Mathematical Institute (University of Oxford)

Multicomponent fluids are mixtures of distinct chemical species (i.e. components) that interact through complex physical processes such as cross-diffusion and chemical reactions. Additional physical phenomena often must be accounted for when modelling these fluids; examples include momentum transport, thermality and (for charged species) electrical effects. Despite the ubiquity of chemical mixtures in nature and engineering, multicomponent fluids have received almost no attention from the finite element community, with many important applications remaining out of reach from numerical methods currently available in the literature. This is in spite of the fact that, in engineering applications, these fluids often reside in complicated spatial regions -- a situation where finite elements are extremely useful! In this talk, we present a novel class of high-order finite element methods for simulating cross-diffusion and momentum transport (i.e. convection) in multicomponent fluids. Our model can also incorporate local electroneutrality when the species carry electrical charge, making the numerical methods particularly desirable for simulating liquid electrolytes in electrochemical applications. We discuss challenges that arise when discretising the partial differential equations of multicomponent flow, as well as some salient theoretical properties of our numerical schemes. Finally, we present numerical simulations involving (i) the microfluidic non-ideal mixing of hydrocarbons and (ii) the transient evolution of a lithium-ion battery electrolyte in a Hull cell electrode.

Contact topology and hyperbolic geometry are two well-established, yet so far largely unrelated subfields of 3-manifold topology. We will discuss a recent result relating phenomena in these two fields. Specifically, we will demonstrate that tightness of certain contact structures on hyperbolic manifolds is detected by the behaviour of geodesics in the underlying hyperbolic geometry. A key geometric tool we will discuss is the deformation theory for hyperbolic manifolds. 

The Ciarlet definition of a finite element has been core to our understanding of the finite element method since its inception. It has proved particularly useful in structuring the implementation of finite element software. However, the definition does not encapsulate all the details required to uniquely implement an element, meaning each user of the definition (whether a researcher or software package) must make further mathematical assumptions to produce a working system. 

The talk presents a new definition built on Ciarlet’s that addresses these concerns. The novel definition forms the core of a new piece of software in development, FUSE, which allows the users to consider the choice of finite element as part of the data they are working with. This is a new implementation strategy among finite element software packages, and we will discuss some potential benefits of the development.

We present a new ensemble framework for boosting the performance of overparameterized unfolding networks solving the compressed sensing problem. We combine a state-of-the-art overparameterized unfolding network with a continuation technique, to warm-start a crucial quantity of the said network's architecture; we coin the resulting continued network C-DEC. Moreover, for training and evaluating C-DEC, we incorporate the log-cosh loss function, which enjoys both linear and quadratic behavior. Finally, we numerically assess C-DEC's performance on real-world images. Results showcase that the combination of continuation with the overparameterized unfolded architecture, trained and evaluated with the chosen loss function, yields smoother loss landscapes and improved reconstruction and generalization performance of C-DEC, consistently for all datasets.

An adaptive regularization algorithm for unconstrained nonconvex optimization is presented in
which the objective function is never evaluated, but only derivatives are used and without prior knowledge of Lipschitz constant.  This algorithm belongs to the class of adaptive regularization methods, for which optimal worst-case complexity results are known for the standard framework where the objective function is evaluated. It is shown in this paper that these excellent complexity bounds are also valid for the new algorithm. Theoretical analysis of both exact and stochastic cases are discussed and  new probabilistic conditions on tensor derivatives are proposed.  Initial experiments on large binary classification highlight the merits of our method.

High-order tensor methods employing local Taylor approximations have attracted considerable attention for convex and nonconvex optimisation. The pth-order adaptive regularisation (ARp) approach builds a local model comprising a pth-order Taylor expansion and a (p+1)th-order regularisation term, delivering optimal worst-case global and local convergence rates. However, for p≥2, subproblem minimisation can yield multiple local minima, and while a global minimiser is recommended for p=2, effectively identifying a suitable local minimum for p≥3 remains elusive.
This work extends interpolation-based updating strategies, originally proposed for p=2, to cases where p≥3, allowing the regularisation parameter to adapt in response to interpolation models. Additionally, it introduces a new prerejection mechanism to discard unfavourable subproblem minimisers before function evaluations, thus reducing computational costs for p≥3.
Numerical experiments, particularly on Chebyshev-Rosenbrock problems with p=3, indicate that the proper use of different minimisers can significantly improve practical performance, offering a promising direction for designing more efficient high-order methods.

Are you thinking of launching your own start-up or considering joining an early-stage company? Navigating the entrepreneurial landscape can be both exciting and challenging. Join Pete for an interactive exploration of the unwritten rules and hidden insights that can make or break a start-up journey.

Drawing from personal experience, Pete's talk will offer practical wisdom for aspiring founders and team members, revealing the challenges and opportunities of building a new business from the ground up.

Whether you're an aspiring entrepreneur, a potential start-up team member, or simply curious about innovative businesses, you'll gain valuable perspectives on the realities of creating something from scratch.

This isn't a traditional lecture – it will be a lively conversation that invites participants to learn, share, and reflect on the world of start-ups. Come prepared to challenge your assumptions and discover practical insights that aren't found in standard business guides.
 

Speaker: Professor Pete Grindrod

Networks provide a powerful language to model and analyse interconnected systems. Their building blocks are  edges, which can  then be combined to form walks and paths, and thus define indirect relations between distant nodes and model flows across the system. In a traditional setting, network models are first-order, in the sense that flow across nodes is made of independent sequences of transitions. However, real-world systems often exhibit higher-order dependencies, requiring more sophisticated models. Here, we propose a variable-order network model that captures memory effects by interpolating between first- and second-order representations. Our method identifies latent modes that explain second-order behaviors, avoiding overfitting through a Bayesian prior. We introduce an interpretable measure to balance model size and description quality, allowing for efficient, scalable processing of large sequence data. We demonstrate that our model captures key memory effects with minimal state nodes, providing new insights beyond traditional first-order models and avoiding the computational costs of existing higher-order models.