Stability of non-linear filter for deterministic dynamics

Anugu Sumith Reddy; Amit Apte

doi:10.3934/fods.2021025

Article Contents

2021, Volume 3, Issue 3: 647-675. Doi: 10.3934/fods.2021025

This issue Previous Article Analysis of the feedback particle filter with diffusion map based approximation of the gain Next Article

Stability of non-linear filter for deterministic dynamics

Anugu Sumith Reddy^1, and
Amit Apte^2, ,

1.
Department of Mathematics, Indian Institute of Technology Bombay, Mumbai, India-400076
2.
International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore, India-560089

^* Corresponding author: Amit Apte
^* Corresponding author: Amit Apte

Received: February 28, 2021

Revised: July 31, 2021

Early access: September 2021

Published: September 2021

The work of ASR was partially supported by Infosys Foundation Excellence Program of ICTS. AA acknowledges support from US Office of Naval Research under grant N00014-18-1-2204. Authors acknowledge the support of the Department of Atomic Energy, Government of India, under projects no.12-R & D-TFR-5.10-1100, and no.RTI4001

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Abstract

This papers shows that nonlinear filter in the case of deterministic dynamics is stable with respect to the initial conditions under the conditions that observations are sufficiently rich, both in the context of continuous and discrete time filters. Earlier works on the stability of the nonlinear filters are in the context of stochastic dynamics and assume conditions like compact state space or time independent observation model, whereas we prove filter stability for deterministic dynamics with more general assumptions on the state space and observation process. We give several examples of systems that satisfy these assumptions. We also show that the asymptotic structure of the filtering distribution is related to the dynamical properties of the signal.

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Mathematics Subject Classification: Primary: 60G35, 93B07, 93E11, 62M20; Secondary: 93C10.

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Figure 1. Dependence of $ \frac{D_N(x, y)}{\sum_{i = 0}^N \rho_{i\tau}} $ vs $ t = N\tau $ with $ \tau = 0.01 $ for $ 100 $ samples. We have $ t = N\tau $ with $ \tau = 0.01 $. We chose $ 100 $ different pairs of $ (x, y) $ for five different choices of $ \rho_t = 1000, \; t+1000, \; \log(t+1000); t^2+1000; t^3 + 1000 $. The initial conditions for the samples are randomly chosen from uniform distribution on $ [-10, 10]^{p} $ where the dimension $ p = 3 $ for Lorenz 63 model (left panel) and $ p = 36 $ for the Lorenz 96 model (right panel). The insets show the plots for large $ t $. (Note that the Lyapunov time scale for both these models is $ O(1) $.)

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