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  abstractAbstractUnderstanding 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 acceptreject 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., 33:102130, 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):16001622, 2014.
[3] A. Abdulle, G. Vilmart, and K.C. Zygalakis, Long time accuracy of LieTrotter splitting methods for Langevin dynamics. SIAM J. Numer. Anal., 53(1):116, 2015.
[4] A. Duncan, G. A. Pavliotis and T. Lelievre, Variance Reduction using Nonreversible Langevin Samplers, J. Stat. Phys., 163(3):457491, 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, 24:259328, 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  abstractAbstractWe 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 nonselfadjointness of linear elliptic operators as well as a discrete perturbation argument hide 

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

Mon 05/02/18 15:00 Fulton G20 Mathematics Seminar  Dr. Sergey Anfinogentov (University of Warwick) Analysis of lowamplitude oscillations in solar corona by motion magnification  abstractAbstractCoronal 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 nonuniformity, a coronal loop can act as a waveguide for MHD waves including standing kink oscillations. These oscillations resemble vibration of a guitar string. MHDseismology 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 decayless regime of kink oscillations has changed the situation. Decayless 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 lowamplitude quasiperiodic 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 timedistance plots.
This approach is found to give good results for harmonic oscillations, including the modulated signals, such as exponentially decaying, multimodal and frequency modulated signals. We applied the motion magnification to the EUV observations of several nonflaring active regions, clearly demonstrating the presence of lowamplitude decayless 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/SergeyAnfinogentov/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. hide 

Mon 12/02/18 14:00 Fulton G20 Mathematics Seminar  Dr. Stephen Metcalfe (University of York) Approximating Blowup Via A Posteriori Error Estimation  abstractAbstractThe numerical approximation of blowup (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 wellbehaved 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. hide 

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  abstractAbstractSolar 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 EastWest orientated polarity inversion lines. To investigate the origin of this hemispheric
pattern in realistic magnetic field configurations, a global evolution model along with a largescale representation
of helicity condensation is applied. Periods in both the rising and decliningphase 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. hide 

Mon 12/03/18 15:00 Fulton G20 Mathematics Seminar  Dr. Thomas Elsden (University of St Andrews) 3D Alfven Resonances in Earth's Magnetosphere  abstractAbstractLow 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 fastAlfven wave coupling, and put this in context in terms of waves in Earth's magnetosphere. hide 

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  abstractAbstractCoherent 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 NavierStokes flows can become singular in a finite time, or whether these remain wellbehaved 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). hide 

Wed 21/03/18 15:00 Fulton G20 Mathematics Seminar  Prof. Thomas Hillen (University of Alberta) Nonlocal Models for Cellular Adhesion  abstractAbstractCellular adhesion is one of the most important interaction force between cells and other tissue components. In 2006, Armstrong, Painter and Sherratt introduced a nonlocal 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 nonlocal model is challenging. In this talk I will present a random walk derivation of the ArmstrongPainterSherratt 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). hide 

Mon 26/03/18 15:00 Fulton G20 Mathematics Seminar  Prof. Miguel Onorato (Università di Torino) New developments in the old FemiPastaUlamTsingou problem  abstractAbstractIn the early fifties E. Fermi in collaboration with J. Pasta, S. Ulam
and M. Tsingou studied numerically a onedimensional 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 FPUTrecurrence). 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 socalled Wave Turbulence theory
that allows one to estimate accurately the relaxation time scale in one
dimensional chains. hide 

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