Professors Alain Goriely, Xue-Mei Li, Coralia Cartis, Edriss S. Titi and Christoph Ortner
10:40 Modelling Dementia (Prof Alain Goriely, MR2)
11:25 Perturbation, Noise, and Averaged Dynamics (Prof Xue-Mei Li, MR2)
12:20 Optimization with expensive and uncertain data – challenges and improvements (Prof Coralia Cartis, MR2)
14:00 Analysis of Oceanic and Tropical Atmospheric with Moisture Models: Global Regularity, Finite-time Blowup and Singular Limit Behavior (Prof Edriss S. Titi, MR2)
14:45 Hybrid Modelling of Interatomic Forces (Physics+Data+Mathematics) (Prof Christoph Ortner, MR2)
15:30 Tea break
16:00 Grad moment method for dilute granular gases of inelastic Maxwell
molecules, Vinay Kumar Gupta (MR2)
16:15 The physics of regularized Stokeslets, Andy Zhao (MR2)
16:30 An Anisotropic Interaction Model for Simulating Fingerprints, Lisa Maria Kreusser (MR2)
16:45 Towards an ideal synthetic system for active matter, Debasish Das (MR2)
Speaker: Prof Alain Goriely Title: Modelling Dementia Abstract: Neurodegenerative diseases such as Alzheimer’s or Parkinson’s are devastating conditions with poorly understood mechanisms and no known cure. Yet a striking feature of these conditions is the characteristic pattern of invasion throughout the brain, leading to well-codified disease stages visible to neuropathology and associated with various cognitive deficits and pathologies. How can we use mathematical modelling to gain insight into this process and, doing so, gain understanding about how the brain works? In this talk, I will show that by linking different theories and methods to recent progress in imaging, we can unravel some of the universal features associated with dementia and, more generally, brain functions.
————————————————————- Speaker: Prof Xue-Mei Li Title: Perturbation, Noise, and Averaged Dynamics Abstract: Perturbation and approximation underly almost all applications of physical mathematical models. The averaging method, first introduced for approximate periodic motions, is now widely used for a large class of problems in both pure and applied mathematics.
The purpose of averaging is easy to describe. Suppose that we have a system of variables interacting with each other and moving at different scales of speed (of order 1 and of order 1/epsilon) with the fast variables `fast oscillatory’. Both slow and fast variables evolve in time according to some rules, for example solving a family of differential or stochastic differential equations. The aim is to determine whether the slow variables can be approximated by an autonomous systems of equations, called the effective dynamic, as epsilon is taken to 0 and the speed of the fast variables tends to infinity. Slow/fast systems arise from perturbations of conservation laws and breaking of symmetries.
We will discuss the latest developments in averaged dynamics, touching on recent work with M. Hairer on averaged dynamics with fractional noise (this leads to very different behaviour from the white noise case and requires new techniques).
————————————————————- Speaker: Prof Coralia Cartis Title: Optimization with expensive and uncertain data – challenges and improvements Abstract: Real-life applications often require the optimization of nonlinear functions with several unknowns or parameters – where the function is the result of highly expensive and complex model simulations involving noisy data (such as climate or financial models, chemical experiments), or the output of a black-box or legacy code, that prevent the numerical analyst from looking inside to find out or calculate problem information such as derivatives. Thus classical optimization algorithms, that use derivatives (steepest descent, Newton’s methods) often fail or are entirely inapplicable in this context. Efficient derivative-free optimization algorithms have been developed in the last 15 years in response to these imperative practical requirements. As even approximate derivatives may be unavailable, these methods must explore the landscape differently and more creatively. In state of the art techniques, clouds of points are generated judiciously and sporadically updated to capture local geometries as inexpensively as possible; local function models around these points are built using techniques from approximation theory and carefully optimised over a local neighbourhood (a trust region) to give a better solution estimate.In this talk, I will describe our implementations and improvements to state-of-the-art methods. In the context of the ubiquitous data fitting/least-squares applications, we have developed a simplified approach that is as efficient as state of the art in terms of budget use, while achieving better scalability. Furthermore, we substantially improved the robustness of derivative-free methods in the presence of noisy evaluations. Theoretical guarantees of these methods will also be provided. Finally, despite derivative-free optimisation methods being able to only provably find local optima, we illustrate that, due to their construction and applicability, these methods can offer a practical alternative to global optimisation solvers, with improved scalability and flexibility. This work is joint with Lindon Roberts (Oxford), Katya Scheinberg (Lehigh), Jan Fiala (NAG Ltd) and Benjamin Marteau (NAG Ltd).
————————————————————- Speaker: Prof Edriss S. Titi Title: Analysis of Oceanic and Tropical Atmospheric with Moisture Models: Global Regularity, Finite-time Blowup and Singular Limit Behavior Abstract: In this talk I will present some recent results concerning global regularity of certain geophysical models. This will include the three-dimensional primitive equations with various anisotropic viscosity and turbulence mixing diffusion, and certain tropical atmospheric models with moisture. Moreover, in the non-viscous (inviscid) case it can be shown that there is a one-parameter family of initial data for which the corresponding smooth solutions of the primitive equations develop finite-time
Capitalizing on the above results, we can provide rigorous justification of the derivation of the Primitive Equations of planetary scale oceanic dynamics from the three-dimensional Navier-Stokes equations, for vanishing small values of the aspect ratio of the depth to horizontal width. Specifically, we can show that the Navier-Stokes equations, after being scaled appropriately by the small aspect ratio parameter of the physical domain, converge strongly to the primitive equations, globally and uniformly in time, and that the convergence rate is of the same order as the aspect ratio parameter. Furthermore, I will also consider the singular limit behavior of a tropical atmospheric model with moisture, as ε → 0, where ε > 0 is a moisture phase transition small convective adjustment relaxation time parameter.
————————————————————- Speaker: Prof Christoph Ortner Title: Hybrid Modelling of Interatomic Forces (Physics+Data+Mathematics) Abstract: Accurate molecular simulation requires quantum chemistry models that accurately capture the interaction between nuclei and electrons. Unfortunately, their high computational cost makes these models unsuitable for simulating complex material phenomena or large molecules. On the other hand, interatomic potentials (IPs) are largely empirical models with poor accuracy and narrow applicability. The past decade has seen a revival of IPs, re-casting their construction as a “machine learning” instead of a “modelling” problem.
In this talk I will backtrack from this trend by returning to simple functional forms, motivated by physics (and somewhat justified by mathematics), but retain a key feature of the data-driven/ML approach: models must be systematically improvable. I will demonstrate how a combination of elementary ideas from analysis, approximation theory, and representation theory leads to a very competitive construction (in terms of fitting error/cost) that has a range of additional advantages, for example that it can be effectively regularised. While my talk will focus specifically on interatomic potentials, the ideas should in principle be applicable to a much wider range of particle models, and at a more conceptual level also to continuum models.
STUDENT/POSTDOC TALK ABSTRACTS
Speaker: Vinay Kumar Gupta Title: Grad moment method for dilute granular gases of inelastic Maxwell
molecules Abstract: A collection of discrete macroscopic particles which dissipate energy during collisions among themselves is termed as a granular material. Granular materials are ubiquitous in nature and industry alike. They occur in all shapes and sizes ranging from few microns to several hundred kilometres. Sand dunes, debris flow, asteroid belt, dust storm, gravels, cement, food grains, sugar, capsules, pills are some typical examples of granular materials. Depending on the energy supplied, they can exist in the solid, liquid or gaseous state. Due to their dissipative nature, granular materials exhibit several interesting —and often counter-intuitive— phenomena. At the same time, this very feature of granular materials poses many difficulties while modelling processes in granular materials due to the non- conservation of energy.
The present work considers the gaseous state of granular materials, for which, analogous to molecular gases, mathematical tools can be developed within the framework of kinetic theory. In this talk, I shall present my recent work on Grad moment method for modelling a dilute granular gaseous flow of d-dimensional smooth, identical, inelastic spheres interacting with Maxwell interaction potential—referred to as inelastic Maxwell molecules (IMM). Here d = 2 means disk flows while d = 3 means sphere flows. This work is a somewhat generalization of my previous work to arbitrary dimensions. To assess the capabilities of the derived models for IMM, the homogeneous cooling state of a freely cooling granular gas of IMM and its stability to small perturbation is studied.
Moreover, the Navier–Stokes level transport coefficients are also obtained from the moment equations and it has been found that the transport coefficients obtained in this work agree exactly with those obtained in previous studies.
————————————————————- Speaker: Andy Zhao Title: The physics of regularized Stokeslets Abstract: Green’s functions (also called fundamental solutions) can be used to construct solutions to several differential equations arising in physics. These solutions, however, cannot be easily implemented on a computer due to the divergence of Green’s functions. One popular way to overcome this difficulty is to remove the singularities in Green’s functions, a process known as regularization or mollification. The amount of regularization is often controlled by a parameter, called the regularization parameter. The significance of the parameter and the physical change to the Green’s function are often not well understood. Nevertheless, they are essential to a good understanding of the error introduced in the regularization process. In this talk, I will discuss a specific Green’s function in fluid mechanics – the Stokeslet. Its regularized versions, called regularized Stokeslets, have been extensively used in many flow problems since the early 2000s. I will elucidate the physical significance of regularized Stokeslets for various regularization schemes. If time permits, I will explain the implication for computation.
————————————————————- Speaker: Lisa Maria Kreusser Title: An Anisotropic Interaction Model for Simulating Fingerprints Abstract: Motivated by the formation of fingerprint patterns we consider a class of interaction models with anisotropic interaction forces whose orientations depend on an underlying tensor field. This class of models can be regarded as a generalization of a gradient flow of a nonlocal interaction potential which has a local repulsion and a long-range attraction structure. In contrast to isotropic interaction models the anisotropic forces in our class of models cannot be derived from a potential. The underlying tensor field introduces an anisotropy leading to complex patterns which do not occur in isotropic models. This anisotropy is characterized by one parameter in the model. We study the variation of this parameter, describing the transition between the isotropic and the anisotropic model, analytically and numerically. We analyze the steady states and their stability by considering the particle model and the associated mean-field equations. Besides, we propose a bio-inspired model to simulate fingerprint patterns as stationary solutions by choosing the underlying tensor field appropriately.
————————————————————- Speaker: Debasish Das Title: Towards an ideal synthetic system for active matter Abstract: How are groups of living organisms such as flocks of birds, sheep, schools of fish and bacterial colonies able to self-organize and display collective motion? This question has fascinated scientists for decades and has given rise to the new field of ‘active matter’. One of the key features of active matter is that it is composed of self-propelled units that move by consuming energy from their surrounding with a direction of self-propulsion typically set by their own anisotropy, either in shape or functionalisation, rather than by an external field. In this talk, I will use an electrohydrodynamic instability called Quincke rotation to develop an ideal self-propelled particle. Dielectric particles suspended in a weakly conducting fluid are known to spontaneously start rotating under the action of a sufficiently strong uniform DC electric field due to Quincke rotation. This rotation can be converted into translation when the particles are placed near a surface providing useful model systems for active matter. Using a combination of numerical simulations and theoretical models, we demonstrate that it is possible to convert the spontaneous Quincke rotation into spontaneous translation even in the absence of surfaces by relying on geometrical asymmetry instead. The resulting novel type of active particle (i) is capable of autonomous self propulsion, i.e. the direction of propulsion is not set and controlled by an external field, (ii) does not require the presence of a surface and (iii) is amenable to theoretical analysis from first principles. Suspensions of randomly-shaped particles under Quincke rotation would thus be expected to perform collective motion by exploring the full three-dimensional space with unspecified swimming direction, opening thereby the door to a potentially new type of active matter.