Mathematics & StatisticsSeminars and colloquia

Title:   Mesh pattern occurrence in random permutations

Date:  3.00pm Tuesday 3rd October

Venue: LT907

Abstract:  We say that the likelihood of a mesh pattern is the asymptotic probability that a random permutation contains an occurrence of the pattern. In this talk we will investigate the likelihood of a variety of patterns, determining their values for every vincular pattern. For bivincular patterns, the Small Anchors Theorem distinguishes between those patterns whose likelihood equals zero, those whose likelihood is positive but less than 1, and those whose likelihood equals 1. We also know how to determine the (rational) likelihood of any bivincular pattern formed of what we call anchored trees. Other bivincular patterns, such as the small ascent and small descent, have irrational likelihoods, whose values can be established using the Chen–Stein method. This is joint work with Jason Smith.

Title: On semi-transitivity of (extended) Mycielski graphs

Date: 3.00pm Tuesday 17th October 2023

Venue: LT908

Abstract: An orientation of a graph is semi-transitive if it is acyclic, and for any directed path v0 → v1 → · · · → vk either there is no arc between v0 and vk, or vi → vj is an arc for all 0 ≤ i < j ≤ k. An undirected graph is semi-transitive if it admits a semi-transitive orientation. Semi-transitive graphs generalize several important classes of graphs, and they are precisely the class of word-representable graphs studied extensively in the literature.

The Mycielski graph of an undirected graph is a larger graph, constructed in a certain way, that maintains the property of being triangle-free but enlarges the chromatic number. These graphs are important as they allow to prove the existence of triangle-free graphs with arbitrarily large chromatic number. An extended Mycielski graph is a certain natural extension of Mycielski graphs.

In this talk, I will discuss a complete characterization of semi-transitive extended Mycielski graphs and comparability Mycielski graphs, as well as a conjectured complete characterization of semi-transitive Mycielski graphs. My results are a far-reaching extension of the result of Kitaev and Pyatkin on non-semi-transitive orientability of the Mycielski graph µ(C5) of the cycle graph C5. Also, I will mention how to use a recent result of Kitaev and Sun to shorten the length of the original proof of non-semi-transitive orientability of µ(C5) from 2 pages to a few lines. 

Title: Eigenvalues of canonical systems

Date: 3.00pm Tuesday 31st October 2023

Venue: LT907

Abstract: In this talk I shall consider eigenvalues of 2x2 canonical systems, which cover one-dimensional Schrödinger equations, Sturm-Liouville equations, Jacobi operators, Dirac systems and (generalised) Krein strings as special cases.  I am particularly interested in the asymptotic behaviour and the density of eigenvalues.  This behaviour is connected with the growth and the smoothness of the coefficients.  It turns out that the results change substantially when one moves from dense to sparse spectrum.  I will also mention a trace formula for the eigenvalues.

Title: Beyond the Hodge Theorem: curl and asymmetric pseudodifferential projections

Date: 3.00pm Tuesday 7th November 2023

Venue: LT907

Abstract: Consider a single photon living in curved space. It is described by Maxwell’s equations. We seek solutions harmonic in time. This reduces to the spectral problem for the operator curl, whose spectrum is, in general, asymmetric about zero (think particle/antiparticle).

Spectral asymmetry is a classical subject in analysis and geometry, whose origins lie in the papers of Atiyah, Patodi and Singer. In this talk I will discuss a new approach to the study of spectral asymmetry based on the use of pseudodifferential techniques developed in a series of recent joint papers by Dmitri Vassiliev and myself.

Emphasis will be placed on ideas and motivation; the talk will include a brief historical overview of the development of the subject area.

Title: Travelling waves in the Rosenau-KdV equation

Date: 3.00pm Tuesday 28th November 2023

Venue: LT907

Abstract: We consider the existence of monotone travelling waves in the Rosenau-KdV equation, which exhibits phenomena that cannot be seen in the well-known Burgers-KdV equation. Many problems are still open. This is joint work with N. Bedjaoui and G. Maypaokha (UPJV).

Title: Mathematical analysis of fracture and related phenomena in atomistic modelling of crystalline materials

Date: 3.00pm, Tuesday 5th December 2023

Venue: LT907

Abstract: The atomistic modelling of fracture and related phenomena in crystalline materials poses a string of mathematically non-trivial and exciting challenges, both on a theoretical and a practical level. At the heart of the problem lies a discrete domain of atoms (a lattice), which exhibits spatial inhomogeneity induced by the crack surface, particularly pronounced in the vicinity of the crack tip. Atoms interact in a highly nonlinear way, resulting in a severely non-convex energy landscape facilitating non-trivial behaviour of atoms such as (i) crack propagation; (ii) near-crack tip plasticity - emission and movement of defects known as dislocations in the vicinity of the crack tip; (iii) surface effects - atoms at the crack surface relaxing or possibly attaining an altogether different crystalline structure. On the practical side, the richness of possible phenomena renders the task of setting up numerical simulations particularly tricky - numerical artefacts, e.g. induced by prescribing a particular boundary condition, can lead to inconsistent results.

In this talk I will aim to summarise on-going efforts aimed at putting the atomistic modelling of fracture on a rigorous mathematical footing. I will begin by introducing a mathematical framework, based in part on bifurcation theory, giving rise to well-defined models for which regularity and stability of solutions can be discussed, followed by describing how the theory can be used to set up numerical continuation-based simulations, e.g. for Mode I crack propagation in silicon on the (111) cleavage plane, using state-of-the-art interatomic potentials. Subsequently I will outline how this framework can be used to rigorously derive upscaled models of near-crack-tip plasticity.

Finally, I will also touch upon recent work on proposing a much more general framework in which a wide range of defect nucleation & migration phenomena in the atomistic modelling of crystalline materials can be treated as bifurcation events.

Title: Thresholds for patterns in random permutations

Date: 3.00pm Tuesday 12th December 2023

Venue: LT420

Abstract: In this talk we will investigate thresholds for the appearance and disappearance of consecutive patterns occurring within large random permutations as the number of inversions increases. We establish these lower and upper thresholds for any fixed consecutive pattern. We also consider thresholds for classical and vincular patterns. To do so, we work with inversion sequences, which we consider to be weak integer compositions. As a result, we introduce a model of random integer compositions which we call the geometric random composition. This talk will focus on how we transfer thresholds for patterns in the geometric random composition to get thresholds for patterns in the uniform random permutation.

Title: Distribution of mesh patterns

Date: 2.00pm Tuesday 23rd January 2024

Venue: LT907

Abstract: The notion of a mesh pattern, generalizing several classes of permutation patterns, was introduced in 2011 by Branden and Claesson to provide explicit expansions for certain permutation statistics as, possibly infinite, linear combinations of (classical) permutation patterns. There is a long line of research papers dedicated to the study of mesh patterns and their generalizations.

In this talk, I will discuss a systematic study of avoidance and distribution of mesh patterns of short length, and also some other more general results

Title: Coagulation and Combinatorics

Date: 2.00pm on Tuesday 20th February 2024

Venue: LT907

Abstract: I will review the recent combinatorics approach to discrete coagulation equations due to Łepek and coworkers and the open problems that arise in that area.

Title: Vortex stretching in Navier-Stokes turbulence 

Date: 3.00pm on Tuesday 27th February 2024

Venue: LT907

Abstract: Turbulence in the Navier-Stokes equations is a major nonlinear problem, that cannot be tackled with existing analytical methods. I discuss key characteristics of turbulence, explaining its physical content and the momentum balance it encodes. I  indicate the key analytical difficulty and past failures to tackle it in the velocity-pressure space. I introduce the vortex dynamical viewpoint and explain the association between the latter and key turbulent physics. By employing the vortex dynamical viewpoint, I compute key aspects of turbulent vortex stretching, including, among other, the Lyapunov exponents of the tangent system, and the correlations between strain rate eigendirections and coherent, filamentary vorticity.

Title: Coagulation, non-associative algebras and combinatorial trees

Date: 2.00pm Tuesday 12th March 2024 

Venue: LT907

Abstract: We consider the classical Smoluchowski coagulation equation with a general frequency kernel. We show that there exists a natural deterministic solution expansion in the non-associative algebra generated by the convolution product of the coalescence term. The non-associative solution expansion is equivalently represented by binary trees. We demonstrate that the existence of such solutions corresponds to establishing the compatibility of two binary-tree generating procedures, by:

• grafting together the roots of all pairs of order-compatible trees at preceding orders, or
• attaching binary branches to all free branches of trees at the previous order.

We then show that the solution represents a linearised flow, and also establish a new numerical simulation method based on truncation of the solution tree expansion and approximating the integral terms at each order by fast Fourier transform. In particular, for general separable frequency kernels, the complexity of the method is linear-loglinear in the number of spatial modes/nodes.

Title: Asymptotic enumeration of monotone permutation grid classes

Date: 2.00pm Tuesday 26th March 2024 

Venue: LT907

Abstract: A monotone grid class Grid(M) is a set of permutations whose shape satisfies constraints specifies by the matrix M, all of whose entries are in {1,−1, 0}. Each entry of the matrix corresponds to a cell in a gridding of a permutation. If the entry is 1, then the points in the cell must increase. If it is −1, they must decrease. If it is 0, the cell must be empty. To find the exact number of permutations of length n in Grid(M) is hard, because a permutation may have more than one gridding, so we only determine the asymptotic enumeration. To do this, we find the number of gridded permutations, the typical proportion of points in each cell, and the ways in which permutations can be gridded. Our focus is mainly on L-shaped, T-shaped, and X-shaped classes.

Title: Resonant free-surface water waves in closed basins

Date:  1.00pm Thursday 25th January 2024

Title: Data-driven design optimisation of chemical reactors

Date:  1.00pm Thursday 1st February 2024

Title: Oscillatory bodily flows: the  eye and the brain

Date:  1.00pm Thursday 8th February 2024

Title: Determination of the index of refraction of anti-reflection coatings

Date:  1.00pm Thursday 22nd February 2024

Title: Electrostatics and variational perturbation theory

Date:  1.00pm Thursday 29th February 2024

Title: TBC

Date:  1.00pm Thursday 7th March 2024

Title: Interplay between zonal jets, waves and turbulence: application to gas giants

Date:  1.00pm Thursday 14th March 2024

Title: Growing in the wind - an interdisciplinary investigation of wind influence on plant growth

Date:  1.00pm Thursday 21st March 2024

Title: Histopathology image analysis of pancreatic ductal adenocarcinoma based on Unsupervised and weakly-supervised learning algorithms

Date: 25th October 2023 at 3.00pm

Venue: LT420 or via Zoom (Link below)

Zoom link: https://strath.zoom.us/j/86543712884

Meeting ID: 865 4371 2884

Password: Gui@Strath

Abstract: Histopathology images are considered the most reliable method for detecting and diagnosing cancer. However, analyzing these images is a complex task that requires a great deal of expertise. Pathologists need to search for useful information in huge pathology images to diagnose diseases in clinical applications. Additionally, analyzing pathology images can predict disease trends and prognosis.

The development of deep neural networks has brought many breakthroughs in automated histopathology image analysis. However, these methods rely heavily on the availability of a large number of pixel-level labels, which is a labor-intensive and time-consuming task. To overcome these challenges, we investigated algorithms for histopathology image analysis using unlabeled or a small number of coarse-grained labels. We applied these algorithms to analyze histopathological images of pancreatic ductal adenocarcinoma (PDAC).

Our experimental results demonstrate that these weakly-supervised and unsupervised-based algorithms accurately segment the components of the pathology images, thus laying the foundation for further image analysis and prognosis prediction studies.

References from https://www.researchgate.net/profile/Luying-Gui

Title: Biological Fluids at Microscale

Date: 3.00pm Wednesday 1st November 2023

Venue: LT908

Abstract:  Fluid flow is a crucial factor in shaping the functionality and behaviour of biological systems. Without fluid flow, life as we know would not exist. In this talk, we'll dive into the world of microscale biological fluids, with a focus on two key aspects: how organisms move and how they transport fluids. When it comes to locomotion, larger creatures take advantage of inertia, a concept familiar from everyday life. But for tiny organisms living in a low Reynolds number world, inertia isn't much help. Instead, they've evolved clever ways to move, like changing their shape (e.g., euglena) or using rotating helices (e.g., bacteria) to get where they need to go. In this talk, we will focus on how bacteria generate propulsion and interact with rigid surfaces from a purely mechanical viewpoint. 

Trying to transport fluid at the microscale, is an equally difficult task because one needs to apply very high-pressure gradients to generate a certain flow rate. For instance, one cannot rely on pressure gradients to remove foreign objects from human lungs or transport cerebrospinal fluid in the brain of smaller organisms like mice. Nature has come with up a solution: cilia, tiny flexible filaments that actively beat, generate flows needed to move fluid efficiently in small spaces. In this talk, we will focus on minimal models that can reproduce cilia beating and look at the effect of external fluid flow on the cilia beating plane.

Title: Understanding large-scale DNA methylation patterns using mathematical models and bioinformatics 

Date: 3.00pm Wednesday 15th November 2023

Venue: LT908

Abstract:  DNA methylation occurs when a chemical methyl group attaches to certain parts of the DNA and is very important for normal cellular function. Large-scale changes in DNA methylation patterns are observed in development, ageing and disease, including large-scale loss of DNA methylation in cancer. Previously, DNA methylation patterns have been examined experimentally but the “gold-standard” experimental approaches used to detect DNA methylation allow only patterns within small genomic regions to be examined. Similarly, while mathematical modelling studies have aided in the understanding of DNA methylation patterns, the computational expense associated with these studies have again limited the use of these models to the study of small-scale patterns. 

During this talk I will discuss my work involving the study of large-scale DNA methylation patterns using both experimental data and mathematical models. I will describe how the analysis of data from new sequencing technologies allows large DNA methylation patterns within single molecules to be examined and allows regions of DNA associated with the most disordered, heterogeneous methylation patterns to be identified. I will also discuss how to construct a mean-field model that can be used to quickly, and accurately, predict statistical properties associated with large-scale DNA methylation patterns. The mathematical model and data analysis pipeline discussed should be valuable tools in future investigations into the mechanisms underlying the large-scale changes to methylation patterns observed in development and disease. The material I will discuss is joint work with Ramon Grima (University of Edinburgh, School of Biological Sciences) and Duncan Sproul (University of Edinburgh, Institute of Genetics and Cancer). 

Title: Models of intermediate complexity for marine ecosystems

Date: 3.00pm Wednesday 22nd November 2023

Venue: LT908

Abstract:  Over 90% of the planet’s habitable volume is marine space. Within it, millions of species are connected in a complex food web. In the face of myriad anthropogenic pressures, the future state of the world’s oceans depends on a cascade of direct and indirect effects through a trophic network. In turn, we will be affected by changes to ecosystems services: provisioning as the availability of seafood; regulation as carbon sequestration; culturally through existence values. Owing to a lack of observed data, mathematical models are necessary to understand the world’s oceans. However, as computing power has increased, the trend has been to ever more complex models. There is a case for models which are only “complex enough”.      

During this talk I will describe the end-to-end ecosystem model StrathE2E and surrounding activities. This is a system I have worked on with Mike Heath and Dougie Speirs since my arrival in the department. I will use this as a case study to highlight useful routines for ecological insight; accelerate the evaluation of the modelled system by orders of magnitude; and possible avenues where we could benefit from increased ecological complexity. I will highlight opportunities to foster links across research groups in the department and the possibility for joint projects.

Title: tbc

Date: 3.00pm Wednesday 7th February 2024

Venue: tbc

Abstract:  tbc

Title: tbc

Date: 3.00pm Wednesday 28th February 2024

Venue: LT908

Abstract:  tbc

Title: The Effect of Rumor and Hoax in Disease Transmission: A Mathematical Model Perspective

Date: Wednesday 6th September, 1.00-2.00pm

Venue: LT908

Abstract: 

In this paper, we present a review of mathematical models for the transmission of rumor and hoax.  We then explore the effect of disinformation such as rumor and hoax and their effect on the spread of disease infection. This phenomenon is not uncommon to occur, such as in the case of the COVID-19 pandemic. There is so much misinformation or even disinformation regarding the nature of the disease and how people should behave toward the disease. In many countries, this led to mismanagement of the disease, in particular, many people exhibited poor behavior or poor disease awareness in responding to the disease. We explore the effect of rumor and hoax in a mathematical model of disease transmission in the form of differential. By modifying the known SIR model, we divide the population into different categories depending on their health status, such as susceptibles, infected, and immune individuals. Furthermore, we also divide the susceptibles into different categories according to the knowledge of the rumor and hoax. We give a  numerical example to show the effect of rumors or hoaxes in the disease dynamics.

Keywords:  Mathematical Model, Differential Equation, Rumor, Hoax, Disease Transmission

Title: Is Harvesting Fish from the Oceanic Twilight Zone Sustainable? A Case Study in Combing Spatial and Physiological Structure in Population Modelling

Date: Wednesday 8th November, 1.00-2.00pm

Venue: LT908

Abstract: Mesopelagic fish species live beyond the continental shelf at depths of about 200m to 1km in the ocean twilight zone. In recent years it has become clear that their abundance is much greater than previously thought, which has prompted thoughts of a ‘gold rush’ of new fisheries. This talk will explore the use of a spatial population model to examine the sustainability of such a potential fishery for two fish species, the glacial lanternfish, and Mueller’s pearlside. These are very widely distributed fish in the north Atlantic and have growth rates that are spatially variable. Moreover, climate change is expected to produce long-term changes in distribution and demography. Modelling the population response to fishing and climate changes therefore requires overcoming technical challenges in combining spatial and physiological structure which will be discussed in the talk. 

Title: The Role of Individuals in Determining the Impacts of Changing Environments on Population Level Dynamics

Date: Wednesday 7th February, 1.00-2.00pm

Venue: LT907

Abstract: 

Climate change is having profound effects on the incidence of vector borne disease, such as dengue, chikungunya and West Nile virus. However, developing effective measures of disease risk on a global scale are challenged by the complex ways in which environmental variation acts in vector-host-pathogen systems. One way in which insect vectors, such as mosquitos, respond to environmental variation is to change their traits this can result in populations comprised of groups of individuals which differ in their traits (e.g. size, fecundity). For example, if food is scarce for juvenile mosquitos then when they become adults they are smaller, and lay fewer eggs to ensure there is less competition for food in the next generation. The environment of the juvenile determines the trait the individual has as an adult. In this way the individuals adapt to the environment as well as shape the environment for future generations.

Current models over-simplify the interaction between individuals, populations and the environment, so risk misestimating predictors of disease risk. Here, we derive a mathematical framework for capturing the interaction of individuals, their traits and the population dynamics. I will show how this new mathematical framework leads to both interesting mathematical questions and can be used to help explain the location, magnitude and timing of historical dengue outbreaks.

Title: Quantifying Performance and Resilience in Marine Assemblages with Complex Life-Histories

Date: Wednesday 21st February, 1.00-2.00pm

Venue: LT908

Abstract:  Ongoing global change challenges our ability to predict how natural populations will both respond to novel climatic regimes and utilise available habitat space. Corals are critical to the functioning of coastal reef ecosystems and, yet, in spite of their intrinsic and economic value, are threatened by an increasing plethora of abiotic and biotic disturbances. Preventing the ensuing loss of coral coverage and diversity calls for a mechanistic understanding of resilience across coral species and populations that is currently lacking in coral reef science. Meanwhile, changing coastal conditions and our ever-expanding exploitation of marine resources, has heralded a perceived increase in the abundance of coastal jellyfish assemblages. While jellyfish are also an important component of coastal marine communities, their public perception is often tainted by their proclivity for aggregating in vast numbers, known as jellyfish blooms, which can disrupt fishing and tourism activities. However, despite the socioeconomic ramifications associated with the formation of these jellyfish blooms, the complex and cryptic lifecycles exhibited by jellyfish species largely precludes accurate predictions into their timing and location, restricting our ability to manage and mitigate their ecological and economic impacts. Here, I will introduce research comprising state-structured population modelling, novel transient demographic approaches, and state-of-the-art hydrodynamic simulations, that offers valuable insight into the performance and resilience of these complex and cryptic marine assemblages under future climate scenarios; frameworks that represent key decision-support tools for informing both the conservation of global coral reefs and our management of the socioeconomic impacts of jellyfish bloom formation.

Title: Some Data-Driven Approaches to Surveillance of Covid-19 in the UK

Date: Wednesday 28th February, 1.00-2.00pm

Venue: LT908

Abstract: Whilst much of the work in modelling transmission of the pandemic was conducted using mathematical transmission models, the quantity of data made available through open data portals, such as the Covid Dashboard, provided alternatives to understanding and intervening to improve public health outcomes. In this talk I will outline some statistical approaches using surveillance data from varying spatial scales to study underlying dynamics of transmission of Covid-19 in the UK. In particular, multivariate flexible regression models and dimension spatial dimension reduction techniques will be used to estimate relative transmissibility of emerging variants of concern, as well as nowcasting current states of pandemic from noisy multivariate time series, respectively.

Title: Estimating the Size of Aedes Aegypti Populations from Dengue Incidence Data: Implications for the Risk of Yellow Fever, Zika Virus and Chikungunya Outbreaks

Date: Wednesday 17th April, 1.00-2.00pm

Venue: TBC

Abstract:  In this talk I present a model to estimate the density of aedes mosquitoes in a community affected by dengue. The model is based on the fitting of a continuous function to the incidence of dengue infections, from which the density of infected mosquitoes is derived straightforwardly. Further derivations allows the calculation of the latent and susceptible mosquitoes' densities, the sum of the three equals the total mosquitoes' density. The model is illustrated with the case of the risk of urban yellow fever resurgence  in dengue infested areas but the same methods apply for other aedes-transmitted infections like Zika and chikungunya viruses.

Title: Using Minimalistic Food-Web Models to Inform Fisheries Management

Date: Wednesday 1st May, 1.00-2.00pm

Venue: TBC

Abstract:  Chance and Necessity (CaN) modelling is a minimalistic food-web modelling framework integrating the existing knowledge about an entire or part of the ecosystem, the available data, and uncertainties. Unlike most of the large ecosystem models, the CaN framework does not aim at predicting the future state of commercial stocks, but rather to reconstruct possible past dynamics of a food-web. Reconstructions result from the delimitation of the possible “state-space” of the food-web based on ecological survey data and expert knowledge, and the exploration of this “state-space”. Here, I will present the underlying concepts of CaN modelling, present case studies of CaN modelling applications and provide examples on how the model outputs can be used to improve the management of commercial fisheries in the future.

Title: Modelling and Inference and Heterogeneity in Bacterial Growth

Date: Wednesday 22nd May, 1.00-2.00pm

Venue: TBC

Abstract:  

Title: TBC

Date: Wednesday 29th May, 1.00-2.00pm

Venue: TBC

Abstract:  

Title: TBC

Date: Wednesday 10th July, 1.00-2.00pm

Venue: LT908

Abstract:  

Title:  DEM applied in the mining and mineral extraction industry 

Date:  1pm Tuesday 19th March 2024

Venue: LT907

Abstract:  As the world transitions towards net zero though electrification our reliance upon critical minerals and ores such as copper, iron, lithium, etc is ever increasing. The mining and  mineral extraction industry currently uses 4-7% of the worlds available energy supply.  Mining and mineral extraction companies are faced with decarbonising their impact on the planet while simultaneously increasing extraction from excavation sites to meet demand using traditional technology. Discrete Element Modelling (DEM) gives us insight into why products behave the way they do and process condition parameter exploration to ultimately drive towards new designs and a more sustainable future.  The presentation will cover who the Weir Advanced Research Centre (WARC) is at UoS, our journey so far in using DEM for Weir products and the future direction of coupling such tools with other mathematical tools.

Title:  Reducible networks of Prandtl-Ishlinskii operators with economic and financial applications

Date:  1pm Tuesday 26th March 2024

Venue: LT907

Abstract:  If the nodes in a network have input-output responses satisfying a certain property (ie. they are Prandtl-Ishlinskii (PI) operators) then remarkable simplifications are possible. For arbitrary network topologies (under mild additional conditions) the entire network can be rigorously reduced to a single aggregated PI operator. This is true even if cascading behaviour, such as bubbles and crashes, can occur in the network.

Two applications will be presented. One is a financial market model incorporating momentum traders. The other is a macroeconomic model with the aggregated PI operator representing inflation expectations in the economy.

Title:  A Numerical Method for an SDE with a WIS Integral

Wednesday 11th October, 2023, 3.00-4.00pm

Venue: Zoom

Abstract:  We consider a one-dimensional SDE with linear multiplicative noise from fractional Brownian motion fBM. The fBM is parametrized by $H\in(0,1)$ the Hurst parameter and we interpret the integral as a WIS integral which allows us to understand the SDE for all values of $H$. This is in contrast to, for example, interpretation through rough paths where $H>1/3$. We will introduce fBM and some of the theory needed to understand the WIS integral such as the Wick product [Mishura].  We will consider the numerical approximation proposed in [Mishura], propose some modifications and indicate how the method may be implemented. We present numerical results and observe for $H<1/2$ faster rates of convergence than predicted from the theory. This is joint work with Roy Schieven and Utku Erdogan.

[Mishura] Y. S. Mishura Stochastic calculus for fractional Brownian motion and related processess. Springer. [2008]

Title:  B-series for Non-Autonomous Semi-Linear SDEs

Friday 25th October, 2023, 3.00-4.00pm

Venue: Zoom

Abstract:  In this talk, we will describe how well known results for B-series applied SDEs can be used to derive order conditions for exponential integrators for non-autonomous semilinear problems. This approach seems to make it possible both to simplify and generalize existing results.

Title:  Numerical Solutions of a Markov-Switching One-Factor Volatility Model with Non-Globally Lipschitz Continuous Coefficients

Friday 25th January, 2024, 4.00-5.00pm

Venue: LT907

Abstract: We extend the one-factor stochastic volatility model to incorporate coefficient terms of super-linear growth under the Markov-switching framework. Since the proposed model is intractable analytically, we develop various mathematical techniques to investigate convergence in probability of the numerical solutions under the local Lipschitz condition. Finally, we perform simulation examples to demonstrate the theoretical results and justify the theoretical results for the valuation of some financial options.

Title:  Explicit Convergence Rates for the M/G/1 Queue under Perturbation

Friday 19th April, 2024, 3.00-4.00pm

Venue: LT907

Abstract: Stochastically ordered Markov process is a topic of special concern to us. As is mentioned by Meyn and Tweedie, many Markov processes are stochastically ordered in their initial state. Thus, we established convergence rates for discrete-time Markov chains on a countable state space that are stochastically ordered starting from a stationary distribution under perturbation. We investigate the explicit criteria to obtain the ordinary ergodicity, geometric ergodicity and polynomial ergodicity for the embedded M/G/1 queue under perturbation. The explicit geometric convergence rates for the original system and the system under perturbation are calculated. Our bounds in the geometric case and polynomial case are closely connected to the first hitting times. Two examples are provided to illustrate our result.

Title:  Truncated Euler-Maruyama Method for Time-Changed SDEs with Super-Linear State Variables and H/"older's Continuous Time Variables

Friday 3rd May, 2024, 3.00-4.00pm

Venue: LT907

Abstract: In this work, an explicit numerical method is developed for a class of non-autonomous time-changed stochastic differential equations, whose coefficients obey H\”older's continuity in terms of the time variables and are allowed to grow super-linearly in terms of the state variables. The strong convergence of the method in the finite time interval is proved and the convergence rate is obtained. Simulations are provided to demonstrate the theoretical results.

Title:  A New Criterion on Stability in Distribution for a Hybrid Stochastic Delay Differential Equation

Friday 17th May, 2024, 3.00-4.00pm

Venue: LT907

Abstract: A new sufficient condition for stability in distribution of a hybrid stochastic delay differential equation (SDDE) has been proposed in this work. Although the new criterion leads to stability for an SDDE, its main component only depends on the coefficients of a corresponding SDE without delay. The Lyapunov method is applied to find an upper bound, so that the SDDE is stable in distribution if the delay is less than the upper bound. Also, the criterion shows that delay terms can be an impetus toward the stability in distribution.

Title: Can We Rely On AI?

Date: 3.00pm Friday 19th January 2024

Venue: LT908

Abstract: Over the last decade, adversarial attack algorithms have revealed instabilities in deep learning tools. These algorithms raise issues regarding safety, reliability and interpretability in artificial intelligence (AI); especially in high risk settings. Mathematics is at the heart of this landscape, with ideas from optimization, numerical analysis and high dimensional stochastic analysis playing key roles. From a practical perspective, there has been a war of escalation between those developing attack and defence strategies. At a more theoretical level, researchers have also studied bigger picture questions concerning the existence and computability of successful attacks. I will present examples of attack algorithms in image classification and optical character recognition. I will also outline recent results on the overarching question of whether, under reasonable assumptions, it is inevitable that AI tools will be vulnerable to attack.

Tea/coffee will also be served in the staff common room (LT911) from 2:30pm.

For those of you who cannot attend in person, the talk will be streamed over zoom. 

https://strath.zoom.us/j/87299195984

Meeting ID: 872 9919 5984

Password: 729852

Title: Solution multiplicity and effects of data and eddy viscosity on Navier-Stokes solutions inferred by physics-informed neural networks

Date: 10am Wednesday 6th December 2023

Venue: Zoom (https://liverpool-ac-uk.zoom.us/j/97615683533?pwd=SWVxVXNnRmRCaUVlUkNINlRxNm94Zz09)  Meeting ID: 976 1568 3533   Passcode: m1LJJ$nU

Abstract: Physics-informed neural networks (PINNs) have emerged as a new simulation paradigm for fluid flows and are especially effective for inverse and hybrid problems. However, vanilla PINNs often fail in forward problems, especially at high Reynolds (Re) number flows. Herein, we study systematically the classical lid-driven cavity flow at Re=2,000, 3,000 and 5,000. We observe that vanilla PINNs obtain two classes of solutions, one class that agrees with direct numerical simulations (DNS), and another that is an unstable solution to the Navier-Stokes equations and not physically realizable. We attribute this solution multiplicity to singularities and unbounded vorticity, and we propose regularization methods that restore a unique solution within 1\% difference from the DNS solution. In particular, we introduce a parameterized entropy-viscosity method as artificial eddy viscosity and identify suitable parameters that drive the PINNs solution towards the DNS solution. Furthermore, we solve the inverse problem by subsampling the DNS solution, and identify a new eddy viscosity distribution that leads to velocity and pressure fields almost identical to their DNS counterparts. Surprisingly, a single measurement at a random point suffices to obtain a unique PINNs DNS-like solution even without artificial viscosity, which suggests possible pathways in simulating high Reynolds number turbulent flows using vanilla PINNs. 

Speaker Bio:  Zhicheng Wang received Ph.D. degree in engineering thermal physics from University of Chinese Academy of Sciences in 2013. He is currently an associate professor at School of Energy and Power Engineering, Dalian University of Technology, Dalian, China. His main research interests include high order numerical methods and scientific machine learning for predicting turbulent flows and multiphase phase flows, as well as the fluid structure interactions. He has published more than 20 papers in Proc. Natl. Acad. Sci. U.S.A., J. Comput. Phys., J. Fluid Mech., Comput. Method Appl. M., J. Fluids Struct., Phys. Fluids.