Events for 03/04/2025 from all calendars

Nonlinear Partial Differential Equations

Time: 3:00PM - 4:00PM

Location: Blocker 302

Speaker: Animikh Biswas, University of Maryland Baltimore County

Title: ACCURACY AND STABILITY OF A CLASS OF APPROXIMATE GAUSSIAN FILTERS FOR THE NAVIER- STOKES EQUATIONS

Abstract: We develop a unified framework for the analysis of several well-known and empirically efficient data assimilation techniques derived from various Gaussian approximations of the Bayesian filtering schemes for geophysical-type dissipative dynamics with quadratic nonlinearities. We establish rigorous results on (time-asymptotic) accuracy and stability of these algorithms with general covariance and observation operators. The accuracy and stability results for EnKF and EnSRKF for dissipative PDEs are, to the best of our knowledge, completely new in this general setting. It turns out that a hitherto unexploited cancellation property involving the ensemble covariance and observation operators and the concept of covariance localization in conjunction with covariance inflation play a pivotal role in the accuracy and stability for EnKF and EnSRKF. Our approach also elucidates the links, via determining functionals, between the approximate-Bayesian and control-theoretic approaches to data assimilation. We consider the ‘‘model’’ dynamics governed by the two-dimensional incompressible Navier-Stokes equations and observations given by noisy measurements of averaged volume elements or spectral/modal observations of the velocity field. In this setup, several continuous-time data assimilation techniques, namely the so-called Nudging (AOT)-algorithm, 3DVar, EnKF and EnSRKF reduce to a stochastically forced Navier-Stokes equations. For the first time, we derive conditions for accuracy and stability of EnKF and EnSRKF. Our analysis reveals an interplay between the resolution of the observations (roughly, the ‘richness’ of the observation space) associated with the observation operator underlying the data assimilation algorithms and covariance inflation and localization which are employed in practice for improved filter performance. This is joint work with Dr. Michal Branicki (University of Edinburgh).