Past Junior Applied Mathematics Seminar

19 October 2021
12:30
Abstract

Many problems in engineering can be understood as controlling the bifurcation structure of a given device. For example, one may wish to delay the onset of instability, or bring forward a bifurcation to enable rapid switching between states. In this talk, we will describe a numerical technique for controlling the bifurcation diagram of a nonlinear partial differential equation by varying the shape of the domain. Our aim is to delay or advance a given branch point to a target parameter value. The algorithm consists of solving a shape optimization problem constrained by an augmented system of equations, called the Moore–Spence system, that characterize the location of the branch points. We will demonstrate the effectiveness of this technique on several numerical experiments on the Allen–Cahn, Navier–Stokes, and hyperelasticity equations.

  • Junior Applied Mathematics Seminar
18 June 2021
13:00
Abstract

Tendon tissue engineering aims to grow functional tissue in the lab. Tissue is grown inside a bioreactor which controls both the mechanical and biochemical environment. As tendon cells alter their behaviour in response to mechanical stresses, designing suitable bioreactor loading regimes forms a key component in ensuring healthy tissue growth.  

Linking the forces imposed by the bioreactor to the shear stress experienced by individual cell is achieved by homogenisation using multiscale asymptotics. We will present a continuum model capturing fluid-structure interaction between the nutrient media and the fibrous scaffold where cells grow. Solutions reflecting different experimental conditions will be discussed in view of the implications for shear stress distribution experienced by cells across the bioreactor.  

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  • Junior Applied Mathematics Seminar
1 June 2021
12:45
Abstract

Neural controlled differential equations (Neural CDEs) are a continuous-time extension of recurrent neural networks (RNNs). They are considered SOTA for modelling functions on irregular time series, outperforming other ODE benchmarks (ODE-RNN, GRU-ODE-Bayes) in offline prediction tasks. However, current implementations are not suitable to be used in online prediction tasks, severely restricting the domains of applicability of this powerful modeling framework. We identify such limitations with previous implementations and show how said limitations may be addressed, most notably to allow for online predictions. We benchmark our online Neural CDE model on three continuous monitoring tasks from the MIMIC-IV ICU database, demonstrating improved performance on two of the three tasks against state-of-the-art (SOTA) non-ODE benchmarks, and improved performance on all tasks against our ODE benchmark.

 

Joint work with Patrick Kidger, Lingyi Yang, and Terry Lyons.

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  • Junior Applied Mathematics Seminar
4 May 2021
12:45
Abstract

The index of a saddle point of a smooth function is the number of descending directions of the saddle. While the index can usually be retrieved by counting the number of negative eigenvalues of the Hessian at the critical point, we may not have the luxury of having second derivatives in data deriving from practical applications. To address this problem, we develop a computational pipeline for estimating the index of a non-degenerate saddle point without explicitly computing the Hessian. In our framework, we only require a sufficiently dense sample of level sets of the function near the saddle point. Using techniques in Morse theory and Topological Data Analysis, we show how the shape of saddle points can help us infer the index of the saddle. Furthermore, we derive an explicit upper bound on the density of point samples necessary for inferring the index depending on the curvature of level sets. 

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  • Junior Applied Mathematics Seminar
9 March 2021
12:45
Edwina Yeo
Abstract

Coronary heart disease is characterised by the formation of plaque on artery walls, restricting blood flow. If a plaque deposit ruptures, blood clot formation (thrombosis) rapidly occurs with the potential to fatally occlude the vessel within minutes. Von Willebrand Factor (vWF) is a shear-sensitive protein which has a critical role in blood clot formation in arteries. At the high shear rates typical in arterial constrictions (stenoses), vWF undergoes a conformation change, unfolding and exposing binding sites and facilitating rapid platelet deposition. 

To understand the effect of  stenosis geometry and blood flow conditions on the unfolding of vWF and subsequent platelet binding, we developed a continuum model for the initiation of thrombus formation by vWF in an idealised arterial stenosis. In this talk I will discuss modelling proteins in flow using viscoelastic fluid models, the insight asymptotic reductions can offer into this complex system and some of the challenges of studying fast arterial blood flows. 

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  • Junior Applied Mathematics Seminar
9 February 2021
12:45
Abstract

Topological data analysis is a growing area of research where topology and geometry meets data analysis. Many data science problems have a geometric flavor, and thus computational tools like persistent homology and Mapper were often found to be useful. Domains of applications include cosmology, material science, diabetes and cancer research. We will discuss some main tools of the field and some prominent applications.

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  • Junior Applied Mathematics Seminar
26 January 2021
12:45
Abstract

Diffusion processes are widely used to model the evolution of random values over time. In many applications, the diffusion process is constrained to a finite domain. We consider the estimation problem of a diffusion process constrained by a polytope, i.e. intersection of finitely many (hyper-)planes, given a discretely observed time series data. Since the boundary behaviours of a diffusion process are characterised by its drift and diffusion functions, we derive sufficient conditions on the drift and diffusion functions for the nonattainablity of a polytope. We use deep learning to estimate the drift and diffusion, and ensure that their constraints are satisfied throughout the training.

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  • Junior Applied Mathematics Seminar
1 December 2020
12:45
Deborah Sulem

Further Information: 

The Junior Applied Mathematics Seminar is intended for students and early career researchers.

Abstract

Multivariate point processes are used to model event-type data in a wide range of domains. One interesting application is to model the emission of electric impulses of biological neurons. In this context, the point process model needs to capture the time-dependencies and interactions between neurons, which can be of two kinds: exciting or inhibiting. Estimating these interactions, and in particular the functional connectivity of the neurons are problems that have gained a lot of attention recently. The general nonlinear Hawkes process is a powerful model for events occurring at multiple locations in interaction. Although there is an extensive literature on the analysis of the linear model, the probabilistic and statistical properties of the nonlinear model are still mainly unknown. In this paper, we consider nonlinear Hawkes models and, in a Bayesian nonparametric inference framework, derive concentration rates for the posterior distribution.  We also infer the graph of interactions between the dimensions of the process and prove that the posterior distribution is consistent on the graph adjacency matrix.

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  • Junior Applied Mathematics Seminar
17 November 2020
12:45
Karel Devriendt
Abstract

What do random spanning trees, graph embeddings, random walks, simplices and graph curvature have in common? As you may have guessed from the title, they are indeed all intimately connected to the effective resistance on graphs! While originally invented as a tool to study electrical circuits, the effective resistance has proven time and again to be a graph characteristic with a variety of interesting and often surprising properties. Starting from a number of equivalent but complementary definitions of the effective resistance, we will take a stroll through some classical theorems (Rayleigh monotonicity, Foster's theorem), a few modern results (Klein's metricity, Fiedler's graph-simplex correspondence) and finally discuss number of recent developments (variance on graphs, discrete curvature and graph embeddings).

 

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  • Junior Applied Mathematics Seminar
3 November 2020
12:45
Michael Coughlan
Abstract

Michael Coughlan (with Sam Howison, Ian Hewitt, Andrew Wells)

Arctic sea ice forms a thin but significant layer at the ocean surface, mediating key climate feedbacks. During summer, surface melting produces considerable volumes of water, which collect on the ice surface in ponds. These ponds have long been suggested as a contributing factor to the discrepancy between observed and predicted sea ice extent. When viewed at large scales ponds have a complicated, approximately fractal geometry and vary in area from tens to thousands of square meters. Increases in pond depth and area lead to further increases in heat absorption and overall melting, contributing to the ice-albedo feedback. 

Previous modelling work has focussed either on the physics of individual ponds or on the statistical behaviour of systems of ponds. In this talk I present a physically-based network model for systems of ponds which accounts for both the individual and collective behaviour of ponds. Each pond initially occupies a distinct catchment basin and evolves according to a mass-conserving differential equation representing the melting dynamics for bare and water-covered ice. Ponds can later connect together to form a network with fluxes of water between catchment areas, constrained by the ice topography and pond water levels. 

I use the model to explore how the evolution of pond area and hence melting depends on the governing parameters, and to explore how the connections between ponds develop over the melt season. Comparisons with observations are made to demonstrate the ways in which the model qualitatively replicates properties of pond systems, including fractal dimension of pond areas and two distinct regimes of pond complexity that are observed during their development cycle. 

Different perimeter-area relationships exist for ponds in the two regimes. The model replicates these relationships and exhibits a percolation transition around the transition between these regimes, a facet of pond behaviour suggested by previous studies. The results reinforce the findings of these studies on percolation thresholds in pond systems and further allow us to constrain pond coverage at this threshold - an important quantity in measuring the scale and effects of the ice-albedo feedback.

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  • Junior Applied Mathematics Seminar

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