Mathematical Institute (University of Oxford)

As part of the internal seminar schedule for Stochastic Analysis for this coming term, DPhil students have been invited to present on their works to date. Student talks are 20 minutes, which includes question and answer time. 

Students presenting are:

Akshay Hegde, supervisor Dmitry Beylaev

Julius Villar, supervisor Dmitry Beylaev

Csaba Toth, supervisor Harald Oberhauser 

Diffusion is one of the most common phenomenon in natural sciences and large part of applied mathematics have been interested in the tools to model it. Trying to study different types of diffusions, the mathematical ways to describe them and the numerical methods to simulate them is an appealing challenge, giving a wide range of applications. The aim of our work is the design of a finite-volume numerical scheme to model non-local diffusion given by the fractional Laplacian and to build numerical solutions for the Lévy-Fokker-Planck equation that involves it. Numerical methods for fractional diffusion have been indeed developed during the last few years and large part of the literature has been focused on finite element methods. Few results have been rather proposed for different techniques such as finite volumes.

 
We propose a new fractional Laplacian for bounded domains, which is expressed as a conservation law. This new approach is therefore particularly suitable for a finite volumes scheme and allows us also to prescribe no-flux boundary conditions explicitly. We enforce our new definition with a well-posedness theory for some cases to then capture with a good level of approximation the action of fractional Laplacian and its anomalous diffusion effect with our numerical scheme. The numerical solutions we get for the Lévy-Fokker-Planck equation resemble in fact the known analytical predictions and allow us to numerically explore properties of this equation and compute stationary states and long-time asymptotics.

Optimization problems constrained to a smooth manifold can be solved via the framework of Riemannian optimization. To that end, a geometrical description of the constraining manifold, e.g., tangent spaces, retractions, and cost function gradients, is required. In this talk, we present a novel approach that allows performing approximate Riemannian optimization based on a manifold learning technique, in cases where only a noiseless sample set of the cost function and the manifold’s intrinsic dimension are available.

Neural SDEs are continuous-time generative models for sequential data. State-of-the-art performance for irregular time series generation has been previously obtained by training these models adversarially as GANs. However, as typical for GAN architectures, training is notoriously unstable, often suffers from mode collapse, and requires specialised techniques such as weight clipping and gradient penalty to mitigate these issues. In this talk, I will introduce a novel class of scoring rules on path space based on signature kernels and use them as an objective for training Neural SDEs non-adversarially. The strict properness of such kernel scores and the consistency of the corresponding estimators, provide existence and uniqueness guarantees for the minimiser. With this formulation, evaluating the generator-discriminator pair amounts to solving a system of linear path-dependent PDEs which allows for memory-efficient adjoint-based backpropagation. Moreover, because the proposed kernel scores are well-defined for paths with values in infinite-dimensional spaces of functions, this framework can be easily extended to generate spatiotemporal data. This procedure permits conditioning on a rich variety of market conditions and significantly outperforms alternative ways of training Neural SDEs on a variety of tasks including the simulation of rough volatility models, the conditional probabilistic forecasts of real-world forex pairs where the conditioning variable is an observed past trajectory, and the mesh-free generation of limit order book dynamics.

Join us for our first Fridays@4 session of Trinity about different academic routes people take post-PhD, with a particular focus on fellowships and grants. We’ll hear from Jason Lotay about his experiences on both sides of the application process, as well as hear about the experiences of ECRs in the South Wing, North Wing, and Statistics. Towards the end of the hour we’ll have a Q+A session with the whole panel, where you can ask any questions you have around this topic!

The basic question in prime number theory is to try to understand the number of primes in some interesting set of integers. Unfortunately many of the most basic and natural examples are famous open problems which are over 100 years old!

We aim to give an accessible survey of (a selection of) the main results and techniques in prime number theory. In particular we highlight progress on some of these famous problems, as well as a selection of our favourite problems for future progress.

Identifying the hidden organizational principles and relevant structures of complex networks is fundamental to understand their properties. To this end, uncovering the structures involving the prominent nodes in a network is an effective approach. In temporal networks, the simultaneity of connections is crucial for temporally stable structures to arise. In this work, we propose a measure to quantitatively investigate the tendency of well-connected nodes to form simultaneous and stable structures in a temporal network. We refer to this tendency as the temporal rich club phenomenon, characterized by a coefficient defined as the maximal value of the density of links between nodes with a minimal required degree, which remain stable for a certain duration. We illustrate the use of this concept by analysing diverse data sets and their temporal properties, from the role of cohesive structures in relation to processes unfolding on top of the network to the study of specific moments of interest in the evolution of the network.

Article link: https://www.nature.com/articles/s41567-022-01634-8

The structure of complex networks can be characterized by counting and analyzing network motifs, which are small graph structures that occur repeatedly in a network, such as triangles or chains. Recent work has generalized motifs to temporal and dynamic network data. However, existing techniques do not generalize to sequential or trajectory data, which represents entities walking through the nodes of a network, such as passengers moving through transportation networks. The unit of observation in these data is fundamentally different, since we analyze observations of walks (e.g., a trip from airport A to airport C through airport B), rather than independent observations of edges or snapshots of graphs over time. In this work, we define sequential motifs in trajectory data, which are small, directed, and sequenced-ordered graphs corresponding to patterns in observed sequences. We draw a connection between counting and analysis of sequential motifs and Higher-Order Network (HON) models. We show that by mapping edges of a HON, specifically a kth-order DeBruijn graph, to sequential motifs, we can count and evaluate their importance in observed data, and we test our proposed methodology with two datasets: (1) passengers navigating an airport network and (2) people navigating the Wikipedia article network. We find that the most prevalent and important sequential motifs correspond to intuitive patterns of traversal in the real systems, and show empirically that the heterogeneity of edge weights in an observed higher-order DeBruijn graph has implications for the distributions of sequential motifs we expect to see across our null models.

ArXiv link: https://arxiv.org/abs/2112.05642