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Filtering dynamical systems using observations of statistics.

Eviatar Bach1,2, Tim Colonius3, Isabel Scherl3

  • 1Department of Environmental Science and Engineering, California Institute of Technology, Pasadena, California 91125, USA.

Chaos (Woodbury, N.Y.)
|March 8, 2024
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Summary
This summary is machine-generated.

This study introduces the ensemble Fokker-Planck filter (EnFPF) for estimating system densities from statistical observations. The EnFPF offers a practical approach to complex filtering problems, improving accuracy and convergence in various dynamical systems.

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Area of Science:

  • Dynamical systems theory
  • Stochastic processes
  • Computational statistics

Background:

  • Standard filtering problems rely on state observations, not statistical ones.
  • Estimating time-evolving densities from noisy statistical data is computationally challenging.
  • Infinite-dimensional filtering in density spaces is often intractable.

Purpose of the Study:

  • To develop a tractable state-space algorithm for filtering dynamical systems using statistical observations.
  • To introduce the ensemble Fokker-Planck filter (EnFPF) as a novel computational methodology.
  • To demonstrate the EnFPF's effectiveness beyond theoretical limitations.

Main Methods:

  • Formulation of a mean-field state-space model.
  • Utilizing interacting particle systems for approximation.
  • Developing an ensemble method based on these approximations.

Main Results:

  • The EnFPF approximates the Kalman-Bucy filter for the Fokker-Planck equation under specific assumptions.
  • Numerical experiments confirm EnFPF's utility in correcting ensemble statistics.
  • The method accelerates convergence to invariant densities for both autonomous and non-autonomous systems.

Conclusions:

  • The ensemble Fokker-Planck filter (EnFPF) provides a viable solution for density estimation in dynamical systems using statistical observations.
  • EnFPF shows promise for applications in climate modeling and turbulence research.
  • The methodology extends the capabilities of filtering techniques to a broader range of complex systems.