Conference Agenda

A nonintrusive model order reduction method for bilinear stochastic differential equations with Gaussian noise is proposed. A reduced order model (ROM) is designed in order to approximate the statistical properties of high-dimensional systems. The drift and diffusion coefficients of the ROM are inferred from state observations by solving appropriate least-squares problems. The closeness of the ROM obtained by the presented approach to the intrusive ROM obtained by the proper orthogonal decomposition (POD) method is investigated. Two generalisations of the snapshot-based dominant subspace construction to the stochastic case are presented. Numerical experiments are provided to compare the developed approach to POD.

Ensemble Kalman Filters (EnKFs) are a class of Monte Carlo algorithms devolped in the 90s for high dimensional stochastic filtering problems. Despite their widespread popularity, especially for numerical weather prediction and data assimilation tasks in the geosciences, a mathematical theory investigating these kinds of algorithms has emerged only recently in the last decade. In particular estimating the asymptotic bias for these kinds of algorithms in the case of nonlinear dynamics and mathematical justifications for their usage in these settings, is a longstanding open problem. This talk is therefore concerned with the elementary mathematical analysis of continuous time versions of EnKFs, which are a particular class of mean field interacting Stochastic Differential Equations. Besides elementary well posedness results, we show a quantitative convergence to the mean field limit with (almost) optimal rates and compare/relate this mean field limit to the optimal filter given by the Kushner-Stratonovich equation.

In this talk we consider an ergodic version of the bounded velocity follower problem, assuming that the decision maker lacks knowledge of the underlying system parameters and must learn them while simultaneously controlling. We propose algorithms based on moving empirical averages and develop a framework for integrating statistical methods with stochastic control theory. Our primary result is a logarithmic expected regret rate. To achieve this, we conduct a rigorous analysis of the ergodic convergence rates of the underlying processes and the risks of the considered estimators.