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Optimal nonlinear filtering using the finite-volume method.

Colin Fox1, Malcolm E K Morrison1, Richard A Norton2

  • 1Department of Physics, University of Otago, Dunedin, New Zealand.

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Summary
This summary is machine-generated.

This study introduces a finite-volume method for optimal sequential inference in deterministic systems. The method conserves probability and ensures positive density functions for accurate state estimation.

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Area of Science:

  • Dynamical Systems and Control Theory
  • Computational Mathematics
  • Statistical Inference

Background:

  • Optimal sequential inference, or filtering, is crucial for estimating the state of deterministic dynamical systems.
  • Simulating the Frobenius-Perron operator, often via a continuity equation, is necessary for optimal filtering.
  • Existing methods may face challenges with probability conservation and maintaining positive density functions.

Purpose of the Study:

  • To present a novel numerical method for optimal sequential inference in deterministic dynamical systems.
  • To demonstrate the probability conservation and positive density properties of the proposed method.
  • To validate the method's performance in nonlinear filtering scenarios, including those with rank-deficient observations.

Main Methods:

  • Utilized the finite-volume numerical method to approximate the solution of the continuity equation governing the Frobenius-Perron operator.
  • Implemented a Courant-Friedrichs-Lewy-type condition to ensure positivity of intermediate discretized solutions.
  • Applied the method to nonlinear filtering of a simple pendulum's state and compared results with the unscented Kalman filter.

Main Results:

  • The finite-volume method demonstrated probability conservation and convergence to optimal continuous-time estimates for smooth, low-dimensional systems.
  • The Courant-Friedrichs-Lewy-type condition successfully maintained positive density functions throughout the simulation.
  • The method effectively handled rank-deficient observations, leading to multimodal probability distributions, outperforming the unscented Kalman filter in certain scenarios.

Conclusions:

  • The finite-volume method offers a robust and accurate approach for optimal sequential inference in deterministic dynamical systems.
  • This method provides a reliable alternative for filtering, especially in cases with complex dynamics or rank-deficient observations.
  • The demonstrated probability conservation and positivity assurance enhance the trustworthiness of state estimation results.