as an independently and identically distributed binary source (to model true vs false statements). Upon observing the news, agent $i$ labels it as true or false that reflects the true validity of the statement with some probability $1-\pi_i$. In other words, agent $i$ misclassified each statement with error probability $\pi_i\in (0,1)$, where the parameter $\pi_i$ models the (un)trustworthiness of the $i$th agent. We present an algorithm to learn these parameters resulting in a distributed fact checking algorithm. We provide various aspects of this algorithm and its convergence properties. This work is a joint work with Ashwin Verma (UCSD) and Prof. Soheil Mohajer (UMN).

The stability of equilibria and asymptotic behaviors of trajectories are often the primary focuses of mathematical modeling.However, many interesting phenomena that we would like to model, such as the ``honeymoon period'' of a disease after the onset of mass vaccination programs, are transient dynamics. Honeymoon periods can last for decades and can be important public health considerations. In many fields of science, especially in ecology, there is growing interest in a systematic study of transient dynamics. In this work we attempt to provide a technical definition of ``long transient dynamics'' such as the honeymoon period and explain how these behaviors arise in systems of ordinary differential equations. We define a transient center, a point in state space that causes long transient behaviors, and derive some of its properties. In the end, we explore some possible extensions of these ideas to systems of delay differential equations.

This presentation revisits Lyapunov function(al)s and their applications in population dynamics. We highlight recent advancements and ongoing challenges in establishing the global stability of both disease-free and endemic equilibria in infectious disease models. Noteworthy among recent advancements is work on discrete-time epidemiological models, offering promising insights that may steer future directions.

Oncolytic viral therapy (OVT) is a promising form of immunotherapy that uses viruses to selectively infect and eliminate cancer cells. These trained viruses replicate within cancer cells, triggering immune responses against cancer. To evaluate the efficacy of OVT, we investigate the dynamics of cancer growth influenced by oncolytic viral activity and the subsequent antibody immune response. Building upon prior research by S. Jahedi, J. Watmuogh, and L. Wang (“Effect of Virus-Specific Immune Response on Outcome of Oncolytic Viral Therapy”), we extend the study by introducing a delay in the immune system’s ability to produce antibodies in response to the virus. As the previous work, we identified two thresholds of virus infection rate: the treatment control threshold ($b_c$) and the treatment optimal threshold ($b_{opt}$) which is delay dependent. Treatment begins to work beyond $b_c$ and achieves full success beyond $b_{opt}$. Further, we assess tumor responses under slow and fast immune systems, exploring various therapy strategies such as single and repeated dosing regimens, varied dosing levels, and timing. Our findings indicate that increasing dosing frequency enhances therapy efficacy, resulting in a higher proportion of infected tumor size relative to the total tumor size.

In this talk, we revisit the notion of infection force from a new angle, which can offer a new perspective to motivate and justify some infection force functions. Our approach can not only explain many existing infection force functions in the literature, but it can also motivate new forms of infection force functions, particularly infection forces depending on disease surveillance of the past. As a demonstration, we propose an SIRS model with delay. We comprehensively investigate the disease dynamics represented by this model, mainly focusing on the local bifurcation caused by the delay and another parameter that reflects the weight of past epidemics in the infection force. We confirm Hopf bifurcations both theoretically and numerically. The results show that, depending on how recent the disease surveillance data are, their assigned weight may have a different impact on disease control measures.

We introduce the second-order Esscher pricing notion for general continuous-time models. This extends the classical notion of Esscher that is used in finance and actuarial sciences. Depending whether we use directly the assets' prices $S$ or their logarithm, we obtain two principal classes of Esscher pricing densities of-order-two. Besides this novel notion, our contributions has three main parts. In the first part, for every class obtained and using the statistical parametrization of the model, we characterize its second-order Esscher densities via pointwise equations. Furthermore, we discuss the relationship between the two classes and the role of the second order factor specific to our Esscher notion. The second main contribution resides in addressing the pricing problem using the obtained Esscher densities. In fact, we show that the bounds of the Esscher stochastic pricing interval, $){Y}^{\rm{inf}},Y^{\rm{up}}($, are solutions to two constrained reflected linear backward stochastic differential equations (RBSDEs hereafter for short), or equivalently reflected BSDEs with singular non Lipschitz generator. This shows that these constrained reflected BSDEs appear also in other setting beyond the cases of constraints on gain-process or constraints on portfolios. For the Merton jump-diffusion model, we derive this Esscher pricing interval and compare it to Merton's pricing formula. The third main contribution lies in conducting numerical and empirical studies to show and test the role of the second order Esscher in many aspects. This talk is based on joint work with Alla Elazkany (University of Alberta) and Michele Vanmaele (Ghent University, Belgium).

Motivated by the recent success of rough volatility models, we introduce the notion of a roughness exponent to measure the roughness of trajectories. We show that it can be estimated in a model-free manner and we discuss its relations to other roughness measures such as weighted quadratic variation and Besov regularity. Then we we introduce a new estimator that measures the roughness exponent of a continuous trajectory based on discrete observations of its antiderivative. This estimator is then applied to estimating the roughness exponent of the volatility in a stochastic volatility model that arises as a nonlinear function of fractional Brownian motion with drift. We provide conditions on the underlying trajectory under which our estimator converges in a strictly pathwise sense. Then we verify that these conditions are satisfied by almost every sample path of fractional Brownian motion with sufficiently regular drift. As a consequence, we obtain strong consistency theorems in the context of a large class of rough volatility models. Numerical simulations show that our estimation procedure performs well after passing to a scale-invariant modification of our estimator. This is joint work with Xiyue Han.

In this work, we examine the joint effect of illiquidity on the dynamics of asset prices and interest rates. Equilibrium is identified by a system of coupled forward-backward stochastic differential equations, which admit explicit asymptotic solutions in the high-liquidity regime. Illiquidity increases price volatility and the risk premium but decreases the interest rates. (Joint work in progress with Paolo Guasoni and Johannes Muhle-Karbe.)

We introduce a new model for a stock price, namely, geometric compound Hawkes process, and show how this model can be applied to solving many problems in finance, including option pricing (European and American) and Merton portfolio optimization problems. This model is a generalization of some well-known models in finance, such as Cox-Ross-Rubinstein model (1976) (geometric binomial process), Aase model (1988) (geometric compound Poisson process) and geometric Markov renewal model (2013), to name a few. Numerical examples are presented as well.

We provide an overview of some results for pattern formation and diffusion-induced synchrony for a 2D PDE-ODE bulk-cell model, where one or more bulk diffusing species are coupled to nonlinear intracellular reactions that are confined within a disjoint collection of small circular compartments. The bulk species are coupled to the spatially segregated intracellular reactions through Robin conditions across the cell boundaries. For this compartmental-reaction diffusion system, we show that symmetry-breaking bifurcations leading to stable asymmetric steady-state patterns, as regulated by a membrane binding rate ratio, occur even when two bulk species have equal bulk diffusivities. This result is in distinct contrast to the usual, and often biologically unrealistic, large differential diffusivity ratio requirement for Turing pattern formation from a spatially uniform state. Secondly, for the case of one-bulk diffusing species in R^2, we derive a new memory-dependent ODE integro-differential system that characterizes how intracellular oscillations in the collection of cells are synchronized through the PDE bulk-diffusion field. By using a fast numerical approach relying on the ``sum-of-exponentials'' method to derive a time-marching scheme for this nonlocal system, diffusion induced synchrony is examined for various spatial arrangements of cells.This theoretical modeling framework, relevant when spatially localized nonlinear oscillators are coupled through a PDE diffusion field, is distinct from the traditional Kuramoto paradigm for studying oscillator synchronization on networks or graphs. Numerical challenges, some still only partially resolved, for numerically implementing our analytical theory are discussed.

The Kuramoto model has been used for the last decades to gain insight into the behaviour of coupled discrete oscillators, as it is simple enough to be analyzed and exhibits a breadth of possible behaviours, such as synchronization, oscillation quenching, and chaos. However, the question arises how one can derive precise coupling terms between spatially localized oscillators that interact through a time-dependent diffusion field. We focus on a compartmental-reaction diffusion system with nonlinear intracellular kinetics of two species inside each small and well-separated cell with reactive boundary conditions. For the case of one-bulk diffusing species in $\mathbb{R}^2$, we derive a new memory-dependent integro-ODE system that characterizes how intracellular oscillations in the collection of cells are coupled through the PDE bulk-diffusion field. By using a fast numerical approach relying on the ``sum-of-exponentials'' method to derive a time-marching scheme for this nonlocal system, diffusion induced synchrony (in-phase, anti-phase, mixed-mode etc.) is examined for various spatial arrangements of cells. This theoretical modelling framework, relevant when spatially localized nonlinear oscillators are coupled through a PDE diffusion field, is distinct from the traditional Kuramoto paradigm for studying oscillator synchronization on networks or graphs. It opens up new avenues for characterizing synchronization phenomena associated with various discrete oscillatory systems in the sciences, such as quorum-sensing behaviour.

We analyze the existence, linear stability, and slow dynamics of localized 1D spike patterns for a Keller-Segel model of chemotaxis that includes the effect of logistic growth of the cellular population. Our analysis of localized patterns for this two-component reaction-diffusion (RD) model is based, not on the usual limit of a large chemotactic drift coefficient, but instead on the singular limit of an asymptotically small diffusivity of the chemoattractant concentration field. In the limit, steady-state and quasi-equilibrium 1D multi-spike patterns are constructed asymptotically. To determine the linear stability of steady-state $N$-spike patterns, we analyze the spectral properties associated with both the “large” $O(1)$ and the “small” $o(1)$ eigenvalues associated with the linearization of the Keller-Segel model. By analyzing a nonlocal eigenvalue problem characterizing the large eigenvalues, it is shown that $N$-spike equilibria can be destabilized by a zero-eigenvalue crossing leading to a competition instability if the cellular diffusion rate exceeds a threshold, or from a Hopf bifurcation if a relaxation time constant is too large. In addition, a matrix eigenvalue problem that governs the stability properties of an $N$-spike steady-state with respect to the small eigenvalues is derived. From an analysis of this matrix problem, an explicit range of cellular diffusion rate where the $N$-spike steady-state is stable to the small eigenvalues is identified. Finally, for quasi-equilibrium spike patterns that are stable on an $O(1)$ time-scale, we derive a differential algebraic system (DAE) governing the slow dynamics of a collection of localized spikes. Unexpectedly, our analysis of the KS model with logistic growth in the small chemical diffusion rate regime is rather closely related to the analysis of spike patterns for the Gierer-Meinhardt RD system.

The fractional order s of a Lévy flight controls the algebraic decay of its corresponding jump length distribution. When $s\gt 1/2$ , $s=1/2$, or $s\lt 1/2$ the corresponding one-dimensional Lévy flights is qualitatively similar to Brownian motion in $N=1$, $N=2$, and $N\geq3$ dimensions. In this talk we describe how this correspondence manifests in the asymptotic analysis of two fractional problems: the characterization of spike equilibrium solutions to singularly perturbed reaction-diffusion systems, and the analysis of the first-hitting-time (FHT) for a Lévy flight to a small target. Our asymptotic results provide insights into how the fractional order affects the linear stability of spike solutions, as well as how target sparsity determines the optimal fractional order minimizing the first-hitting-time.

Real-time delivery of credible information is critical throughout disease outbreaks to predict changes in the outbreak as early as possible and guide public health measures. However, too little data, collected too slowly, can affect the accuracy of these predictions and the efficacy of mitigation efforts. Advances in artificial intelligence (AI) can fill gaps in data and improve disease outbreak response at every stage. AI-powered tools have enabled monitoring the spread of diseases at local, state, and national levels; predicting upcoming peaks and their intensities; identifying hot spots; guiding the purchase and allocation of healthcare resources; informing decisions and policies, both for closing down facilities and for reopening them; and optimizing vaccination rollout strategies. In this talk, I will discuss harnessing AI to fill gaps in data and improve disease outbreak response at every stage. In particular, I will present some of the AI-based frameworks that we designed and deployed during COVID-19 to support the efforts of decision-makers, the medical community, and the public in managing every stage of the COVID-19 crisis. These frameworks utilize innovative AI models to integrate both conventional (e.g., historical data) and unconventional data (e.g., Google Trends, Google Trends Rate, social media, satellite data, and economic activity data).

Multivariate joint modeling that integrates multiple longitudinal data and a terminal event for clustered data has increased interest in medical research. In the study of periodontal disease, probing pocket depth and recession serve as tooth-level biomarkers that are associated with the risk of tooth loss. Considering the natural clustering of the teeth within individuals, we propose a clustered multivariate joint model to assess the effect of multiple tooth-level longitudinal biomarkers on the risk of tooth loss. We estimate our joint model using a Bayesian estimation approach. We showed that our method outperforms the alternative joint model that ignores the cluster effects. We further use our proposed joint model to make dynamic prediction on the probability of tooth loss within a specified time horizon, given a set of baseline and longitudinal predictors, and assess the performance of dynamic prediction for clustered data.

Artificial intelligence (AI) is increasingly being integrated into every phase of drug development and patient care. The ongoing innovations in AI-driven methodologies are largely contingent on the availability of extensive, high-quality data sets. However, data access is often hindered by rigorous regulatory and privacy constraints, especially concerning real-world data routinely collected in healthcare. This challenge is especially pronounced in the realms of healthcare and drug discovery, where it significantly impedes progress. Data sets are frequently assembled independently, with minimal overlap, leading to data scarcity. This scarcity complicates data curation for researchers attempting to address pivotal research questions that span multiple data sets. Synthetic data, especially those produced using generative AI techniques, offer potential solutions to these privacy concerns and help mitigate the data deficiency. In her presentation, Dr. Chen will discuss her research in generative AI, focusing on two specific applications: 1) the use of synthetic data as a privacy-enhancing tool for the Public Health Agency of Canada, and 2) synthetic data for drug discovery. She will share insights from her experience in generating, evaluating, and distributing large volumes of synthetic data sets. Additionally, Dr. Chen will introduce a groundbreaking diffusion GNN model, Syngand, designed for the end-to-end generation of ligand and pharmacokinetic data.

The rapid and precise differentiation between influenza and COVID-19 is significant, given their similar early symptoms but significantly different effects on public health. At this mini-symposium, I will present how artificial intelligence can tackle this challenge through a supervised machine learning (ML) model that effectively distinguishes between these infections. Validated by strong performance indicators (ROC AUC ? 91%), this model uses synthetic data of viral infections to identify unique disease markers. Feature selection has pinpointed essential factors such as viral load and productively infected cells, which are vital for predicting disease progression and guiding specific interventions. Additionally, I will briefly discuss our ongoing research into developing an ML framework that explains the intricacies among various cellular entities. This study seeks to decode the mechanisms of cancer cachexia and immune dysregulation by focusing on the immune balances influencing tumour and muscle outcomes. Generally, these endeavours highlight the profound impact of AI in advancing disease diagnosis and deepening our understanding of complex biological systems, which aligns with Canada’s objectives to enhance healthcare outcomes via cutting-edge technologies.

Mathematical biology has advanced to a level where it can mechanistically model many complex ecological processes. Nevertheless, some time-varying variables are often assumed to be fixed as they may depend on several ecological covariates, the governing dynamics of which are not exactly known. This simplification can be overcome by using hybrid mechanistic machine-learning models, where the dynamics of already well-studied variables are modelled using mechanistic models, such as differential equations, and the potentially complex and yet not fully developed dynamics of the remaining variables are modelled using machine learning algorithms. This approach has proven successful in predicting ecological phenomena; however, the machine learning part remains a black box, limiting the gained insight. We propose using causal Bayesian networks for the machine learning part. This way, while often maintaining or improving the prediction accuracy of previous models, causal relationships between the variables can be partly identified and used for developing mechanistic sub-models to replace the machine-learning part. While this approach can be readily applied to ecological processes, several challenges still need to be addressed to realize its full potential.

Blooms of cyanobacteria (CB) can disrupt lake ecosystems by killing large numbers of lake organisms, including members of key fish species. Climate change is projected to exacerbate this problem by increasing water temperatures to be closer to those that facilitate maximum CB growth. This raises the question of how temperature increases, and the more severe CB blooms that result from them, will affect the viability and health of other lake species. To address this, we construct a stoichiometric model that includes common components of an Alberta lake food web (CB, algae, daphnia, yellow perch, walleye), and features the effects of oxygen depletion caused by CB blooms and uptake of microcystin secreted by CB. We fit this model using field data from CB observations in Alberta lakes, and simulate it under different warming scenarios to predict the magnitude and effects of CB blooms in the future.

Mosquitoes are important vectors for the transmission of some major infectious diseases of humans, i.e., malaria, dengue, west Nile virus and Zika virus. The burden of these diseases is different for different regions, being highest in tropical and subtropical areas, which have high annual rainfall, warm temperatures, and less pronounced seasonality. The life cycle of mosquitoes consists of four distinct stages: eggs, larvae, pupae, and adults. These life stages have different mortality rates and only adults can reproduce. Seasonal weather may affect the population dynamics of mosquitoes, and the relative abundance of different mosquito stages, since the maturation rate to the next stage depends on temperature, and because egg survival depends on rainfall. We developed a stage-structured model that considers laboratory experiments describing how temperature and rainfall affects the reproduction, maturation and survival of different Anopheles mosquito stages, the species that transmits the parasite that causes malaria. We consider seasonal temperature and rainfall patterns and describe the stage-structured population dynamics of the Anopheles mosquito in Ain Mahbel, Algeria, Cape Town, South Africa, Nairobi, Kenya and Kumasi, Ghana. We find that regional differences in seasonal weather patterns affect mosquito population dynamics. Control strategies often target one specific life stage, for example, applying larvicides to kill mosquito larvae, or spraying insecticides to kill adult mosquitoes. Our findings suggest that differences in seasonal weather patterns affect mosquito stage structure, and best approaches to vector control may vary between regions.

Bylot Island, in the Canadian Arctic, is home to a predator-prey system that includes snowy owls, arctic foxes, and lemmings. The latter are prey for both the snowy owls and foxes. Snowy owls reside periodically on the island, their presence or absence depending on the availability of the prey only. Unlike many other predator-prey systems, the composition of the food web (whether it includes one or two predators) depends on the internal dynamics of the system. We begin with an ordinary differential equation model, that we reduce to a simple one-species discrete map. The simplicity of this map allows us to prove several key aspects of solution behaviour, and fully characterise the cyclic and pseudo-cyclic dynamics in the owl-fox-lemming system. Furthermore, we observe a bistability-like behaviour between two- and three-cycle solutions. The persistence of arctic predator-prey systems is under threat due to climate change, and our work contributes to efforts to understand the dynamics of these important species.

Flow through local constrictions has important applications in various areas of biology and engineering. In blood flow applications, for example, knowledge of more detailed flow behaviour through constricted flow geometries can allow for assessment and treatment of conditions such as atherosclerosis, in which stenosed (or constricted) vessels can arise from plaque deposits along the interior of the vessel walls. In general, particle-based methods are computationally expensive, even to date. Efficient methodologies at the mesoscopic scale, rather than fully detailed microscopic options, can provide the trade-off between computational cost and more detailed assessments, compared to solving the macroscopic Navier-Stokes equations. This talk focuses on recent advances in using a numerically efficient mesoscopic particle-based method for flow applications, called Multi-particle collision dynamics (MPC), that was originally introduced by Malevanets and Kapral in the late 1990s. Theoretical and numerical results will be presented for flow through constricted cylinders using MPC. The particle-based simulations are shown to exhibit built-in compressibility that can, in part, be quantified by a pressure-dependent viscosity in the Navier-Stokes equations. Approximate analytical solutions are provided, that are based on the Karman-Pohlhausen method applied to the compressible Navier-Stokes equations with variable viscosity. The resulting nonlinear ODEs for the pressure gradient are solved numerically with Maple/Matlab, and compared to the particle-based MPC simulation results. The effects of compressibility, slip, pressure-dependent viscosity parameter and geometry changes are assessed, through appropriate velocity/pressure/density curve comparisons between the MPC and the analytical results.

We use a novel implicit second-order projection method together with a superconsistent collocation scheme (Funaro, 1993; Fatone et al., 2005; De l’Isle and Owens, 2021) for the solution of the primitive variable formulation of the Navier–Stokes equations at very high Reynolds numbers. In particular, we apply a superconsistent collocation scheme to the convection–diffusion equation arising from one step of the projection method and this represents the first time that superconsistent collocation methods have been employed for the solution of a convection–diffusion equation having unsteady convection velocity. In order to evidence the second-order (in space and time) convergence of our scheme for both the velocity and pressure fields we choose to solve the two-dimensional unsteady Taylor–Green vortex problem (Taylor and Green, 1937). The numerical results presented for the solution of the two-dimensional square lid-driven cavity problem are in excellent agreement over the whole range of Reynolds numbers considered (5000$\leq{\textit{Re}}\leq$1000) with some others in the literature (Ghia et al., 1982; Wang and Liu, 2019; Bruneau and Saad, 2006; Auteri et al., 2002; Pan and Glowinski, 2000).

This work will focus on the development and analysis of data-driven system identification methods and applications in fluid dynamics. The objective is aiming to uncover the underlying pattern of the flow based on model-order reduction methods, in addition to the insights of the flow mechanics. The nature of fluid is very intricate mainly because of being nonlinear, immense in spatial-temporal dimensions, sensitive to the boundary conditions and it’s interfered by the noise of irrelevant behaviors during the data measurement. The reduced-order models achieve to identify and reconstruct the principal dynamics indicating the physics of the respective flow. In addition, they filter the noise to prevent from the over-fitting problem. The understanding of the nature of fluid dynamics bridges from the broad existence to the numerous applications in diverse fields, such as performance improvement and energy saving in manufacturing which is a priority area in Canada.

In this talk, we propose and analyze a second order accurate (in both time and space) numerical scheme for the Poisson-Nernst-Planck-Navier-Stokes (PNPNS) system, which describes the ion electrodiffusion in fluids. The marker and cell (MAC) finite difference method is taken as the spatial discretization. In the temporal discretization, the mobility function is updated with a second order accurate extrapolation formula for the sake of unique solvability, and a modified Crank-Nicolson approximation is applied to the singular logarithmic nonlinear term. Nonlinear artificial regularization terms are added in the numerical scheme, so that the positivity-preserving property could be theoretically proved, which comes from the singularity of these artificial terms. Meanwhile, a second order accurate, semi-implicit approximation is applied to the convective term in the PNP evolutionary equation, and the fluid momentum equation is similarly computed. Furthermore, an optimal rate convergence analysis is provided in this paper, in which the higher order asymptotic expansion for the numerical solution, the rough error estimate and refined error estimate techniques have to be included to accomplish such an analysis. This work combines the following theoretical properties for a second order accurate numerical scheme for the PNPNS system: (i) second order accuracy in both time and space, (ii) unique solvability and positivity, (iii) energy stability, and (iv) optimal rate convergence. A few numerical results are presented to validate the theoretical analysis.

Solid particle dissolution is an important physicochemical process occurring in many everyday scenarios, including consumption of pharmaceutical and food products, application of granular fertilizers and preparation of chemical solutions for cleaning or in a laboratory setting. One important application is the dissolution of drug particles in the body to make active pharmaceutical ingredients available for absorption for a therapeutic purpose. In this context, in vitro dissolution testing is an important procedure to gain an understanding of the rate and extent of drug release in different physiologically relevant solvents and under different hydrodynamic conditions. One tool for this purpose is the United States Pharmacopeia (USP) apparatus 4 [1], a flow-through cell in which dissolution of particulate systems in a controlled flow environment is evaluated under a range of conditions. In this study, we consider the canonical problem of dissolution of a solid initially spherical particle in a solvent. The dissolution rate and shape evolution of the particle may be influenced by mass transfer mechanisms including molecular diffusion of the dissolved drug, forced convection induced by the pumped solvent and free convection resulting from density gradients arising from drug concentration differences around the particle. Recently published work describing dissolution dominated by forced [2] and free convection [3] will be presented. The dissolution models are derived from fundamental conservation laws accounting for the role of the different mass transfer mechanisms and used to study the shape evolution of the particle over time. For the purposes of modelling dissolution of large populations of polydisperse drug particles in systems such as the apparatus 4, simpler approaches involving mass-balance arguments are typically necessary. The ability of some specific mass transfer correlations used in these models, which are based on steady flow past a sphere, to capture the overall complex dissolution behaviour in different regimes is assessed. [1] United States Pharmacopeial Convention. The United States Pharmacopeia 2011 : USP 34 ; the National Formulary : NF 29. United States Pharmacopeial Convention, 2010. [2] Assunção, M., M. Vynnycky, and K. M. Moroney. "On the dissolution of a solid spherical particle." Physics of Fluids 35.5 (2023). [3] Assunção, M., M. Vynnycky, and K. M. Moroney. "Free-convective dissolution of a solid spherical particle." Physics of Fluids 36.4 (2024).

Gas leakage is a common problem in hydrocarbon wells, occurring through fractures, fissures, and microannuli forming along cement-casing and cement-formation interfaces. The geometrical features of micro-annuli exhibit high variability due to irregularities in the operational parameters. To prevent gas leakage, a process called squeeze cementing is used, which involves pumping a thin cement slurry into the microannulus under pressure. In our previous work, we developed a numerical framework to investigate the idealized problem of radial invasion of viscoplastic fluids, representing cement slurry, into Hele-Shaw cell geometries filled with Newtonian fluid. The Hele-Shaw of this work has randomized gap thickness and represents microannulus. We further proposed a multi-perforation invasion scenario. This model allows us to model the cement invasion into a section of the wellbore and to explore the effect of different operational parameters.  This work extends our previous model by introducing two critical physical phenomena at the small scale: capillary forces and pore blockage. This work is joint with I. A. Frigaard from the University of British Columbia.

Primary cementing is the process of placing a cement sheath in the annulus between the casing and the formation in an open hole of an oil or gas well. Our group at UBC has been at the forefront of developing models for primary cementing over the past 20 years. Our current goal is to bring our models from research codes and laboratory experiments, directly into the hands of Canadian stakeholders, in the form of useable predictive software. In this talk we will describe our primary cementing models, and our predictive software development project. We model the process as the displacement of viscoplastic fluids in an inclined, non-concentric, narrow annulus. Our 2D model uses a Hele-Shaw approach to describe the displacement, and the Herschel-Bulkley model to describe the fluids. We will overview the evolution of our 2D model, including as a purely laminar, immiscible displacement, the addition of transitional and turbulent regimes, and adding dispersion to the interface to better match our 3D simulations and laboratory experiments results.

Gas dispersion in non-Newtonian fluids is widely applicable in numerous chemical and biochemical processes. However, the investigation into the effect of the power-law model constants describing the rheological behavior of shear-thinning fluids has not yet been undertaken. The utilization of computational fluid dynamics (CFD) techniques is considered a promising approach for exploring the impact of fluid rheological behavior on gas dispersion within complex fluids. In this study, a numerical model was developed to simulate the hydrodynamics of gas dispersion in non-Newtonian fluids using a coaxial mixer. Subsequently, a series of experiments was conducted to assess the mass transfer efficacy of a coaxial mixer to validate the numerical model. Various methods, including dynamic gassing-in and electrical resistance tomography (ERT), were employed to quantify mass transfer and gas hold-up profiles. The influence of fluid rheological properties, gas flow number, and rotating mode on power consumption, mass transfer coefficient, bubble size profile, and hydrodynamics was examined experimentally and numerically. A response surface model (RSM) was employed to explore the individual effects of power-law model constants on mass transfer in an aerated coaxial mixer. The RSM model utilized five levels for the consistency index (k), five levels for the flow index (n), and three levels for the gas flow number. This study revealed that the co-rotating coaxial mixer was well-suited for dispersing gas within a fluid with high consistency. In contrast, the counter-rotating mixer proved effective in enhancing gas dispersion within a fluid with a lower flow index. This phenomenon occurred because the central and anchor impellers, rotating in opposing directions, intensified the shear rate in areas distant from the central impeller, thereby reducing the stagnant fluid zones within the coaxial mixer.

A spectacular array of 20th century technologies were developed by scientists and engineers who relied on integral transform techniques. Standard practice in control theory, signal processing, and many other fields leverages the paradoxical fact that many problems are far more tractable when they are mapped out of their original context a dual representation via integral transforms. Integral transforms rule many domains, however there are other domains, such as optimization where they are not standard practice. In this talk, we describe a set of integral transform techniques that can be applied to problems that are conventionally studied through the lens of optimization. We show that integral transforms of open geometric, statistical, and physical lines of attack on optimization. We review some recent applications of this framework to problems in designing materials at the nanoscale and point to applications for systems at human scales and above.

Gradient-based methods are a cornerstone of mathematical optimization. However, strictly following the gradient to a local optimum is problematic for non-convex optimization, which has led to a zoo-full of methods that augment gradients to avoid pitfalls. In this talk we argue that information theory provides a unique way to augment gradient descent that makes minimally-biased assumptions about the structure of the solution space of optimization problems. We will show that information-theoretic augmented gradient descent is mathematically equivalent to physical systems at finite temperature. We will describe implementations of this approach that show established techniques from molecular dynamics provide powerful tools for optimization problems that bear little resemblance to molecular systems.

For a given wave number $\mu\in\mathbb{R}$ and bounded domain $\Omega \subset \mathbb{R}^2$ with boundary $M$, the Steklov-Helmholtz eigenvalue problem is to find eigenfunctions $u$ and eigenvalues $\sigma\in\mathbb{R}$ such that, \[ \begin{cases} -\Delta u -\mu^2 u = 0 \text{ in } \Omega,\\ \partial_\nu u = \sigma u \text{ on } M, \end{cases} \] where $\nu$ is the outward unit normal to $M$. We suggest a robust and exponentially converging numerical method to approximate the Steklov-Helmholtz eigenvalues for analytic domains in the plane. The robustness is with respect to the choice of the wave number $\mu_D$ (exceptional) which could correspond to a Dirichlet-Laplace eigenvalue. In this case, as $\mu^2$ approaches a Dirichlet-Laplace eigenvalue of multiplicity $\ell$ from the left, the first $\ell$ $\sigma$s approach $-\infty$. Our method involves converting the PDE into a BIE and the use of a single layer potential ansatz for $u$ involving the fundamental solution and some unknown density $\phi$. We are led to a generalized eigenvalue problem where the matrices correspond to the discretized layer potentials. In case of exceptional wave numbers $\mu_D$ we look at the SVD of the discretized single layer matrix. Using the single layer, we reconstruct the eigenfunctions $u$ from the eigendensities $\phi$. We establish a scaling property for the eigenvalues and propose a conjecture for the number of negative eigenvalues. Using the layer potentials we are also able to solve a variety of boundary and eigenvalue problems for the Laplace and Helmholtz equations. We also extend the method to include domains of genus 1. Currently, we are working on a shape optimization problem and relaxing the the regularity requirements on the boundary $M$.

The direct numerical simulation (DNS) of a fluid flow resolves all the flow structures with a properly refined mesh. When the convection dominates the dynamics, as in numerous applications, very fine meshes are required, making DNS computationally unaffordable for practical purposes. A possible way to reduce the computational cost without sacrificing accuracy is to solve for the flow average using a mesh coarser than that required by DNS and introduce a model for the effects not captured by the flow average alone. Among such alternatives to DNS, Large Eddy Simulation (LES) is one of the most widely used for practical applications in science and engineering. We will discuss an LES appraoch that applies a differential nonlinear low-pass filter to the (non-physical) solution computed with the fluid dynamics model and a coarse mesh. Since the nonlinear filter is added sequentially to the fluid dynamics model, the implementation of this LES model does not require any major modification of a legacy solver. Through applications to the incompressible Navier-Stokes equations, weakly compressible Euler equations, and the quasi-geostrophic equations, we will demostrate the efficacy and efficiency of our LES approach.

We present an informal outline of how ChatGPT works and describe the underlying mathematical concepts behind it. For instance, graph theory, vector calculus, linear algebra, Markov models, and probabilistic inference are all used in a fundamental way. The talk will not be too technical and is suitable for a general audience.