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.

Gaussian processes provide a powerful probabilistic kernel learning framework, which allows high-quality nonparametric learning via methods such as Gaussian process regression. Nevertheless, its learning phase requires unrealistic massive computations for large datasets. In this talk, we present a quadrature-based approach for scaling up Gaussian process regression via a low-rank approximation of the kernel matrix. The low-rank structure is utilized to achieve effective hyperparameter learning, training, and prediction. Our Gauss-Legendre features method is inspired by the well-known random Fourier features approach, which also builds low-rank approximations via numerical integration. However, our method is capable of generating high-quality kernel approximation using a number of features that is poly-logarithmic in the number of training points, while similar guarantees will require an amount that is at the very least linear in the number of training points when using random Fourier features. The utility of our method for learning with low-dimensional datasets is demonstrated using numerical experiments.

The Ciarlet definition of a finite element has been used for many years to describe the requisite parts of a finite element. In that time, finite element theory and implementation have both developed and improved, which has left scope for a redefinition of the concept of a finite element. In this redefinition, we look to encapsulate some of the assumptions that have historically been required to complete Ciarlet’s definition, as well as incorporate more information, in particular relating to the symmetries of finite elements, using concepts from Group Theory. This talk will present the machinery of the proposed new definition, discuss its features and provide some examples of commonly used elements.

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.

The PDE formulation of Mean Field Games (MFG) is described by nonlinear systems in which a Hamilton—Jacobi—Bellman (HJB) equation and a Kolmogorov—Fokker—Planck (KFP) equation are coupled. The advective term of the KFP equation involves a partial derivative of the Hamiltonian that is often assumed to be continuous. However, in many cases of practical interest, the underlying optimal control problem of the MFG may give rise to bang-bang controls, which typically lead to nondifferentiable Hamiltonians. In this talk we present results on the analysis and numerical approximation of second-order MFG systems for the general case of convex, Lipschitz, but possibly nondifferentiable Hamiltonians.
In particular, we propose a generalization of the MFG system as a Partial Differential Inclusion (PDI) based on interpreting the partial derivative of the Hamiltonian in terms of subdifferentials of convex functions.

We present theorems that guarantee the existence of unique weak solutions to MFG PDIs under a monotonicity condition similar to one that has been considered previously by Lasry & Lions. Moreover, we introduce a monotone finite element discretization of the weak formulation of MFG PDIs and prove the strong convergence of the approximations to the value function in the H¹-norm and the strong convergence of the approximations to the density function in L^q-norms. We conclude the talk with some numerical experiments involving non-smooth solutions. 

This is joint work with my supervisor Iain Smears. 

Classically, topological vector bundles are classified by homotopy classes of maps into infinite Grassmannians. This allows us to study topological vector bundles using obstruction theory: we can detect whether a vector bundle has a trivial subbundle by means of cohomological invariants. In the context of algebraic geometry, one can ask whether algebraic vector bundles over smooth affine varieties can be classified in a similar way. Recent advances in motivic homotopy theory give a positive answer, at least over an algebraically closed base field. Moreover, the behaviour of vector bundles over general base fields has surprising connections with the theory of quadratic forms.