Mathematics colloquia

$\Gamma$-convergence of Onsager--Machlup functionals and MAP estimation in non-parametric Bayesian inverse problems

Speaker: Tim Sullivan (Warwick and Alan Turing Institute)
Date: Wednesday 25 May, 14:00-15:00

Abstract: The Bayesian solution to a statistical inverse problem can be summarised by a mode of the posterior distribution, i.e.\ a MAP estimator. The MAP estimator essentially coincides with the (regularised) variational solution to the inverse problem, seen as minimisation of the Onsager--Machlup functional of the posterior measure. An open problem in the stability analysis of inverse problems is to establish a relationship between the convergence properties of solutions obtained by the variational approach and by the Bayesian approach. To address this problem, we propose a general convergence theory for modes that is based on the $\Gamma$-convergence of Onsager--Machlup functionals, and apply this theory to Bayesian inverse problems with Gaussian and edge-preserving Besov priors. Joint work with Birzhan Ayanbayev, Ilja Klebanov, and Han Cheng Lie.

Slow travelling wave solutions of the nonlocal Fisher-KPP equation

Speaker: John Billingham (Nottingham)
Date: Wednesday 06 April 2022, 14:00-15:00

Abstract: In this talk I will discuss travelling wave solutions, u = U(x-ct), of the nonlocal Fisher-KPP equation in one spatial dimension,
u_t = D u_xx + u(1-phi*u)
with D << 1 and c << 1, where phi*u is the spatial convolution of the population density, u(x,t), with a continuous, symmetric, strictly positive kernel, phi(x), which is decreasing for x>0 and has a finite derivative as x -> 0+.
The formal method of matched asymptotic expansions and numerical methods can be used to solve the travelling wave equation for various kernels, phi(x), when c << 1. The most interesting feature of the leading order solution behind the wavefront is a sequence of tall, narrow spikes with O(1) weight, separated by regions where U is exponentially small. The regularity of phi(x) at x=0 is a key factor in determining the number and spacing of the spikes, and the spatial extent of the region where spikes exist.

Studying dynamics using computational polynomial optimization

Speaker: David Goluskin (University of Victoria, Canada)
Date: Wednesday 02 February 2022, 14:00-15:00

Abstract: For nonlinear ODEs and PDEs that cannot be solved exactly, various properties can be inferred by constructing functions that satisfy suitable inequalities. Although the most familiar example is proving stability of an equilibrium by constructing Lyapunov functions, similar approaches can produce many other types of mathematical statements, including for systems with chaotic behavior. Such statements include bounds on attractor properties or on transient behavior, estimates of basins of attraction, and design of nonlinear controls. Analytical results of these types often trade precision for tractability. Much greater precision can be achieved by using computational methods of polynomial optimization to construct functions that satisfy the desired inequalities. This talk will provide an overview of the different ways in which polynomial optimization can be used to study dynamics. I will show various examples in which polynomial optimization produces arbitrarily sharp results while other methods do not. I will focus on the ODE case, where theory and computational methods are more complete. Extensions to PDEs may be discussed briefly.

Deep learning of conjugate mappings

Speaker: Jason Bramburger (George Mason University, Virginia)
Date: Wednesday 27 October 2021, 2-3pm

Abstract: Despite many of the most common chaotic dynamical systems being continuous in time, it is through discrete time mappings that much of the understanding of chaos is formed. Henri Poincaré first made this connection by tracking consecutive iterations of the continuous flow with a lower-dimensional, transverse subspace. The mapping that iterates the dynamics through consecutive intersections of the flow with the subspace is now referred to as a Poincaré map, and it is the primary method available for interpreting and classifying chaotic dynamics. Unfortunately, in all but the simplest systems, an explicit form for such a mapping remains outstanding. In this talk I present a method of discovering explicit Poincaré mappings using deep learning to construct an invertible coordinate transformation into a conjugate representation where the dynamics are governed by a relatively simple chaotic mapping. The invertible change of variable is based on an autoencoder, which allows for dimensionality reduction, and has the advantage of classifying chaotic systems using the equivalence relation of topological conjugacies. We illustrate with low-dimensional systems such as the Rössler and Lorenz systems, while also demonstrating the utility of the method on the infinite-dimensional Kuramoto--Sivashinsky equation.

Minimal Lagrangians and where to find them

Speaker: Jason Lota (Oxford)
Date: Wednesday 27 May 2020, 3:00 p.m.

Abstract: A classical problem going back to ancient Greece is to find the shortest curve in the plane enclosing a given area: the isoperimetric problem. A similar question is whether given a curve on a surface it can be deformed to a shortest one. Whilst the solutions to these classical problems are well-known, natural generalisations in higher dimensions are mostly unsolved. I will explain how this leads us to the study of minimal Lagrangians and the question of how to find them, which will take us to the interface between symplectic topology, Riemannian geometry and analysis of nonlinear PDEs, with links to theoretical physics.

How directed is a directed network?

Speaker: R.S.MacKay (University of Warwick)
Date: Wednesday 29 April 2020, 3:00 p.m.

Abstract: Many systems can be represented by directed graphs, e.g. food webs, supply networks, social networks, metabolic networks, language networks, financial networks. The nodes represent the objects and a directed edge indicates a flow from one node to another or influence of one node on another.

In some networks, the edges line up in an overall direction; in others, they do not. The old notion of “trophic level” from ecology, and its more recent analogue “upstreamness” in economics, provide one way to quantify this, but they require basal or top nodes and have various other shortcomings.

In joint work with Samuel Johnson and Bazil Sansom, we present an improved notion of trophic level and a resulting notion of trophic coherence. We illustrate their application to a wide variety of real-world networks and we derive some nice mathematical relationships of trophic coherence with other significant network properties like non-normality, stability of contagion processes, and cyclicality.

The work was supported by the Economic and Social Research Council via the Instability hub of the Rebuilding Macroeconomics programme of the National Institute for Economic and Social Research.

Is dispersion a stabilising or destabilising mechanism? Landau-damping induced by fast background flows
Speaker: Edriss Titi (Cambridge, Texas A&M Univ, Weizmann Institute of Science)
Date: Wednesday 4 March 2020

Abstract: In this talk Edriss Titi will present a unified approach for the effect of fast rotation and dispersion as an averaging mechanism for, on the one hand, regularising and stabilising certain evolution equations, such as the Navier-Stokes and Burgers equations. On the other hand, Edriss will present some results in which large dispersion acts as a destabilising mechanism for the long-time dynamics of certain dissipative evolution equations, such as the Kuramoto-Sivashinsky equation. In addition, they will present some new results concerning two- and three-dimensional turbulent flows with high Reynolds numbers in periodic domains, which exhibit ``Landau-damping" mechanism due to large spatial average in the initial data.

Weighing the Fog of War: A physicist's adventures in operations research and history
Speaker: Niall MacKay (York)
Date: Wednesday 11 December 2019

Abstract: A tour through ten years' work with operations researchers and historians on combat modelling, military history and their interaction. I'll begin with some elementary ideas from Lanchester theory and present some neat results for multilateral models. Then I shall describe some of our historical work, including on the First World War at sea and the Second World War in the air.

Global stability properties of the climate: Melancholia states, invariant measures, and phase transitions
Speaker: Valerio Lucarini (Reading)
Date: Tuesday 3 December 2019

Abstract: For a wide range of values of the incoming solar radiation, the Earth features at least two attracting states, which correspond to competing climates. The warm climate is analogous to the present one; the snowball climate features global glaciation and conditions that can hardly support life forms. Paleoclimatic evidences suggest that in past our planet flipped between these two states. The main physical mechanism responsible for such instability is the ice-albedo feedback. Following an idea developed by Eckhardt and co. for the investigation of multistable turbulent flows, we study the global instability giving rise to the snowball/warm multistability in the climate system by identifying the climatic edge state, a saddle embedded in the boundary between the two basins of attraction of the stable climates. We refer to these states as Melancholia States. We then introduce random perturbations as modulations to the intensity of the incoming solar radiation. We observe noise-induced transitions between the competing basins of attractions. In the weak noise limit, large deviation laws define the invariant measure and the statistics of escape times. By empirically constructing the instantons, we show that the Melancholia states are the gateways for the noise-induced transitions. In the region of multistability, in the zero-noise limit, the measure is supported only on one of the competing attractors. For low (high) values of the solar irradiance, the limit measure is the snowball (warm) climate. The changeover between the two regimes corresponds to a first order phase transition in the system. The framework we propose seems of general relevance for the study of complex multistable systems. Finally, we propose a new method for constructing Melancholia states from direct numerical simulations, thus bypassing the need to use the edge-tracking algorithm.
Refs.

On extreme and record events in dynamical systems
Speaker: Mark Holland (Exeter)
Date: Wednesday 6 November 2019

Abstract: Understanding extreme events, such as severe weather, climatic or financial events is a major challenge. In this talk I will explain some of the mathematical approaches used in understanding extreme events, such as finding their probability distribution. We will assume that the underlying time series process is modelled by a deterministic dynamical system, such as a discrete time map or differential equation. These latter processes have some dependency, and so any results known about extremes for independent, identically distributed (i.i.d) random processes cannot immediately be transferred to dynamical systems. As part of the talk, I will review relevant extreme value theory associated to the i.i.d case. I will then discuss this theory for dynamical systems, illustrating with examples from low dimensional chaotic maps.

Seminar details

Upcoming seminars

Past seminars

Cookies

Mathematics colloquia

Seminar details

Upcoming seminars

Past seminars

2021/22

2020/21

2019/20

2018/19

2017/18