### Abstract

Two types of filtering failure are the well known filter divergence where errors may exceed the size of the corresponding true chaotic attractor and the much more severe catastrophic filter divergence where solutions diverge to machine infinity in finite time. In this paper, we demon-strate that these failures occur in filtering the L-96 model, a nonlinear chaotic dissipative dynamical system with the absorbing ball property and quasi-Gaussian unimodal statistics. In particular, catas-trophic filter divergence occurs in suitable parameter regimes for an ensemble Kalman filter when the noisy turbulent true solution signal is partially observed at sparse regular spatial locations. With the above documentation, the main theme of this paper is to show that we can suppress the catastrophic filter divergence with a judicious model error strategy, that is, through a suitable linear stochastic model. This result confirms that the Gaussian assumption in the Kalman filter formulation, which is violated by most ensemble Kalman filters through the nonlinearity in the model, is a necessary condition to avoid catastrophic filter divergence. In a suitable range of chaotic regimes, adding model errors is not the best strategy when the true model is known. However, we find that there are several parameter regimes where the filtering performance in the presence of model errors with the stochastic model supersedes the performance in the perfect model simulation of the best ensemble Kalman filter considered here. Secondly, we also show that the advantage of the reduced Fourier domain filtering strategy [A. Majda and M. Grote, Proceedings of the National Academy of Sciences, 104, 1124-1129, 2007], [E. Castronovo, J. Harlim and A. Majda, J. Comput. Phys., 227(7), 3678-3714, 2008], [J. Harlim and A. Majda, J. Comput. Phys., 227(10), 5304-5341, 2008] is not simply through its numerical efficiency, but significant filtering accuracy is also gained through ignoring the correlation between the appropriate Fourier coefficients when the sparse observations are available in regular space locations.

Original language | English (US) |
---|---|

Pages (from-to) | 27-43 |

Number of pages | 17 |

Journal | Communications in Mathematical Sciences |

Volume | 8 |

Issue number | 1 |

State | Published - 2010 |

### Fingerprint

### Keywords

- Filter divergence
- Kalman filter
- Lorenz-96 model

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications in Mathematical Sciences*,

*8*(1), 27-43.

**Catastrophic filter divergence in filtering nonlinear dissipative systems.** / Harlim, John; Majda, Andrew J.

Research output: Contribution to journal › Article

*Communications in Mathematical Sciences*, vol. 8, no. 1, pp. 27-43.

}

TY - JOUR

T1 - Catastrophic filter divergence in filtering nonlinear dissipative systems

AU - Harlim, John

AU - Majda, Andrew J.

PY - 2010

Y1 - 2010

N2 - Two types of filtering failure are the well known filter divergence where errors may exceed the size of the corresponding true chaotic attractor and the much more severe catastrophic filter divergence where solutions diverge to machine infinity in finite time. In this paper, we demon-strate that these failures occur in filtering the L-96 model, a nonlinear chaotic dissipative dynamical system with the absorbing ball property and quasi-Gaussian unimodal statistics. In particular, catas-trophic filter divergence occurs in suitable parameter regimes for an ensemble Kalman filter when the noisy turbulent true solution signal is partially observed at sparse regular spatial locations. With the above documentation, the main theme of this paper is to show that we can suppress the catastrophic filter divergence with a judicious model error strategy, that is, through a suitable linear stochastic model. This result confirms that the Gaussian assumption in the Kalman filter formulation, which is violated by most ensemble Kalman filters through the nonlinearity in the model, is a necessary condition to avoid catastrophic filter divergence. In a suitable range of chaotic regimes, adding model errors is not the best strategy when the true model is known. However, we find that there are several parameter regimes where the filtering performance in the presence of model errors with the stochastic model supersedes the performance in the perfect model simulation of the best ensemble Kalman filter considered here. Secondly, we also show that the advantage of the reduced Fourier domain filtering strategy [A. Majda and M. Grote, Proceedings of the National Academy of Sciences, 104, 1124-1129, 2007], [E. Castronovo, J. Harlim and A. Majda, J. Comput. Phys., 227(7), 3678-3714, 2008], [J. Harlim and A. Majda, J. Comput. Phys., 227(10), 5304-5341, 2008] is not simply through its numerical efficiency, but significant filtering accuracy is also gained through ignoring the correlation between the appropriate Fourier coefficients when the sparse observations are available in regular space locations.

AB - Two types of filtering failure are the well known filter divergence where errors may exceed the size of the corresponding true chaotic attractor and the much more severe catastrophic filter divergence where solutions diverge to machine infinity in finite time. In this paper, we demon-strate that these failures occur in filtering the L-96 model, a nonlinear chaotic dissipative dynamical system with the absorbing ball property and quasi-Gaussian unimodal statistics. In particular, catas-trophic filter divergence occurs in suitable parameter regimes for an ensemble Kalman filter when the noisy turbulent true solution signal is partially observed at sparse regular spatial locations. With the above documentation, the main theme of this paper is to show that we can suppress the catastrophic filter divergence with a judicious model error strategy, that is, through a suitable linear stochastic model. This result confirms that the Gaussian assumption in the Kalman filter formulation, which is violated by most ensemble Kalman filters through the nonlinearity in the model, is a necessary condition to avoid catastrophic filter divergence. In a suitable range of chaotic regimes, adding model errors is not the best strategy when the true model is known. However, we find that there are several parameter regimes where the filtering performance in the presence of model errors with the stochastic model supersedes the performance in the perfect model simulation of the best ensemble Kalman filter considered here. Secondly, we also show that the advantage of the reduced Fourier domain filtering strategy [A. Majda and M. Grote, Proceedings of the National Academy of Sciences, 104, 1124-1129, 2007], [E. Castronovo, J. Harlim and A. Majda, J. Comput. Phys., 227(7), 3678-3714, 2008], [J. Harlim and A. Majda, J. Comput. Phys., 227(10), 5304-5341, 2008] is not simply through its numerical efficiency, but significant filtering accuracy is also gained through ignoring the correlation between the appropriate Fourier coefficients when the sparse observations are available in regular space locations.

KW - Filter divergence

KW - Kalman filter

KW - Lorenz-96 model

UR - http://www.scopus.com/inward/record.url?scp=77953316289&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77953316289&partnerID=8YFLogxK

M3 - Article

VL - 8

SP - 27

EP - 43

JO - Communications in Mathematical Sciences

JF - Communications in Mathematical Sciences

SN - 1539-6746

IS - 1

ER -