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Estimating and quantifying uncertainty in unknown system parameters from limited data remains a challenging inverse problem in a variety of real-world applications. While many approaches focus on estimating constant parameters, a subset of these problems includes time-varying parameters with unknown evolution models that often cannot be directly observed. This work develops a systematic particle filtering approach that reframes the idea behind artificial parameter evolution to estimate time-varying parameters in nonstationary inverse problems arising from deterministic dynamical systems. Focusing on systems modeled by ordinary differential equations, we present two particle filter algorithms for time-varying parameter estimation: one that relies on a fixed value for the noise variance of a parameter random walk; another that employs online estimation of the parameter evolution noise variance along with the time-varying parameter of interest. Several computed examples demonstrate the capability of the proposed algorithms in estimating time-varying parameters with different underlying functional forms and different relationships with the system states (i.e. additive vs. multiplicative).
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Fourier series-based approximation of time-varying parameters in ordinary differential equations
Abstract Many real-world systems modeled using differential equations involve unknown or uncertain parameters. Standard approaches to address parameter estimation inverse problems in this setting typically focus on estimating constants; yet some unobservable system parameters may vary with time without known evolution models. In this work, we propose a novel approximation method inspired by the Fourier series to estimate time-varying parameters in deterministic dynamical systems modeled with ordinary differential equations. Using ensemble Kalman filtering in conjunction with Fourier series-based approximation models, we detail two possible implementation schemes for sequentially updating the time-varying parameter estimates given noisy observations of the system states. We demonstrate the capabilities of the proposed approach in estimating periodic parameters, both when the period is known and unknown, as well as non-periodic time-varying parameters of different forms with several computed examples using a forced harmonic oscillator. Results emphasize the importance of the frequencies and number of approximation model terms on the time-varying parameter estimates and corresponding dynamical system predictions.
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- Award ID(s):
- 1819203
- PAR ID:
- 10487368
- Publisher / Repository:
- IOP Publishing Ltd.
- Date Published:
- Journal Name:
- Inverse Problems
- ISSN:
- 0266-5611
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1088/1361-6420/ad1fe5
