- Date: 6 Sep. (Tue.)
- Place: W.W. 6th-floor, Colloquium Room and on the Web (Zoom)
- Time: 16:30-18:00
- Speaker: Dan Crisan (Imperial College London)
- Title: Classical and modern results in the theory and applications of stochastic filtering

Onwards from the mid-twentieth century, the stochastic filtering problem has caught the attention of thousands of mathematicians, engineers, statisticians, and computer scientists. Its applications span the whole spectrum of human endeavour, including satellite tracking, credit risk estimation, human genome analysis, and speech recognition. Stochastic filtering has engendered a surprising number of mathematical techniques for its treatment and has played an important role in the development of new research areas, including stochastic partial differential equations, stochastic geometry, rough paths theory, and Malliavin calculus. It also spearheaded research in areas of classical mathematics, such as Lie algebras, control theory, and information theory. The aim of this talk is to give a historical account of the subject concentrating on the continuous-time framework. I will also present a recent application of filtering to the estimation of partially observed high dimensional fluid dynamics models. In particular, I will introduce a so-called particle filter that incorporates a nudging mechanism. The nudging procedure is used in the prediction step. In the absence of nudging, the particles have trajectories that are independent solutions of the model equations. The nudging presented here consists in adding a drift to the trajectories of the particles with the aim of maximising the likelihood of their positions given the observation data. This introduces a bias in the system that is corrected during the resampling step. The methodology is tested on a two-layer quasi-geostrophic model for a beta-plane channel flow with O(10^6) degrees of freedom out of which only a minute fraction are noisily observed.

The talk is based on the papers:

[1] D Crisan, The stochastic filtering problem: a brief historical account, Journal of Applied Probability 51 (A), 13-22

[2] C Cotter, D Crisan, D Holm, W Pan, I Shevchenko, Data assimilation for a quasi-geostrophic model with circulation-preserving stochastic transport noise, Journal of Statistical Physics, 1-36, 2020.

[3] D Crisan, I Shevchenko, Particle filters with nudging, work in progress.