Brett, Charles E. A., Lam, Kei Fong, Law, K. J. H., McCormick, D. S., Scott, Michael R. and Stuart, A. M. (2013) Accuracy and stability of filters for dissipative PDEs. Physica D: Nonlinear Phenomena, Volume 245 (Number 1). pp. 34-45. doi:10.1016/j.physd.2012.11.005 ISSN 0167-2789.

Research output not available from this repository.

Request-a-Copy directly from author or use local Library Get it For Me service.

Request Changes to record.

Abstract

Data assimilation methodologies are designed to incorporate noisy observations of a physical system into an underlying model in order to infer the properties of the state of the system. Filters refer to a class of data assimilation algorithms designed to update the estimation of the state in an on-line fashion, as data is acquired sequentially. For linear problems subject to Gaussian noise, filtering can be performed exactly using the Kalman filter. For nonlinear systems filtering can be approximated in a systematic way by particle filters. However in high dimensions these particle filtering methods can break down. Hence, for the large nonlinear systems arising in applications such as oceanography and weather forecasting, various ad hoc filters are used, mostly based on making Gaussian approximations. The purpose of this work is to study the accuracy and stability properties of these ad hoc filters. We work in the context of the 2D incompressible Navier–Stokes equation, although the ideas readily generalize to a range of dissipative partial differential equations (PDEs). By working in this infinite dimensional setting we provide an analysis which is useful for the understanding of high dimensional filtering, and is robust to mesh-refinement. We describe theoretical results showing that, in the small observational noise limit, the filters can be tuned to perform accurately in tracking the signal itself (filter accuracy), provided the system is observed in a sufficiently large low dimensional space; roughly speaking this space should be large enough to contain the unstable modes of the linearized dynamics. The tuning corresponds to what is known as variance inflation in the applied literature. Numerical results are given which illustrate the theory. The positive results herein concerning filter stability complement recent numerical studies which demonstrate that the ad hoc filters can perform poorly in reproducing statistical variation about the true signal.

Item Type:	Journal Article
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Journal or Publication Title:	Physica D: Nonlinear Phenomena
Publisher:	Elsevier BV
ISSN:	0167-2789
Official Date:	2013
Dates:	Date Event 2013 Published
Volume:	Volume 245
Number:	Number 1
Page Range:	pp. 34-45
DOI:	10.1016/j.physd.2012.11.005
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access

Request changes or add full text files to a record

Repository staff actions (login required)

View Item