Stability and Pattern formation in Nonlocal Interaction Models

Geometric methods based on PDEs have revolutionized the field of image processing and image analysis. I will discuss recent work to develop these ideas for machine learning applications involving “big data”. The main idea is to pose variational problems involving graph cuts in terms of total variation minimization problems. We then develop both phase field and mean curvature methods to solve these problems quickly. I will introduce the notion of the Ginzburg-Landau functional on graphs and the related dynamic thresholding method. Unlike numerical methods for PDEs, the graph problems are able to exploit dramatic spectral truncation of the graph Laplacian, sometimes with a tiny fraction of the eigenfunctions. I will show examples from semi-supervised learning, nonlocal means image processing, and modularity optimization for unsupervised learning and community detection in networks. 2 Stability and Pattern formation in Nonlocal Interaction Models José Antonio Carrillo, Imperial College London, Great Britain Abstract. I will review some recent results for first and second order models of swarming in terms of patterns, stationary states, and qualitative properties. I will discuss the stability of these patterns for the continuum and discrete particle cases. These non-local models appear in collective behavior for animals, control engineering, and molecular structures among others. We first concentrate in the spatial shape of these patterns and the dynamics when inertia terms are neglected. The mathematical question behind consists in finding properties about local minimizers of the total interaction energy. Concerning 2nd order models, we will discuss particular properties of two patterns: flocks and mills. We will discuss the stability of these patterns in the discrete case. In both cases, we will describe the properties obtained for the continuum limits. I will review some recent results for first and second order models of swarming in terms of patterns, stationary states, and qualitative properties. I will discuss the stability of these patterns for the continuum and discrete particle cases. These non-local models appear in collective behavior for animals, control engineering, and molecular structures among others. We first concentrate in the spatial shape of these patterns and the dynamics when inertia terms are neglected. The mathematical question behind consists in finding properties about local minimizers of the total interaction energy. Concerning 2nd order models, we will discuss particular properties of two patterns: flocks and mills. We will discuss the stability of these patterns in the discrete case. In both cases, we will describe the properties obtained for the continuum limits.