Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Variational Pseudo-Observation Guided Nudged Particle Filter

Theofania Karampela, Ryne Beeson

Abstract

Nonlinear filtering with standard PF methods requires mitigative techniques to quell weight degeneracy, such as resampling. This is especially true in high-dimensional systems with sparse observations. Unfortunately, such techniques are also fragile when applied to systems with exceedingly rare events. Nonlinear systems with these properties can be assimilated effectively with a control-based PF method known as the nPF, but this method has a high computational cost burden. In this work, we aim to retain this strength of the nudged method while reducing the computational cost by introducing a variational method into the algorithm that acts as a continuous pseudo-observation path. By maintaining a PF representation, the resulting algorithm continues to capture an approximation of the filtering distribution, while reducing computational runtime and improving robustness to the "rare" event of switching phases. Preliminary testing of the new approach is demonstrated on a stochastic variant of the nonlinear and chaotic L63 model, which is used as a surrogate for mimicking "rare" events. The new approach helps to overcome difficulties in applying the nPF for realistic problems and performs favorably with respect to a standard PF with a higher number of particles.

A Variational Pseudo-Observation Guided Nudged Particle Filter

Abstract

Nonlinear filtering with standard PF methods requires mitigative techniques to quell weight degeneracy, such as resampling. This is especially true in high-dimensional systems with sparse observations. Unfortunately, such techniques are also fragile when applied to systems with exceedingly rare events. Nonlinear systems with these properties can be assimilated effectively with a control-based PF method known as the nPF, but this method has a high computational cost burden. In this work, we aim to retain this strength of the nudged method while reducing the computational cost by introducing a variational method into the algorithm that acts as a continuous pseudo-observation path. By maintaining a PF representation, the resulting algorithm continues to capture an approximation of the filtering distribution, while reducing computational runtime and improving robustness to the "rare" event of switching phases. Preliminary testing of the new approach is demonstrated on a stochastic variant of the nonlinear and chaotic L63 model, which is used as a surrogate for mimicking "rare" events. The new approach helps to overcome difficulties in applying the nPF for realistic problems and performs favorably with respect to a standard PF with a higher number of particles.
Paper Structure (13 sections, 2 theorems, 18 equations, 5 figures, 3 tables)

This paper contains 13 sections, 2 theorems, 18 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Standard and Nudged Particle Filter
  3. Standard Particle Filter
  4. Nudged Particle Filter
  5. Adaptive Nudging Calculation
  6. Additional Remarks
  7. Variational Nudged Particle Filter
  8. The Variational Nudged Particle Filter Algorithm
  9. Numerical Experiments
  10. The Stochastic Lorenz 1963 Model
  11. Simulation Parameters
  12. Simulation Results
  13. Conclusions

Key Result

Theorem 2.1

Given a sample $X^{k, x}_{t_k} \sim p^N_k(dx | \mathcal{F}^Y_k)$ and observation $Y_{k + 1}$, the solution of the OCP with dynamic constraint for $X^{k, x}_t$, $t \in [t_k, t_{k + 1})$ given by equation: controlled SDE, $g$ the log-likelihood of the observation, $R_s$ the diffusion matrix of the stochastic forcing in equation: controlled SDE, is (under reasonable assumptions) a feedback control l

Figures (5)

  • Figure 1: L63 deterministic attractor with initial conditions (see Table \ref{['tab:mc_ic_sweep_ess_rmse']}) used in numerical experiments (see § \ref{['section: numerical experiments']}).
  • Figure 2: Schematic evolution of a single particle within one observation interval for the nPF (left) and the Var-nPF (right).
  • Figure 3: Initial particle samples for all methods shown with gray circle markers. Prior states of each method (triangles) at the first observation $t = 0.5$ scaled to the normalized weight.
  • Figure 4: Time evolution of the first state component (i.e., $x$).
  • Figure 5: Nudging magnitude relative to BM forcing (green), ensemble mean for this relative control (blue), and the nESS.

Theorems & Definitions (2)

  • Theorem 2.1: Nudged Particle Filter (nPF) Lingala:2014buYeong:2020
  • Corollary 2.1: nPF Radon-Nikodym Derivative Yeong:2020