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Filtering Dynamical Systems Using Observations of Statistics

Eviatar Bach, Tim Colonius, Isabel Scherl, Andrew Stuart

TL;DR

The experiments show that the EnFPF is able to correct ensemble statistics, to accelerate convergence to the invariant density for autonomous systems, and to accelerate convergence to time-dependent invariant densities for non-autonomous systems.

Abstract

We consider the problem of filtering dynamical systems, possibly stochastic, using observations of statistics. Thus, the computational task is to estimate a time-evolving density $ρ(v, t)$ given noisy observations of the true density $ρ^\dagger$; this contrasts with the standard filtering problem based on observations of the state $v$. The task is naturally formulated as an infinite-dimensional filtering problem in the space of densities $ρ$. However, for the purposes of tractability, we seek algorithms in state space; specifically, we introduce a mean-field state-space model, and using interacting particle system approximations to this model, we propose an ensemble method. We refer to the resulting methodology as the ensemble Fokker-Planck filter (EnFPF). Under certain restrictive assumptions, we show that the EnFPF approximates the Kalman-Bucy filter for the Fokker-Planck equation, which is the exact solution to the infinite-dimensional filtering problem. Furthermore, our numerical experiments show that the methodology is useful beyond this restrictive setting. Specifically, the experiments show that the EnFPF is able to correct ensemble statistics, to accelerate convergence to the invariant density for autonomous systems, and to accelerate convergence to time-dependent invariant densities for non-autonomous systems. We discuss possible applications of the EnFPF to climate ensembles and to turbulence modeling.

Filtering Dynamical Systems Using Observations of Statistics

TL;DR

The experiments show that the EnFPF is able to correct ensemble statistics, to accelerate convergence to the invariant density for autonomous systems, and to accelerate convergence to time-dependent invariant densities for non-autonomous systems.

Abstract

We consider the problem of filtering dynamical systems, possibly stochastic, using observations of statistics. Thus, the computational task is to estimate a time-evolving density given noisy observations of the true density ; this contrasts with the standard filtering problem based on observations of the state . The task is naturally formulated as an infinite-dimensional filtering problem in the space of densities . However, for the purposes of tractability, we seek algorithms in state space; specifically, we introduce a mean-field state-space model, and using interacting particle system approximations to this model, we propose an ensemble method. We refer to the resulting methodology as the ensemble Fokker-Planck filter (EnFPF). Under certain restrictive assumptions, we show that the EnFPF approximates the Kalman-Bucy filter for the Fokker-Planck equation, which is the exact solution to the infinite-dimensional filtering problem. Furthermore, our numerical experiments show that the methodology is useful beyond this restrictive setting. Specifically, the experiments show that the EnFPF is able to correct ensemble statistics, to accelerate convergence to the invariant density for autonomous systems, and to accelerate convergence to time-dependent invariant densities for non-autonomous systems. We discuss possible applications of the EnFPF to climate ensembles and to turbulence modeling.
Paper Structure (37 sections, 2 theorems, 56 equations, 8 figures, 1 table)

This paper contains 37 sections, 2 theorems, 56 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Assimilating Statistical Observations
  3. Motivation and Literature Review
  4. Obtaining observations of statistics
  5. Acceleration of convergence to a (possibly time-dependent) invariant measure
  6. Parameter estimation
  7. Correcting for model error
  8. Assimilation of time-averaged observations
  9. Contributions and Paper Outline
  10. Problem and Algorithm
  11. Problem Statement
  12. The Standard Filtering Problem
  13. The Fokker--Planck Filtering Problem
  14. Mean-Field Equation
  15. Particle Approximation of Mean-Field Equation
  16. ...and 22 more sections

Key Result

Lemma 1

Assume that $\rho(0)\sim \mu(0) = \mathcal{N}(m_0, C_0)$ with Then, for $m(t)$ and $C(t)$ satisfying equations eq:kb_mean--eq:kb_init,

Figures (8)

  • Figure 1: The density of an Ornstein--Uhlenbeck process evolving in time (top panel). At regular intervals, we make observations of this density and use them to inform the evolution of an ensemble (bottom panel).
  • Figure 2: The impact of filtering on the root-mean-square error (RMSE) in the mean and second moment in the Lorenz63 model.
  • Figure 3: The estimated Wasserstein distance to the invariant density in Lorenz63, in unfiltered and filtered cases. For the filtered case, the first and second moments are assimilated. Each curve is averaged over 10 different initializations.
  • Figure 4: Top panel: an ensemble evolving in time from left to right, superimposed on the invariant density of Lorenz63 in the $x$--$z$ plane. Orange corresponds to higher probability density and blue to lower. Bottom panel: the same but with the EnFPF applied.
  • Figure 5: The estimated Wasserstein distance to the invariant density in Lorenz63, in unfiltered and filtered cases when different moments are assimilated. The curves are averaged over 25 initial conditions, and the shaded areas correspond to $\pm$ the standard error over the initializations. Here, for the filtered cases, the EnFPF is applied at every cycle.
  • ...and 3 more figures

Theorems & Definitions (6)

  • Remark 1
  • Lemma 1
  • proof
  • Remark 2
  • Theorem 1
  • proof