Matthew Thorpe (The University of Warwick)

Discrete-To-Continuum Limits in Graph-Based Semi-Supervised Learning

Semi-supervised learning (SSL) is the problem of finding missing labels from a partially labelled data set. The heuristic one uses is that “similar feature vectors should have similar labels”. The notion of similarity between feature vectors explored in this talk comes from a graph-based geometry where an edge is placed between feature vectors that are closer than some connectivity radius. A natural variational solution to the SSL is to minimise a Dirichlet energy built from the graph topology. And a natural question is to ask what happens as the number of feature vectors goes to infinity? In this talk I will give results on the asymptotics of graph-based SSL using an optimal transport topology. The results will include a lower bound on the number of labels needed for consistency.

Riccardo Cristoferi (Radboud University)

Shape optimization for attractive-repulsive energies

Patterns are everywhere! Their regularity is both fascinating and intriguing: if from an aesthetic point of view we are pleased when admiring them, from a mathematical point of view we want to explain this regularity.

In this talk we will shed some light on mechanisms behind pattern formation. In particular, we focus on the formation of a single shape. We work in a variational framework, namely we view patterns as configurations having least energy. The optimal shape will then be the compromise of the effects of competing forces. The prototype is having an attractive and a repulsive force. We will consider several variants of such terms having different features, and investigate how this affect the optimal shape.

This talk is based on a series of works in collaboration with Marco Bonacini (Università di Trento), Maria Giovanna Mora (Università di Pavia), Lucia Scardia (Heriot-Watt University), and Ihsan Topaloglu (Virginia Commonwealth University).

Bertram Düring (The University of Warwick)

A Lagrangian scheme for the solution of nonlinear diffusion equations

Many nonlinear diffusion equations can be interpreted as gradient flows whose dynamics are driven by internal energies and given external potentials, examples include the heat equation and the porous medium equation. When solving these equations numerically, schemes that respect the equations’ special structure are of particular interest. In this talk we present a Lagrangian scheme for nonlinear diffusion equations.

For discretisation of the Lagrangian map, we use a finite subspace of linear maps in space and a variational form of the implicit Euler method in time. We present numerical experiments for the porous medium equation in two space dimensions.