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2017/18 semester 2 events in Mathematics

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Mon 15/01/18 14:00
Fulton G20
Mathematics Seminar
Dr. Kostas Zygalakis (University of Edinburgh)
Ergodic Stochastic Differential Equations and Sampling: A numerical analysis perspective
abstract

Abstract

Understanding the long time behaviour of solutions to ergodic stochastic differential equations is an important question with relevance in many field of applied mathematics and statistics. Hence, designing appropriate numerical algorithms that are able to capture such behaviour correctly is extremely important. A recently introduced framework [1,2,3] using backward error analysis allows us to characterise the bias with which one approximates the invariant measure (in the absence of the accept/reject correction). These ideas will be used to design numerical methods exploiting the variance reduction of recently introduced nonreversible Langevin samplers [4,5]. Finally if there is time we will discuss, how things ideas can be combined with the idea of Multilevel Monte Carlo [6] to produce unbiased estimates of ergodic averages without the need the of an accept-reject correction [7] and optimal computational cost.

[1] K.C. Zygalakis. On the existence and applications of modified equations for stochastic differential equations. SIAM J. Sci. Comput., 33:102-130, 2011.
[2] A. Abdulle, G. Vilmart, and K. C. Zygalakis. High order numerical approximation of the invariant measure of ergodic sdes. SIAM J. Numer. Anal., 52(4):1600-1622, 2014.
[3] A. Abdulle, G. Vilmart, and K.C. Zygalakis, Long time accuracy of Lie-Trotter splitting methods for Langevin dynamics. SIAM J. Numer. Anal., 53(1):1-16, 2015.
[4] A. Duncan, G. A. Pavliotis and T. Lelievre, Variance Reduction using Nonreversible Langevin Samplers, J. Stat. Phys., 163(3):457-491, 2016.
[5] A. Duncan, G. A. Pavliotis and K. C. Zygalakis, Nonreversible Langevin Samplers: Splitting Schemes, Analysis and Implementation, arXiv:1701.04247.
[6] M.B. Giles, Mutlilevel Monte Carlo methods, Acta Numerica, 24:259-328, 2015.
[7] L. Szpruch, S. Vollmer, K. C. Zygalakis and M. B. Giles, Multi Level Monte Carlo methods for a class of ergodic stochastic differential equations. arXiv:1605.01384.

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Mon 22/01/18 14:00
Fulton G20
Mathematics Seminar
Prof. Georgios Akrivis (University of Ioannina)
Stability of implicit and implicit–explicit multistep methods for nonlinear parabolic equations in Hilbert spaces
abstract

Abstract

We consider the discretization of a class of nonlinear parabolic equations in Hilbert spaces by both implicit and implicit–explicit multistep methods, and establish local stability under best possible and best possible linear stability conditions, respectively. Our approach is based on a spectral and Fourier stability technique and uses a suitable quantification of the non-self-adjointness of linear elliptic operators as well as a discrete perturbation argument

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Mon 29/01/18 14:00
Fulton G20
Mathematics Seminar
Dr. Peter Stewart (University of Galsgow)
The fluid mechanics of the optic nerve
abstract

Abstract

The optic nerve is formed by a dense collection of nerve fibres which connect the photoreceptors in the retina to the thalamus in the brain. This nerve is surrounded by a sheath, which contains a thin layer of cerebrospinal fluid (CSF) at the intracranial pressure. The central retinal artery and vein, which supply the retinal circulation, pass through the centre of the optic nerve as they enter the eye, but about half way back from the globe they deviate and pass through the nerve sheath into the surrounding fatty tissue. Hence, these blood vessels form an interesting point of coupling between the eye and the brain. In this talk I will show how modelling of the flow of CSF along the nerve sheath and the flow of blood in the central retinal artery and vein can be used to quantify the condition of the brain, suggesting a non-invasive method for estimating the intracranial pressure and providing a methodology for quantifying traumatic brain injuries. In particular, I will demonstrate how a large increase in CSF pressure is transmitted into the retinal artery and vein, leading to a spreading shock wave through the retinal circulation and the possible rupture of retinal blood vessels (ie retinal haemorrhages).

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Mon 05/02/18 15:00
Fulton G20
Mathematics Seminar
Dr. Sergey Anfinogentov (University of Warwick)
Analysis of low-amplitude oscillations in solar corona by motion magnification
abstract

Abstract

Coronal loops are the basic building elements of solar corona, appearing in EUV images as bright elongated structures apparently oriented along the magnetic field lines. Being a plasma non-uniformity, a coronal loop can act as a waveguide for MHD waves including standing kink oscillations. These oscillations resemble vibration of a guitar string. MHD-seismology can use such a vibrating coronal loop as a source of information about the coronal magnetic field that controls almost all processes in the corona, including the accumulation of energy and its release during flares and coronal mass ejections. Up to the present time, observations of decaying kink oscillations of coronal loops have been relatively rare. These events are mainly associated with solar flares and CMEs, and hence occur occasionally. Therefore, they are not suitable for the routine seismological diagnostics. The recent discovery of the decay-less regime of kink oscillations has changed the situation. Decay-less kink oscillations are a ubiquitous phenomenon and can be observed in almost any active region. Hence, they can be used as a seismological tool for routine diagnostic of the coronal magnetic field. Unfortunately, analysis of these oscillations is a challenging task because of their small displacements amplitude, ~0.2 Mm, that is less than the pixel size of modern EUV imaging instruments like SDO/AIA. We present a technique that solves this problem. Motion magnification is a method for artificial magnification of low-amplitude quasi-periodic transverse displacements in the image sequence (imaging data cubes or videos). It acts like a magnifying glass for low amplitude transverse motions, making them much better visible in animations and time-distance plots. This approach is found to give good results for harmonic oscillations, including the modulated signals, such as exponentially decaying, multi-modal and frequency modulated signals. We applied the motion magnification to the EUV observations of several non-flaring active regions, clearly demonstrating the presence of low-amplitude decay-less oscillations in the majority of coronal loops. At least for one case, we also revealed the presence of multiple harmonics. Our implementation of the motion magnification algorithm is written in Python 3 and is freely available at https://github.com/Sergey-Anfinogentov/motion_magnification. Though, the code is designed to work with EUV imaging observations of the Sun, it can be applied to any time sequence of images, i.e. an imaging data cube or video.

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Mon 12/02/18 14:00
Fulton G20
Mathematics Seminar
Dr. Stephen Metcalfe (University of York)
Approximating Blow-up Via A Posteriori Error Estimation
abstract

Abstract

The numerical approximation of blow-up (singularities which develop in solutions to certain differential equations) is a difficult problem due to the high temporal and spatial resolution needed close to the singularity; this necessitates the design of special numerical methods for this class of problem. Indeed, many such numerical methods have been designed but they often make quite restrictive assumptions. Additionally, there is no rigorous way of knowing whether the resulting computations from existing methods are quantitatively reasonable. In this talk, I will discuss recent developments in an alternative approach to the approximation of such problems which is based on a posteriori error estimation. In principle, providing such error bounds are well-behaved close to the singularity, this will allow us to select the computational parameters (time steps, mesh sizes) in our numerical method as well as provide the secondary advantage of giving a quantitative bound on the size of the error.

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Mon 19/02/18 15:00
Fulton G20
Mathematics Seminar
Prof. John Mason (The Open University)
What makes an example exemplary for students?
Mon 26/02/18 15:00
Fulton G20
Mathematics Seminar
Prof. Duncan Mackay (University of St Andrews)
Helicity condensation and the solar cycle
abstract

Abstract

Solar filaments exhibit a global chirality pattern that is closely related to the injection and transport of magnetic helicity across the Sun. Dextral/sinistral filaments, corresponding to negative/positive magnetic helicity dominant in the northern/southern hemisphere. This pattern is inconsistent with the sign of magnetic helicity injected by differential rotation along East-West orientated polarity inversion lines. To investigate the origin of this hemispheric pattern in realistic magnetic field configurations, a global evolution model along with a large-scale representation of helicity condensation is applied. Periods in both the rising and declining-phase of the solar cycle are simulated. In the helicity condensation model positive/negative values of the vorticity of the convective cells is used in the northern/southern hemisphere to inject negative/positive helicity. The magnitude of the vorticity is then varied as a free parameter. To reproduce the correct percentage relationship between the dominant and minority filament chirality in each hemisphere, a vorticity of magnitude 2. 5e−6 s−1 is required. This rate is however insufficient to produce the 16 correct unimodal profile of chirality with latitude. To achieve this, a vorticity of 5e−6 s−1 or higher is needed. This places a lower limit on the magnitude of the vorticity in the helicity condensation model, such that it can dominate over differential rotation and reproduce the observed hemispheric pattern. Future observational studies need to establish if such a consistent vorticity pattern of this magnitude exists within convective cells across all latitudes of the Sun.

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Mon 12/03/18 15:00
Fulton G20
Mathematics Seminar
Dr. Thomas Elsden (University of St Andrews)
3D Alfven Resonances in Earth's Magnetosphere
abstract

Abstract

Low frequency periodic perturbations to the Earth's background magnetic field, known as ultra low frequency (ULF) waves, have been observed on the ground and in space for many decades. With a large scale length and low frequency, these waves can be effectively studied using magnetohydrodynamics (MHD). In this talk I'll discuss some of our recent numerical work on the resonant coupling of two MHD wave modes, namely fast-Alfven wave coupling, and put this in context in terms of waves in Earth's magnetosphere.

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Mon 19/03/18 15:00
Fulton G20
Mathematics Seminar
Prof. Andrew Gilbert (University of Exeter)
Fast Stretching of Paired Vortex Filaments in Axisymmetric and General Geometry
abstract

Abstract

Coherent vortices are a component of many high Reynolds number flows including turbulence. A fundamental question is how quickly vortex stretching can occur, linked to the thorny question of whether Euler or Navier--Stokes flows can become singular in a finite time, or whether these remain well-behaved indefinitely. I will first discuss work with Steve Childress on vortex stretching in axisymmetric geometry --- collision of two vortex rings --- where stretching is accelerated giving a power law growth. Then, I will outline work on extending these calculations to a vortex dipole pair with arbitrary centre line. In simplified models similarity solutions can be obtained that become singular in a finite time, but these neglect axial flow, which complicates any analysis and may suppress this behaviour. (With Steve Childress, New York University).

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Wed 21/03/18 15:00
Fulton G20
Mathematics Seminar
Prof. Thomas Hillen (University of Alberta)
Non-local Models for Cellular Adhesion
abstract

Abstract

Cellular adhesion is one of the most important interaction force between cells and other tissue components. In 2006, Armstrong, Painter and Sherratt introduced a non-local PDE model for cellular adhesion, which was able to describe known experimental results on cell sorting and cancer growth. While the numerical implementation leads to nice results, the analysis of this non-local model is challenging. In this talk I will present a random walk derivation of the Armstrong-Painter-Sherratt adhesion model, I will discuss local and global existence of solutions, and the underlying bifurcation structure of steady states. (joint work with A. Buttenschoen, K. Painter, A. Gerisch, M. Winkler).

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Mon 26/03/18 15:00
Fulton G20
Mathematics Seminar
Prof. Miguel Onorato (Università di Torino)
New developments in the old Femi-Pasta-Ulam-Tsingou problem
abstract

Abstract

In the early fifties E. Fermi in collaboration with J. Pasta, S. Ulam and M. Tsingou studied numerically a one-dimensional chain of equal masses connected by a weakly nonlinear spring. One of their goals was to establish the time scale needed for the system to reach a thermalized state. However, instead of thermalization, they observed numerically a recurrence to the initial condition (this is known as the FPUT-recurrence). This unexpected result has lead to the development of the modern nonlinear physics (discovery of solitons, integrability etc).

After an historical introduction, I will discuss new developments based on the so-called Wave Turbulence theory that allows one to estimate accurately the relaxation time scale in one dimensional chains.

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Mon 21/05/18 14:00
Fulton J20
Mathematics Seminar
Dr. Carlo Mercuri (University of Swansea)
Variations on a Schrodinger-Poisson system