Mathematical strategies for filtering turbulent dynamical systems

Pages: 441 - 486, Volume 27, Issue 2, June 2010

doi:10.3934/dcds.2010.27.441       Abstract        Full Text (1261.9K)       Related Articles

Andrew J. Majda - Department of Mathematics and Center for Atmosphere and Ocean Science, Courant Institute for Mathematical Sciences, New York University, New York, NY 10012-1110, United States (email)
John Harlim - Depatment of Mathematics, North Carolina State University, Raleigh, NC 27695, United States (email)
Boris Gershgorin - Department of Mathematics and Center for Atmosphere and Ocean Science, Courant Institute for Mathematical Sciences, New York University, New York, NY 10012-1110, United States (email)

Abstract: The modus operandi of modern applied mathematics in developing very recent mathematical strategies for filtering turbulent dynamical systems is emphasized here. The approach involves the synergy of rigorous mathematical guidelines, exactly solvable nonlinear models with physical insight, and novel cheap algorithms with judicious model errors to filter turbulent signals with many degrees of freedom. A large number of new theoretical and computational phenomena such as "catastrophic filter divergence" in finite ensemble filters are reviewed here with the intention to introduce mathematicians, applied mathematicians, and scientists to this remarkable emerging scientific discipline with increasing practical importance.

Keywords:  stochastic parameter estimation, Kalman filter, filtering turbulent systems, data assimilation, model error.
Mathematics Subject Classification:  Primary: 93E11, 62L12; Secondary: 65C20.

Received: October 2009;      Revised: February 2010;      Published: February 2010.