University of Warwick

If a dynamical system has a conservation law, i.e. a constant along the trajectory of the motion, the study of its evolution along the trajectories of a perturbed system becomes interesting. Conservation laws can be seen everywhere, especially at the level of probability distributions of a reduced dynamic.  We explain this with a number of models, in which we see a singular perturbation problem and identify a conservation law, the latter is used to seek out the correct scale to work with and to reduce the complexity of the system. The reduced dynamic consists of a family of  ODEs with rapidly oscillating right hands side from which in the limit we obtain a Markov process. For stochastic completely integrable system, the limit describes the evolution of the level sets of the family of Hamiltonian functions over a very large time scale.

Identifying correlations within multiple streams of high-volume time series is a general but challenging problem.  A simple exact solution has cost that is linear in the dimensionality of the data, and quadratic in the number of streams.  In this work, we use dimensionality reduction techniques (sketches), along with ideas derived from coding theory and fast matrix multiplication to allow fast (subquadratic) recovery of those pairs that display high correlation.

Joint work with Jacques Dark

 I will give a light introduction to the theory of regularity structures and then discuss recent developments with regards to renormalization within the theory - in particular I will describe joint work with Martin Hairer where multiscale techniques from constructive field theory are adapted to provide a systematic method of obtaining needed stochastic estimates for the theory. 

 Monte Carlo methods are one of the main tools of modern statistics and applied mathematics. They are commonly used to approximate integrals, which allows statisticians to solve many tasks of interest such as making predictions or inferring parameter values of a given model. However, the recent surge in data available to scientists has led to an increase in the complexity of mathematical models, rendering them much more computationally expensive to evaluate. This has a particular bearing on Monte Carlo methods, which will tend to be much slower due to the high computational costs.

This talk will introduce a Monte Carlo integration scheme which makes use of properties of the integrand (e.g. smoothness or periodicity) in order to obtain fast convergence rates in the number of integrand evaluations. This will allow users to obtain much more precise estimates of integrals for a given number of model evaluations. Both theoretical properties of the methodology, including convergence rates, and practical issues, such as the tuning of parameters, will be discussed. Finally, the proposed algorithm will be illustrated on a Bayesian inverse problem for a PDE model of subsurface flow.