AC2MB Seminar: Fall 2016

## Applied and Computational Mathematics

and

Mathematical Biology seminar

**Time:**Wednesdays at 2:15pm

**Location:**CM.T01

**12.10, 2:15pm in CM.T01**

__Sinking bubbles, climbing drops, and parachuting seeds__

CATHAL CUMMINS (UoE)

__Abstract:__*In this talk I consider three problems that arise in fluid mechanics: Sinking bubbles (Computational Mathematics), climbing drops (Mathematical Analysis), and parachuting seeds (Mathematical Biology).*

**Sinking bubbles:**When a pint of Guinness is freshly poured, there is a wait of about two minutes until the pint is fully "settled". In this time, the Guinness bubbles can be seen sinking along the sides of the glass. This phenomenon has been known for a long time, but what caused these bubbles to sink remained a mystery. In this talk, I will present a simple model that reveals the mechanism that drives bubbles downward in a pint of Guinness.

**Climbing drops:**A liquid drop deposited on a substrate inclined to the horizontal will slide down under the action of gravity, provided it is not pinned due to contact angle hysteresis. If the substrate is harmonically shaken in the vertical direction, intuition would suggest that such a drop will continue to slide down the substrate. In a set of remarkable experiments, however, researchers have found that such drops can actually climb up the substrate, against gravity. In this second part, I will present an asymptotic analysis of a three-dimensional model of a liquid drop on an oscillating substrate, in the limit of small and weak vibration. We find that drops can climb uphill due to the non-linear interaction of two orthogonal vibration modes.

**Parachuting fruit:**The fluid mechanical principles that allow a passenger jet to lift off the ground are not applicable to the flight of small plant fruit (the seed-bearing structure in flowering plants). The reason for this is scaling: human flight requires very large Reynolds numbers, while plant fruit have comparatively small Reynolds numbers. At this small scale, there are a variety of modes of flight available to fruit: from parachuting to gilding and autorotation. In this final part, I will focus on the aerodynamics of small plumed fruit that utilise the parachuting mode of flight. If a parachute-type fruit is picked up by the breeze, it can be carried over formidable distances. Incredibly, these parachutes are mostly empty space; making this an extremely efficient mode of transport. Moreover, the fruit can become more or less streamlined depending on the environmental conditions; in this way, they behave as a smart technology. I will discuss some of the research that we are conducting to reveal the fruit's underlying flight mechanism.

**19.10, 2:15pm in CM.T01**

__Spatial disorder in physiological transport models__

OLIVER JENSEN (U. MANCHESTER)

__Abstract:__*Exchange organs such as the lung and placenta exhibit a high degree of spatial disorder at many different spatial scales, both within and between individuals. This presents considerable modelling challenges, particularly if computational models are to be of clinical utility. I will describe two simple systems where methods of uncertainty quantification have been applied to physiological transport processes, illustrating how asymptotic techniques can be used to predict the variance of outcomes. The first problem, loosely motivated by maternal blood flow in the human placenta, concerns the strengths and limitations of homogenization in describing the transport of a solute over a spatially disordered array of sinks (https://arxiv.org/abs/1606.05600); the second problem concerns the interaction between an inhaled aerosol drop and a spatially heterogeneous layer of mucus lining an airway, where the disorder evolves dynamically as the drop spreads (http://dx.doi.org/10.1098/rspa.2016.0270).*

**26.10, 2:15pm in CM.T01**

__Estimation of sparse precision matrices in paired gene expression data__

NATALIA BOCHKINA (UoE)

__Abstract:__*We consider the problem of joint estimation of two similar sparse precision matrices and the corresponding conditional dependence graphs for high dimensional data where observations of these matrices are dependent. We propose a new method to estimate simultaneously these precision matrices, a weighted fused graphical lasso estimator which encourages both sparsity and similarity in the estimated precision matrices. The tuning parameters controlling sparsity of the matrices are automatically selected by controlling the estimated expected number of false positive edges, and the penalty term controlling similarity of the matrices is weighted for every pair of variables to account for linear dependence between datasets. We observe overestimation of triangular motifs in the corresponding conditional dependence graphs, that are common to other fused graphical lasso methods, so we incorporate an additional step to remove such edges. We conduct a simulation study to show that the proposed methodology recovers the true conditional dependence graphs well for different types of networks and different combinations of sample size and dimension. We apply the suggested approaches to high-dimensional case studies of gene expression data with samples in two medical conditions, non-lesional and psoriasis lesional tissues (first dataset) as well as healthy and lung cancer tissues (second dataset), to estimate common networks of genes and also the differentially connected genes that interact differently in the two types of tissues. In both cases the data is paired, as both types of tissues are taken from the same individuals. Our findings indicate denser graph structures for lesional (and tumor) samples than for healthy samples, with subgroups of genes interacting together.*

This is joint work with Adria Caballe (University of Edinburgh) and Claus Mayer (Biomathematics and Statistics Scotland).

**31.10., 2:00pm in CM.T01 CHANGES: TIME and DAY**

__Certainty and Assumptions__

ALISTAIR B FORBES (National Physical Laboratory)

__Abstract:__*Abraham Maslow: If the only tool we have is a hammer, all problems tend to look like nails. George Box: All models are wrong, but some models are useful. Donald Rumsfeld: There are known knowns and known unknowns. But there are also unknown unknowns. Theodore Micceri: The Unicorn, The Normal Curve and Other Improbable Creatures. The National Physical Laboratory is the UKâs National Metrology Institute (NMI), responsible for ensuring that all measurement in the UK can be made traceable to standard units, e.g. to the metre, etc. A measurement result is traceable only if it is associated with an uncertainty statement. All uncertainty statements are derived from an underlying model of the measurement system describing the functional relationship between the various variables and the statistical characterisation associated with the measurements. The Guide to the Expression of Uncertainty in Measurement (the GUM) provides a methodology for evaluating measurement uncertainty. It starts with an input-output model Y = f(X) in which the measurand Y is expressed as a function of influence quantities X. Once a probability distribution p(X) is assigned to X, the distribution p(Y) for Y is defined. Uncertainties associated with X are propagated forward through the model to determine the uncertainty associated with Y and there are straightforward computational tools for performing this type of uncertainty propagation. The current uncertainty propagation methodologies used by NMIs assume i) the input-output model Y = f(X) exists, ii) that the function f is known with certainty and iii) the distributions associated to X are known with certainty. In practice, all three assumptions can be challenged. Measurement problems naturally arise as inverse problems and, in order to apply the uncertainty propagation tools, have to be reformulated to look like forward problems. Models are often approximations or have empirical components that try to account for our lack of complete knowledge of the underlying physical system. Distributions are often assigned on the basis of assumptions of normality, independence and expert judgment. In this talk, I will discuss some of the issues associated with distributional assumptions and how we can use hierarchical models to arrive at more comprehensive uncertainty statements that are based on a more realistic assessment of what we know, what we donât know or even what we donât know we donât know.*

Alistair Forbes joined the National Physical Laboratory in 1985 after studying mathematics at the universities of Aberdeen, Newcastle upon Tyne and Pennsylvania. He is a Fellow in the Mathematics and Modelling Group and Science Area Leader for Data Science and Uncertainty Quantification. He is a Chartered Mathematician, a fellow of the Royal Statistical Society and Visiting Professor at the University of Huddersfield.

**02.11., 2:15pm in CM.T01**

__Atomistic and Multi-Scale Materials Modelling__

CHRISTOPH ORTNER (U. WARWICK)

__Abstract:__*In this talk, I describe some ideas related to multi-scale materials simulation at the atomistic scale, taking into account both chemistry (electronic structure) and mechanics (lattice elasticity), from an analysis and numerical analysis perspective. The accepted ``correct'' model for most use cases is Kohn-Sham density functional theory (DFT). Due to its cubic scaling cost, it cannot be employed to describe complex material behavior such as dislocation dynamics or crack propagation. Instead, users employ interatomic potentials (computational cheap but low accuracy), or more recently multi-scale methods that try to combine the two models to better balance the cost/accuracy ratio. Such multi-scale techniques are playing an increasing role in a wide range of sciences and engineering disciplines; see in particular Nobel Prize in Chemistry 2013. The implicit premise of molecular simulation and of virtually all multi-scale methods is locality and separability of the potential energy. A classical, and well-understood example of locality is locality of the density matrix in insulators. But for multi-scale modelling mechanical response we need even stronger results. In this talk, I will show in detail how the required locality of forces and potential energy arises in some simple electronic structure models. I will then exploit these results in the construction of new interatomic potentials and QM/MM multi-scale algorithms with rigorous rates of convergence in terms of the QM region size.*

**09.11, 2:15pm in CM.T01**

__Complex behaviour of interfacial flows: Dynamics of fattening/thinning sessile droplets and falling liquid films__

MARK PRADAS (OPEN UNIV.)

__Abstract:__*Interfacial flows are found in a vast spectrum of natural and engineered systems and they are often characterised by the presence of a wide range of different scales which are nonlinearly interacting with each other. As a result, these systems may exhibit complex behaviour and generic features, such as collective motion and self-organisation processes; or hysteresis and stick-slip dynamics. In this talk I will show two different problems exhibiting these types of behaviour: droplets with a time-dependent volume spreading on chemically disordered substrates, and falling liquid films, which is an example of what is known as interfacial turbulence. We have developed a number of novel methodologies for the study of these systems which I will outline in the talk.*

**2:15pm in CM.T01 Moved to next semester: Feb. 8, 2017**

__TITLE__

MARIA BRUNA (OXFORD)

__Abstract:__*TBA*

JOINT WORK?

**30.11., 3:00pm, Maxwell Institute Applied Maths Seminar at ICMS**

(jointly with ACM seminar, UoE)

__Complex dynamics in multiscale systems__

SERAFIM KALLIADASIS (IMPERIAL COLLEGE)

__Abstract:__*Understanding the evolution of complex multiscale systems is crucial from a fundamental but also the applications' point of view. For instance, many engineering systems are complex and multiscale and understanding their dynamics has the potential to predict a specific system's behavior, engineer its design and build-in response to arrive at a highly optimal and robust system. We combine elements from homogenization theory, nonlinear science, statistical physics, critical phenomena and information theory to develop a number of novel and generic methodologies that enable us to undertake the rigorous and systematic study of the emergence of complex behavior in multiscale systems. The methodologies are exemplified with paradigmatic prototypes from different classes of complex systems such as interface dynamics in disordered media, convectively unstable open flows and stochastic multiscale phenomena in noisy spatially extended systems and diffusion in multiscale potentials.*

[Joint work with Marc Pradas (Open), Markus Schmuck (Heriot-Watt), Dmitri Tseluiko (Loughborough), Grigorios A. Pavliotis (Imperial) and Andrew Duncan (Sussex)]

**3:50pm Coffee break**

**4:10pm 2nd talk at ICMS**

__Computing physically relevant solutions in Nonlinear PDEs and models across scales__

CHARALAMBOS MAKRIDAKIS (Sussex)

__Abstract:__*The computation of singular phenomena (shocks, defects, dislocations, interfaces, cracks) arises in many complex systems. For computing such phenomena, it is natural to seek methods that are able to detect them and to devote the necessary computational recourses to their accurate resolution. Often weak solutions of PDEs related to these problems are not unique. Since numerical methods perturb the mathematical model, mathematical analysis emerges as a necessary tool providing mathematical guarantees ensuring that our computational methods approximate physically relevant solutions. Our purpose in this talk is to review results and discuss related computational and analytical challenges for such nonlinear problems modelled by PDEs. In addition we shall discuss related issues emerging in adaptive modelling across scales.*

**5:00pm Wine reception**

**07.12., 2:15pm in CM.T01 (HWU)**

__Three-field formulation leading to locking-free finite element discretisations of poroelasticity__

RICARDO RUIZ BAIER (OXFORD)

__Abstract:__*In this talk we present a stable and convergent conforming finite element method for the discretisation of the linear poroelasticity equations in a new formulation, where the volumetric contributions to the total stress are merged into an additional unknown. The resulting saddle point formulation can be analysed by means of a Fredholm alternative, after realizing that the problem is a compact perturbation of a Stokes-like invertible system. A generic Galerkin scheme is constructed, whose solvability properties follow closely those from the continuous variational form, and more importantly, given that specific finite dimensional spaces are chosen adequately, it is stable even in the incompressible limit.*