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
Summary
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.
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.
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