Communications in Mathematical Sciences

Volume 8 (2010)

Number 1

Special Issue on the Occasion of Andrew Majda’s Sixtieth Birthday: Part I

Filtering a nonlinear slow-fast system with strong fast forcing

Pages: 67 – 92



Boris Gershgorin

Andrew Majda


A three-mode nonlinear slow-fast system with fast forcing is studied here as a model for filtering turbulent signals from partial observations. The model describes the interaction of two externally driven fast modes with a slow mode through catalytic nonlinear coupling. The special structure of the nonlinear interaction allows for the analytical solution for the first and second order statistics even with fast forcing. These formulas are used for testing the exact Nonlinear Extended Kalman Filter for the slow-fast system with fast forcing. Various practical questions such as the influence of the strong fast forcing on the slowly varying wave envelope, the role of observations, the frequency and variance of observations, and the model error due to linearization are addressed here.


Nonlinear model; slow-fast system; extended Kalman filter; fast forcing

2010 Mathematics Subject Classification

34A05, 93E11

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