A nonlinear test model for filtering slow-fast systems

Boris Gershgorin, Andrew Majda

Research output: Contribution to journalArticlepeer-review


A nonlinear test model for filtering turbulent signals from partial observations of nonlinear slow-fast systems with multiple time scales is developed here. This model is a nonlinear stochastic real triad model with one slow mode, two fast modes, and catalytic nonlinear interaction of the fast modes depending on the slow mode. Despite the nonlinear and non-Gaussian features of the model, exact solution formulas are developed here for the mean and covariance. These formulas are utilized to develop a suite of statistically exact extended Kalman filters for the slow-fast system. Important practical issues such as filter performance with partial observations, which mix the slow and fast modes, model errors through linear filters for the fast modes, and the role of observation frequency and observational noise strength are assessed in unambiguous fashion in the test model by utilizing these exact nonlinear statistics.

Original languageEnglish (US)
Pages (from-to)611-649
Number of pages39
JournalCommunications in Mathematical Sciences
Issue number3
StatePublished - 2008


  • Extended Kalman filter
  • Nonlinear model
  • Slow-fast system

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


Dive into the research topics of 'A nonlinear test model for filtering slow-fast systems'. Together they form a unique fingerprint.

Cite this