Applied Mathematics Seminars in 2018

Abstract: Deep feedforward neural networks (DFNNs) are a powerful tool for functional approximation. We describe flexible versions of generalized linear and generalized linear mixed models incorporating basis functions formed by a DFNN. Efficient computational methods for high-dimensional Bayesian inference are developed using Gaussian variational approximation, with a parsimonious but flexible factor parametrization of the covariance matrix. We implement natural gradient methods for the optimization, exploiting the factor structure of the variational covariance matrix in computation of the natural gradient. Our flexible DFNN models and Bayesian inference approach lead to a regression and classification method that has a high prediction accuracy, and is able to quantify the prediction uncertainty in a principled and convenient way. We also describe how to perform variable selection in our deep learning method. The proposed methods are illustrated in a wide range of simulated and real-data examples, and the results compare favourably to a state of the art flexible regression and classification method in the statistical literature, the Bayesian additive regression trees (BART) method. User-friendly software packages in Matlab and R implementing the proposed methods are available at https://github.com/VBayesLab.

Abstract: Where do you place pollutant capture units? When objects move through heterogeneous flow environments, such as oceanic micro-plastics, the answer is not obvious. We formulate flow capture problems, involving flows and sinks, and, using dynamical systems techniques, show that blindly positioning capture units carries high risk of failure. Capture efficiency depends on capture rate: long-term efficiency decreases as the number of capture units increases, whereas short-term efficiency increases. Doubling numbers of capture units can more than double the capture rate. The formal description of flow capture problems will impact engineering solutions ranging from atmospheric CO2 capture to oceanic micro-plastic pollution.

Abstract: Robustness, and the ability to function and thrive amid changing and unfavourable environments, is a fundamental requirement for all living systems. Moreover, it has been a long-standing mystery how the extraordinarily complex communication networks inside living cells, comprising thousands of different interacting molecules, are able to exhibit such remarkable robustness since complexity is generally associated with fragility. In this talk I will give an overview of our recent research on robustness in cellular signalling networks, with an emphasis on the robust functionality known as Robust Perfect Adaptation (RPA). This work is now published in Nature Communications, and is available here: https://rdcu.be/M46K. This work has suggested a resolution to the complexity-robustness paradox through the discovery that robust adaptive signalling networks must be decomposable into topological basis modules of just two possible types. This newly-discovered modularisation of complex bionetworks has important implications for evolutionary biology, embryology and development, cancer research and drug development.

Abstract: We present a matched asymptotics framework for constructing and analysing the stability of localised patterns that arise in singularly perturbed activator-inhibitor reaction-diffusion systems. In two spatial dimensions, by way of analyses of nonlocal eigenvalue problems, we resolve two long-standing problems regarding 1) the stability of spot patterns to oscillatory instabilities, and 2) the stability of stripe patterns to break-up instabilities, the latter motivated by the persistence of striped vegetation patterns on steep hillsides. In three spatial dimensions, we calculate explicit stability thresholds for self-replication and annihilation of spots, and derive a gradient flow that governs their slow dynamics. Joint work with Theodore Kolokolnikov, Michael J. Ward, and Shuangquan Xie.

Abstract: We will consider the inverse problem of determining the sound speed or index of refraction of a medium by measuring the travel times of waves going through the medium. This problem arises in global seismology in an attempt to determine the inner structure of the Earth by measuring travel times of earthquakes. It has also several applications in optics and medical imaging among others.

The problem can be recast as a geometric problem: Can one determine the Riemannian metric of a Riemannian manifold with boundary by measuring the distance function between boundary points? This is the boundary rigidity problem. We will also consider the problem of determining the metric from the scattering relation, the so-called lens rigidity problem. The linearization of these problems involve the integration of a tensor along geodesics, similar to the X-ray transform.

We will also describe some recent results, join with Plamen Stefanov and Andras Vasy, on the partial data case, where you are making measurements on a subset of the boundary. No previous knowledge of Riemannian geometry will be assumed.

Abstract: After a short introduction to the mechanism and politics of fracking, the talk will concentrate on the fluid mechanics and elastodynamics of driving fluid into cracks and the quite different response when the pressure is released and the fluid flows back out. Development of the governing equations will be presented along with their numerical solution and asymptotic analysis in certain useful limits. Videos of laboratory experiments will be shown and the results compared with the theoretical predictions.

Abstract: In classic van der Pol-type oscillator theory, a relaxation cycle consists of two slow and two fast orbit segments per period (slow-fast-slow-fast). A possible alternative relaxation oscillator type consists of one slow and one fast segment only. In electrical circuit theory, Le Corbeiller (published in IEEE 1960) termed this type a two-stroke oscillator (compared to the four-stroke vdP oscillator). I will provide examples of two-stroke relaxation oscillators and discuss these problems from a geometric singular perturbation theory point of view "beyond the standard form". It is worth mentioning that Fenichel's seminal work on geometric singular perturbation theory (published in JDE 1979) discusses this more general setting, but it has not received much attention in the literature.

Abstract: Rogue periodic waves stand for gigantic waves on a periodic background. The nonlinear Schrodinger equation in the focusing case admits two families of periodic wave solutions expressed by the Jacobian elliptic functions dn and cn. Both periodic waves are modulationally unstable with respect to long-wave perturbations. Exact solutions for the rogue periodic waves are constructed by using the explicit expressions for the periodic eigenfunctions of the Zakharov–Shabat spectral problem and the Darboux transformations. These exact solutions generalize the classical rogue wave (the so-called Peregrine’s breather). Computations of rogue periodic waves rely on properties of the nonlinear Schrodinger equation due to its integrability.