PDEA Minisymposium: Kinetic equations, fractional porous medium equations, random fields on Riemannian manifolds, and large deviations

The symposium consists of the following presentations:

10.15-11, Å101150, Andrea Pascucci (Bologna)

Title: Functional analysis for kinetic Fokker-Plank equations

Abstract: We give an overview of functional spaces of Hölder and intrinsic Sobolev-Slobodeckij type that naturally arise in the study of a class of ultra-parabolic Kolmogorov type operators satisfying the weak Hörmander condition. We prove continuous embeddings into Lorentz and intrinsic Hölder spaces. We also prove approximation and interpolation inequalities by means of an intrinsic Taylor expansion, extending analogous results for Hölder spaces.

11.15-12, Å101150, Espen R. Jakobsen (NTNU)

Title: A convergent discretisation of the porous medium equation with fractional pressure.

Abstract: see attatched PDF/LaTeX.

13.15-14, Å101127, Annika Lang (Chalmers)

Title: Simulation of random fields on Riemannian manifolds

Abstract: Random fields are important building blocks in spatial models disturbed by randomness such as solutions to stochastic partial differential equations. The fast simulation of random fields is therefore crucial for efficient algorithms in uncertainty quantification. In this talk I present numerical methods for Gaussian random fields on Riemannian manifolds and discuss their convergence. Simulations illustrate the theoretical findings. This talk is based on joint work with Erik Jansson, Mihály Kovács, and Mike Pereira.

14.15-15, Å101127, Henrik Hult (KTH)

Title: On large deviations for stochastic approximations.

Abstract: Stochastic approximation is a general and useful random iterative  root finding algorithm originating from the work of Robbins and Monro in the 1950s. Many popular training algorithms in statistics, machine learning, and statistical physics can be formulated as stochastic approximations, including stochastic gradient descent, reinforcement learning, contrastive divergence, adaptive MCMC, and various adapted extended ensamble methods such as Wang-Landau and accelerated weight histograms. In this talk we will present on-going work on large deviations for stochastic approximations and provide a new representation of the rate function. An interpretation that learning algorithms can forget will be discussed and the rate function reveals how the forgetting occurs. The talk is based on joint work with Adam Lindhe, Pierre Nyquist and Guo-Jhen Wu.