University College London

The formulation of Mean Field Games (MFG) via partial differential equations typically requires continuous differentiability of the Hamiltonian in order to determine the advective term in the Kolmogorov--Fokker--Planck equation for the density of players. However, in many cases of practical interest, the underlying optimal control problem may exhibit bang-bang controls, which typically lead to nondifferentiable Hamiltonians. In this talk we will present results on the analysis and numerical approximation of stationary second-order MFG systems for the general case of convex, Lipschitz, but possibly nondifferentiable Hamiltonians. In particular, we will propose a generalization of the MFG system as a Partial Differential Inclusion (PDI) based on interpreting the derivative of the Hamiltonian in terms of subdifferentials of convex functions. We present results that guarantee the existence of unique weak solutions to the stationary MFG PDI under a monotonicity condition similar to one that has been considered previously by Lasry and Lions. Moreover, we will propose a monotone finite element discretization of the weak formulation of the MFG PDI, and present results that confirm the strong H^1-norm convergence of the approximations to the value function and strong L^q-norm convergence of the approximations to the density function. The performance of the numerical method will be illustrated in experiments featuring nonsmooth solutions. This talk is based on joint work with my supervisor Iain Smears.

The design and analysis of finite element methods for high Reynolds flow remains a challenging task, not least because of the difficulties associated with turbulence. In this talk we will first revisit some theoretical results on interior penalty methods using equal order interpolation for smooth solutions of the Navier-Stokes’ equations at high Reynolds number and show some recent computational results for turbulent flows.

Then we will focus on so called pressure robust methods, i.e. methods where the smoothness of the pressure does not affect the upper bound of error estimates for the velocity of the Stokes’ system. We will discuss how convection can be stabilized for such methods in the high Reynolds regime and, for the lowest order case, show an interesting connection to turbulence modelling.

In this talk we will look at the different ways to define city boundaries, and the relevance to consider socio-demographic and spatial connectivity in urban systems, in particular if interventions are to be considered.

How can one integrate singular functions over fractals? And why would one want to do this? In this talk I will present a general approach to numerical quadrature on the compact attractor of an iterated function system of contracting similarities, where integration is with respect to the relevant Hausdorff measure. For certain singular integrands of logarithmic or algebraic type the self-similarity of the integration domain can be exploited to express the singular integral exactly in terms of regular integrals that can be approximated using standard techniques. As an application we show how this approach, combined with a singularity-subtraction technique, can be used to accurately evaluate the singular double integrals that arise in Hausdorff-measure Galerkin boundary element methods for acoustic wave scattering by fractal screens. This is joint work with Andrew Gibbs (UCL) and Andrea Moiola (Pavia).

In this talk, we describe some recent work on applying tools from category theory in finite model theory, descriptive complexity, constraint satisfaction, and combinatorics.

The motivations for this work come from Computer Science, but there may be something of interest for model theorists and other logicians.

The basic setting involves studying the category of relational structures via a resource-indexed family of adjunctions with some process category - which unfolds relational structures into treelike forms, allowing natural resource parameters to be assigned to these unfoldings.

One basic instance of this scheme allows us to recover, in a purely structural, syntax-free way:

- the Ehrenfeucht-Fraisse game

- the quantifier rank fragments of first-order logic

- the equivalences on structures induced by (i) the quantifier rank fragments, (ii) the restriction to the existential-positive part, and (iii) the extension with counting quantifiers

- the combinatorial parameter of tree-depth (Nesetril and Ossona de Mendez).

Another instance recovers the k-pebble game, the finite-variable fragments, the corresponding equivalences, and the combinatorial parameter of treewidth.

Other instances cover modal, guarded and hybrid fragments, generalized quantifiers, and a wide range of combinatorial parameters.

This whole scheme has been axiomatized in a very general setting, of arboreal categories and arboreal covers.

Beyond this basic level, a landscape is beginning to emerge, in which structural features of the resource categories, adjunctions and comonads are reflected in degrees of logical and computational tractability of the corresponding languages.

Examples include semantic characterisation and preservation theorems, Lovasz-type results on  isomorphisms, and classification of constraint satisfaction problems.

We develop random graph models where graphs are generated by connecting not only pairs of vertices by edges but also larger subsets of vertices by copies of small atomic subgraphs of arbitrary topology. This allows the for the generation of graphs with extensive numbers of triangles and other network motifs commonly observed in many real world networks. More specifically we focus on maximum entropy ensembles under constraints placed on the counts and distributions of atomic subgraphs and derive general expressions for the entropy of such models. We also present a procedure for combining distributions of multiple atomic subgraphs that enables the construction of models with fewer parameters. Expanding the model to include atoms with edge and vertex labels we obtain a general class of models that can be parametrized in terms of basic building blocks and their distributions that includes many widely used models as special cases. These models include random graphs with arbitrary distributions of subgraphs, random hypergraphs, bipartite models, stochastic block models, models of multilayer networks and their degree corrected and directed versions. We show that the entropy for all these models can be derived from a single expression that is characterized by the symmetry groups of atomic subgraphs.

Materials made from a mixture of liquid and solid are, instinctively, very obviously complex. From dilatancy (the reason wet sand becomes dry when you step on it) to extreme shear-thinning (quicksand) or shear-thickening (cornflour oobleck) there is a wide range of behaviours to explain and predict. I'll discuss the seemingly simple case of solid spheres suspended in a Newtonian fluid matrix, which still has plenty of surprises up its sleeve.

We consider random graph models where graphs are generated by connecting not only pairs of nodes by edges but also larger subsets of
nodes by copies of small atomic subgraphs of arbitrary topology. More specifically we consider canonical and microcanonical ensembles
corresponding to constraints placed on the counts and distributions of atomic subgraphs and derive general expressions for the entropy of such
models. We also introduce a procedure that enables the distributions of multiple atomic subgraphs to be combined resulting in more coarse
grained models. As a result we obtain a general class of models that can be parametrized in terms of basic building blocks and their
distributions that includes many widely used models as special cases. These models include random graphs with arbitrary distributions of subgraphs (Karrer & Newman PRE 2010, Bollobas et al. RSA 2011), random hypergraphs, bipartite models, stochastic block models, models of multilayer networks and their degree corrected and directed versions. We show that the entropy expressions for all these models can be derived from a single expression that is characterized by the symmetry groups of their atomic subgraphs.

Cities are now central to addressing global changes, ranging from climate change to economic resilience. There is a growing concern of how to measure and quantify urban phenomena, and one of the biggest challenges in quantifying different aspects of cities and creating meaningful indicators lie in our ability to extract relevant features that characterize the topological and spatial patterns of urban form. Many different models that can reproduce large-scale statistical properties observed in systems of streets have been proposed, from spatial random graphs to economical models of network growth. However, existing models fail to capture the diversity observed in street networks around the world. The increased availability of street network datasets and advancements in deep learning models present a new opportunity to create more accurate and flexible models of urban street networks, as well as capture important characteristics that could be used in downstream tasks.  We propose a simple approach called Convolutional-PCA (ConvPCA) for both creating low-dimensional representations of street networks that can be used for street network classification and other downstream tasks, as well as a generating new street networks that preserve visual and statistical similarity to observed street networks.

Link to the preprint