Importance sampling for rare event tracking within the ensemble Kalman filtering framework
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| Format: | Preprint |
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2024
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| author | Rached, Nadhir Ben von Schwerin, Erik Shaimerdenova, Gaukhar Tempone, Raul |
| author_facet | Rached, Nadhir Ben von Schwerin, Erik Shaimerdenova, Gaukhar Tempone, Raul |
| contents | In this work we employ importance sampling (IS) techniques to track a small over-threshold probability of a running maximum associated with the solution of a stochastic differential equation (SDE) within the framework of ensemble Kalman filtering (EnKF). Between two observation times of the EnKF, we propose to use IS with respect to the initial condition of the SDE, IS with respect to the Wiener process via a stochastic optimal control formulation, and combined IS with respect to both initial condition and Wiener process. Both IS strategies require the approximation of the solution of Kolmogorov Backward equation (KBE) with boundary conditions. In multidimensional settings, we employ a Markovian projection dimension reduction technique to obtain an approximation of the solution of the KBE by just solving a one dimensional PDE. The proposed ideas are tested on three illustrative examples: Double Well SDE, Langevin dynamics and noisy Charney-deVore model, and showcase a significant variance reduction compared to the standard Monte Carlo method and another sampling-based IS technique, namely, multilevel cross entropy. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_12793 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Importance sampling for rare event tracking within the ensemble Kalman filtering framework Rached, Nadhir Ben von Schwerin, Erik Shaimerdenova, Gaukhar Tempone, Raul Numerical Analysis Dynamical Systems Optimization and Control Probability 35Q93, 60G35, 60H35, 65C05, 93E20 In this work we employ importance sampling (IS) techniques to track a small over-threshold probability of a running maximum associated with the solution of a stochastic differential equation (SDE) within the framework of ensemble Kalman filtering (EnKF). Between two observation times of the EnKF, we propose to use IS with respect to the initial condition of the SDE, IS with respect to the Wiener process via a stochastic optimal control formulation, and combined IS with respect to both initial condition and Wiener process. Both IS strategies require the approximation of the solution of Kolmogorov Backward equation (KBE) with boundary conditions. In multidimensional settings, we employ a Markovian projection dimension reduction technique to obtain an approximation of the solution of the KBE by just solving a one dimensional PDE. The proposed ideas are tested on three illustrative examples: Double Well SDE, Langevin dynamics and noisy Charney-deVore model, and showcase a significant variance reduction compared to the standard Monte Carlo method and another sampling-based IS technique, namely, multilevel cross entropy. |
| title | Importance sampling for rare event tracking within the ensemble Kalman filtering framework |
| topic | Numerical Analysis Dynamical Systems Optimization and Control Probability 35Q93, 60G35, 60H35, 65C05, 93E20 |
| url | https://arxiv.org/abs/2403.12793 |