Mathematical Institute (University of Oxford)

Cryptocurrencies such as Bitcoin have only recently become a significant part of the financial landscape. Many billions of dollars are now traded daily on limit order-book markets such as Binance, and these are probably among the most open, liquid and transparent markets there are. They therefore make an interesting platform from which to investigate myriad questions to do with market microstructure. I shall talk about a few of these, including live-trading experiments to investigate the difference between on-paper strategy analysis (typical in the academic literature) and actual trading outcomes. I shall also mention very recent work on the new Hyperliquid exchange which runs on a blockchain basis, showing how to use this architecture to obtain datasets of an unprecendented level of granularity. This is joint work with Jakob Albers, Mihai Cucuringu and Alex Shestopaloff.

High-order tensor methods for solving both convex and nonconvex optimization problems have recently generated significant research interest, due in part to the natural way in which higher derivatives can be incorporated into adaptive regularization frameworks, leading to algorithms with optimal global rates of convergence and local rates that are faster than Newton's method. On each iteration, to find the next solution approximation, these methods require the unconstrained local minimization of a (potentially nonconvex) multivariate polynomial of degree higher than two, constructed using third-order (or higher) derivative information, and regularized by an appropriate power of the change in the iterates. Developing efficient techniques for the solution of such subproblems is currently, an ongoing topic of research,  and this talk addresses this question for the case of the third-order tensor subproblem. In particular, we propose the CQR algorithmic framework, for minimizing a nonconvex Cubic multivariate polynomial with  Quartic Regularisation, by sequentially minimizing a sequence of local quadratic models that also incorporate both simple cubic and quartic terms.

The role of the cubic term is to crudely approximate local tensor information, while the quartic one provides model regularization and controls progress. We provide necessary and sufficient optimality conditions that fully characterise the global minimizers of these cubic-quartic models. We then turn these conditions into secular equations that can be solved using nonlinear eigenvalue techniques. We show, using our optimality characterisations, that a CQR algorithmic variant has the optimal-order evaluation complexity of $O(\epsilon^{-3/2})$ when applied to minimizing our quartically-regularised cubic subproblem, which can be further improved in special cases.  We propose practical CQR variants that judiciously use local tensor information to construct the local cubic-quartic models. We test these variants numerically and observe them to be competitive with ARC and other subproblem solvers on typical instances and even superior on ill-conditioned subproblems with special structure.

Spectro-microscopy is an experimental technique with great potential to science challenges such as the observation of changes over time in energy materials or environmental samples and investigations of the chemical state in biological samples. However, its application is often limited by factors like long acquisition times and radiation damage. We present two measurement strategies that significantly reduce experiment times and applied radiation doses. These strategies involve acquiring only a small subset of all possible measurements and then completing the full data matrix from the sampled measurements. The methods are data-driven, utilizing spectral and spatial importance subsampling distributions to select the most informative measurements. Specifically, we use data-driven leverage scores and adaptive randomized pivoting techniques. We explore raster importance sampling combined with the LoopASD completion algorithm, as well as CUR-based sampling where the CUR approximation also serves as the completion method. Additionally, we propose ideas to make the CUR-based approach adaptive. As a result, capturing as little as 4–6% of the measurements is sufficient to recover the same information as a conventional full scan.

Low-rank approximation of parameter-dependent matrices A(t) is an important task in the computational sciences, with applications in areas such as dynamical systems and the compression of series of images. In this talk, we introduce AdaCUR, an efficient randomised algorithm for computing low-rank approximations of parameter-dependent matrices using the CUR decomposition. The key idea of our approach is the ability to reuse column and row indices for nearby parameter values, improving efficiency. The resulting algorithm is rank-adaptive, provides error control, and has complexity that compares favourably with existing methods. This is joint work with Yuji Nakatsukasa.

Many applications in electromagnetism, magnetohydrodynamics, and pour media flow are well-posed in spaces from the 3D de Rham complex involving $H^1$, $H(curl)$, $H(div)$, and $L^2$. Discretizing these spaces with the usual conforming finite element spaces typically leads to discrete problems that are both structure-preserving and uniformly stable with respect to the mesh size and polynomial degree. Robust preconditioners/solvers usually require the inversion of subproblems or auxiliary problems on vertex, edge, or face patches of elements. For high-order discretizations, the cost of inverting these patch problems scales like $\mathcal{O}(p^9)$ and is thus prohibitively expensive. We propose a new set of basis functions for each of the spaces in the discrete de Rham complex that reduce the cost of the patch problems to $\mathcal{O}(p^6)$ complexity. By taking advantage of additional properties of the new basis, we propose further computationally cheaper variants of existing preconditioners. Various numerical examples demonstrate the performance of the solvers.

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.