In this talk, we will discuss a multilevel coarsening method for weighted graph Laplacians formulated in a mixed saddle-point form. In this formulation, there are two sets of degrees of freedom (dofs), which are respectively associated with the vertices and edges of the underlying graph. We construct coarse spaces for both sets of dofs based on aggregation of vertices exploiting local spectral decompositions. The coarse spaces have provable approximation property, and are constructed such that the inf-sup stability on coarse levels is maintained. The coarsening procedure is local with high computation-to-communication ratio, the benefit of which is illustrated in a parallel scaling test. An efficient linear solver for the coarse problems is also developed based on the hybridization technique. For applications of the coarsening method, we focus on graph Laplacians coming from the finite volume discretization problems in oil reservoir simulations. Numerical examples demonstrating its applications in constructing eigensolvers for graph Laplacians, multilevel samplers, multilevel Monte Carlo simulations, and nonlinear multigrid solvers will be presented.

Modern machine learning models, such as neural networks, have a number of theoretically puzzling but empirically robust properties. Chief among them are: (a) neural networks are trained on datasets which are much smaller than the total number of model parameters; (b) training proceeds by empirical risk minimization via a first order method from a random starting point and, despite the non-convexity of the risk, typically returns a global minimizer; (c) this minimizer of the risk not only fits interpolates the data precisely but also performs well on unseen data (i.e. generalizes). The purpose of this talk is to introduce these fascinating properties and give some basic intuitions for why they might be possible. The emphasis will be on heuristics rather than on precise theorems.