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Online learning of both state and dynamics using ensemble Kalman filters
Ensemble Kalman Inversion for nonlinear problems: Weights, consistency, and variance bounds
1. | Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706 USA |
2. | Department of Mathematics, Duke University, Durham, NC 27708 USA |
Ensemble Kalman Inversion (EnKI) [
We propose WEnKI and WEnSRF, the weighted versions of EnKI and EnSRF in this paper. It follows the same gradient flow as that of EnKI/EnSRF with weight corrections. Compared to the classical methods, the new methods are unbiased, and compared with IS, the method has bounded weight variance. Both properties will be proved rigorously in this paper. We further discuss the stability of the underlying Fokker-Planck equation. This partially explains why EnKI, despite being inconsistent, performs well occasionally in nonlinear settings. Numerical evidence will be demonstrated at the end.
References:
[1] |
C. Andrieu, N. Freitas, A. Doucet and M. Jordan, An introduction to MCMC for machine learning, Machine Learning, 50 (2003), 5-43. Google Scholar |
[2] |
A. Bain and D. Crisan, Fundamentals of Stochastic Filtering, Stochastic Modelling and Applied Probability, Springer New York, 2008.
doi: 10.1007/978-0-387-76896-0. |
[3] |
K. Bergemann and S. Reich,
A localization technique for ensemble Kalman filters, Quarterly Journal of the Royal Meteorological Society, 136 (2010), 701-707.
doi: 10.1002/qj.591. |
[4] |
K. Bergemann and S. Reich,
A mollified ensemble Kalman filter, Quarterly Journal of the Royal Meteorological Society, 136 (2010), 1636-1643.
doi: 10.1002/qj.672. |
[5] |
D. Blömker, C. Schillings, P. Wacker and S. Weissmann, Well posedness and convergence analysis of the ensemble Kalman inversion, Inverse Problems, 35 (2019), 085007.
doi: 10.1088/1361-6420/ab149c. |
[6] |
D. Blomker, C. Schillings and P. Wacker,
A strongly convergent numerical scheme from ensemble Kalman inversion, SIAM Journal on Numerical Analysis, 56 (2018), 2537-2562.
doi: 10.1137/17M1132367. |
[7] |
J. A. Cañizo, J. A. Carrillo and J. Rosado,
A well-posedness theory in measures for some kinetic models of collective motion, Mathematical Models and Methods in Applied Sciences, 21 (2011), 515-539.
doi: 10.1142/S0218202511005131. |
[8] |
N. Chada, A. Stuart and X. Tong,
Tikhonov regularization within ensemble Kalman inversion, SIAM Journal on Numerical Analysis, 58 (2019), 1263-1294.
doi: 10.1137/19M1242331. |
[9] |
N. Chada and X. T. Tong, Convergence acceleration of ensemble Kalman inversion in nonlinear settings, preprint, arXiv: 1911.02424. Google Scholar |
[10] |
N. K. Chada, M. A. Iglesias, L. Roininen and A. M. Stuart, Parameterizations for ensemble kalman inversion, Inverse Problems, 34 (2018), 055009.
doi: 10.1088/1361-6420/aab6d9. |
[11] |
J. de Wiljes, S. Reich and W. Stannat,
Long-time stability and accuracy of the ensemble Kalman–Bucy filter for fully observed processes and small measurement noise, SIAM Journal on Applied Dynamical Systems, 17 (2018), 1152-1181.
doi: 10.1137/17M1119056. |
[12] |
Z. Ding and Q. Li, Ensemble Kalman inversion: Mean-field limit and convergence analysis, arXiv: 1908.05575. Google Scholar |
[13] |
Z. Ding and Q. Li, Ensemble Kalman sampling: Mean-field limit and convergence analysis, preprint, arXiv: 1910.12923. Google Scholar |
[14] |
A. Doucet, N. De Freitas and N. Gordon, An Introduction to Sequential Monte Carlo Methods, Springer New York, New York, NY, 2001.
doi: 10.1007/978-1-4757-3437-9_1. |
[15] |
A. Doucet, N. De Freitas and N. Gordon, Sequential Monte Carlo Methods in Practice, Springer New York, London, 2001.
doi: 10.1007/978-1-4757-3437-9. |
[16] |
O. G. Ernst, B. Sprungk and H.-J. Starkloff,
Analysis of the ensemble and polynomial chaos Kalman filters in bayesian inverse problems, SIAM/ASA Journal on Uncertainty Quantification, 3 (2015), 823-851.
doi: 10.1137/140981319. |
[17] |
G. Evensen, Data Assimilation: The Ensemble Kalman Filter, Springer-Verlag, Berlin, Heidelberg, 2006.
doi: 10.1007/978-3-642-03711-5. |
[18] |
N. Fournier and A. Guillin,
On the rate of convergence in Wasserstein distance of the empirical measure, Probability Theory and Related Fields, 162 (2015), 707-738.
doi: 10.1007/s00440-014-0583-7. |
[19] |
A. Garbuno-Inigo, F. Hoffmann, W. Li and A. M. Stuart,
Interacting Langevin diffusions: Gradient structure and ensemble Kalman sampler, SIAM Journal on Applied Dynamical Systems, 19 (2020), 412-441.
doi: 10.1137/19M1251655. |
[20] |
A. Garbuno-Inigo, N. Nüsken and S. Reich, Affine invariant interacting Langevin dynamics for Bayesian inference, CoRR, abs/1912.02859. Google Scholar |
[21] |
J. Geweke,
Bayesian inference in econometric models using Monte Carlo integration, Econometrica, 57 (1989), 1317-1339.
doi: 10.2307/1913710. |
[22] |
M. Herty and G. Visconti, Continuous limits for constrained ensemble Kalman filter, Inverse Problems, 36 (2020), 075006.
doi: 10.1088/1361-6420/ab8bc5. |
[23] |
M. A. Iglesias, K. J. H. Law and A. M. Stuart, Ensemble Kalman methods for inverse problems, Inverse Problems, 29 (2013), 045001.
doi: 10.1088/0266-5611/29/4/045001. |
[24] |
T. Lange and W. Stannat, On the continuous time limit of the ensemble Kalman filter, preprint, arXiv: 1901.05204.
doi: 10.1090/mcom/3588. |
[25] |
K. J. H. Law, H. Tembine and R. Tempone, Deterministic mean-field ensemble Kalman filtering, SIAM Journal on Scientific Computing, 38 (2016), A1251–A1279.
doi: 10.1137/140984415. |
[26] |
D. M. Livings, S. L. Dance and N. K. Nichols,
Unbiased ensemble square root filters, Physica D: Nonlinear Phenomena, 237 (2008), 1021-1028.
doi: 10.1016/j.physd.2008.01.005. |
[27] |
Y. Lu, J. Lu and J. Nolen, Accelerating Langevin sampling with birth-death, preprint, arXiv: 1905.09863. Google Scholar |
[28] |
J. Martin, L. Wilcox, C. Burstedde and O. Ghattas, A stochastic newton MCMC method for large-scale statistical inverse problems with application to seismic inversion, SIAM Journal on Scientific Computing, 34 (2012), A1460–A1487.
doi: 10.1137/110845598. |
[29] |
Y. M. Marzouk, H. N. Najm and L. A. Rahn,
Stochastic spectral methods for efficient Bayesian solution of inverse problems, Journal of Computational Physics, 224 (2007), 560-586.
doi: 10.1016/j.jcp.2006.10.010. |
[30] |
A. Muntean, J. Rademacher and A. Zagaris, Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, LAMM, 3, Springer, Cham, 2016.
doi: 10.1007/978-3-319-26883-5. |
[31] |
N. Papadakis, E. Mémin, A. Cuzol and N. Gengembre, Data assimilation with the weighted ensemble Kalman filter, Tellus A, 62 (2010), 673-697. Google Scholar |
[32] |
S. Reich,
A dynamical systems framework for intermittent data assimilation, BIT Numerical Mathematics, 51 (2011), 235-249.
doi: 10.1007/s10543-010-0302-4. |
[33] |
S. Reich and C. Cotter, Probabilistic Forecasting and Bayesian Data Assimilation, Cambridge University Press, 2015.
doi: 10.1017/CBO9781107706804.![]() ![]() |
[34] |
S. Reich and S. Weissmann, Fokker-planck particle systems for bayesian inference: Computational approaches, preprint, arXiv: 1911.10832. Google Scholar |
[35] |
C. Schillings and A. M. Stuart,
Analysis of the ensemble Kalman filter for inverse problems, SIAM Journal on Numerical Analysis, 55 (2017), 1264-1290.
doi: 10.1137/16M105959X. |
[36] |
M. Tippett, J. Anderson, C. Bishop, T. Hamill and J. Whitaker,
Ensemble square root filters, Monthly Weather Review, 131 (2003), 1485-1490.
doi: 10.1175/1520-0493(2003)131<1485:ESRF>2.0.CO;2. |
show all references
References:
[1] |
C. Andrieu, N. Freitas, A. Doucet and M. Jordan, An introduction to MCMC for machine learning, Machine Learning, 50 (2003), 5-43. Google Scholar |
[2] |
A. Bain and D. Crisan, Fundamentals of Stochastic Filtering, Stochastic Modelling and Applied Probability, Springer New York, 2008.
doi: 10.1007/978-0-387-76896-0. |
[3] |
K. Bergemann and S. Reich,
A localization technique for ensemble Kalman filters, Quarterly Journal of the Royal Meteorological Society, 136 (2010), 701-707.
doi: 10.1002/qj.591. |
[4] |
K. Bergemann and S. Reich,
A mollified ensemble Kalman filter, Quarterly Journal of the Royal Meteorological Society, 136 (2010), 1636-1643.
doi: 10.1002/qj.672. |
[5] |
D. Blömker, C. Schillings, P. Wacker and S. Weissmann, Well posedness and convergence analysis of the ensemble Kalman inversion, Inverse Problems, 35 (2019), 085007.
doi: 10.1088/1361-6420/ab149c. |
[6] |
D. Blomker, C. Schillings and P. Wacker,
A strongly convergent numerical scheme from ensemble Kalman inversion, SIAM Journal on Numerical Analysis, 56 (2018), 2537-2562.
doi: 10.1137/17M1132367. |
[7] |
J. A. Cañizo, J. A. Carrillo and J. Rosado,
A well-posedness theory in measures for some kinetic models of collective motion, Mathematical Models and Methods in Applied Sciences, 21 (2011), 515-539.
doi: 10.1142/S0218202511005131. |
[8] |
N. Chada, A. Stuart and X. Tong,
Tikhonov regularization within ensemble Kalman inversion, SIAM Journal on Numerical Analysis, 58 (2019), 1263-1294.
doi: 10.1137/19M1242331. |
[9] |
N. Chada and X. T. Tong, Convergence acceleration of ensemble Kalman inversion in nonlinear settings, preprint, arXiv: 1911.02424. Google Scholar |
[10] |
N. K. Chada, M. A. Iglesias, L. Roininen and A. M. Stuart, Parameterizations for ensemble kalman inversion, Inverse Problems, 34 (2018), 055009.
doi: 10.1088/1361-6420/aab6d9. |
[11] |
J. de Wiljes, S. Reich and W. Stannat,
Long-time stability and accuracy of the ensemble Kalman–Bucy filter for fully observed processes and small measurement noise, SIAM Journal on Applied Dynamical Systems, 17 (2018), 1152-1181.
doi: 10.1137/17M1119056. |
[12] |
Z. Ding and Q. Li, Ensemble Kalman inversion: Mean-field limit and convergence analysis, arXiv: 1908.05575. Google Scholar |
[13] |
Z. Ding and Q. Li, Ensemble Kalman sampling: Mean-field limit and convergence analysis, preprint, arXiv: 1910.12923. Google Scholar |
[14] |
A. Doucet, N. De Freitas and N. Gordon, An Introduction to Sequential Monte Carlo Methods, Springer New York, New York, NY, 2001.
doi: 10.1007/978-1-4757-3437-9_1. |
[15] |
A. Doucet, N. De Freitas and N. Gordon, Sequential Monte Carlo Methods in Practice, Springer New York, London, 2001.
doi: 10.1007/978-1-4757-3437-9. |
[16] |
O. G. Ernst, B. Sprungk and H.-J. Starkloff,
Analysis of the ensemble and polynomial chaos Kalman filters in bayesian inverse problems, SIAM/ASA Journal on Uncertainty Quantification, 3 (2015), 823-851.
doi: 10.1137/140981319. |
[17] |
G. Evensen, Data Assimilation: The Ensemble Kalman Filter, Springer-Verlag, Berlin, Heidelberg, 2006.
doi: 10.1007/978-3-642-03711-5. |
[18] |
N. Fournier and A. Guillin,
On the rate of convergence in Wasserstein distance of the empirical measure, Probability Theory and Related Fields, 162 (2015), 707-738.
doi: 10.1007/s00440-014-0583-7. |
[19] |
A. Garbuno-Inigo, F. Hoffmann, W. Li and A. M. Stuart,
Interacting Langevin diffusions: Gradient structure and ensemble Kalman sampler, SIAM Journal on Applied Dynamical Systems, 19 (2020), 412-441.
doi: 10.1137/19M1251655. |
[20] |
A. Garbuno-Inigo, N. Nüsken and S. Reich, Affine invariant interacting Langevin dynamics for Bayesian inference, CoRR, abs/1912.02859. Google Scholar |
[21] |
J. Geweke,
Bayesian inference in econometric models using Monte Carlo integration, Econometrica, 57 (1989), 1317-1339.
doi: 10.2307/1913710. |
[22] |
M. Herty and G. Visconti, Continuous limits for constrained ensemble Kalman filter, Inverse Problems, 36 (2020), 075006.
doi: 10.1088/1361-6420/ab8bc5. |
[23] |
M. A. Iglesias, K. J. H. Law and A. M. Stuart, Ensemble Kalman methods for inverse problems, Inverse Problems, 29 (2013), 045001.
doi: 10.1088/0266-5611/29/4/045001. |
[24] |
T. Lange and W. Stannat, On the continuous time limit of the ensemble Kalman filter, preprint, arXiv: 1901.05204.
doi: 10.1090/mcom/3588. |
[25] |
K. J. H. Law, H. Tembine and R. Tempone, Deterministic mean-field ensemble Kalman filtering, SIAM Journal on Scientific Computing, 38 (2016), A1251–A1279.
doi: 10.1137/140984415. |
[26] |
D. M. Livings, S. L. Dance and N. K. Nichols,
Unbiased ensemble square root filters, Physica D: Nonlinear Phenomena, 237 (2008), 1021-1028.
doi: 10.1016/j.physd.2008.01.005. |
[27] |
Y. Lu, J. Lu and J. Nolen, Accelerating Langevin sampling with birth-death, preprint, arXiv: 1905.09863. Google Scholar |
[28] |
J. Martin, L. Wilcox, C. Burstedde and O. Ghattas, A stochastic newton MCMC method for large-scale statistical inverse problems with application to seismic inversion, SIAM Journal on Scientific Computing, 34 (2012), A1460–A1487.
doi: 10.1137/110845598. |
[29] |
Y. M. Marzouk, H. N. Najm and L. A. Rahn,
Stochastic spectral methods for efficient Bayesian solution of inverse problems, Journal of Computational Physics, 224 (2007), 560-586.
doi: 10.1016/j.jcp.2006.10.010. |
[30] |
A. Muntean, J. Rademacher and A. Zagaris, Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, LAMM, 3, Springer, Cham, 2016.
doi: 10.1007/978-3-319-26883-5. |
[31] |
N. Papadakis, E. Mémin, A. Cuzol and N. Gengembre, Data assimilation with the weighted ensemble Kalman filter, Tellus A, 62 (2010), 673-697. Google Scholar |
[32] |
S. Reich,
A dynamical systems framework for intermittent data assimilation, BIT Numerical Mathematics, 51 (2011), 235-249.
doi: 10.1007/s10543-010-0302-4. |
[33] |
S. Reich and C. Cotter, Probabilistic Forecasting and Bayesian Data Assimilation, Cambridge University Press, 2015.
doi: 10.1017/CBO9781107706804.![]() ![]() |
[34] |
S. Reich and S. Weissmann, Fokker-planck particle systems for bayesian inference: Computational approaches, preprint, arXiv: 1911.10832. Google Scholar |
[35] |
C. Schillings and A. M. Stuart,
Analysis of the ensemble Kalman filter for inverse problems, SIAM Journal on Numerical Analysis, 55 (2017), 1264-1290.
doi: 10.1137/16M105959X. |
[36] |
M. Tippett, J. Anderson, C. Bishop, T. Hamill and J. Whitaker,
Ensemble square root filters, Monthly Weather Review, 131 (2003), 1485-1490.
doi: 10.1175/1520-0493(2003)131<1485:ESRF>2.0.CO;2. |







WEnKI | WEnSRF | |||
Moments | Est. | Re. Error | Est. | Re. Error |
3.82 | 0.0056 | 3.88 | 0.0098 | |
14.73 | 0.0114 | 15.19 | 0.0192 | |
57.19 | 0.0177 | 59.86 | 0.0281 | |
223.79 | 0.0243 | 237.75 | 0.0366 | |
882.83 | 0.0312 | 951.95 | 0.0447 | |
EnKI | EnSRF | |||
Moments | Est. | Re. Error | Est. | Re. Error |
3.69 | 0.0413 | 3.70 | 0.0391 | |
13.66 | 0.0833 | 13.73 | 0.0785 | |
50.90 | 0.1258 | 51.35 | 0.1181 | |
190.68 | 0.1687 | 193.24 | 0.1575 | |
718.31 | 0.2117 | 732.17 | 0.1965 | |
WEnKF | IS | |||
Moments | Est. | Re. Error | Est. | Re. Error |
3.40 | 0.1156 | 3.52 | 0.0858 | |
11.65 | 0.2181 | 12.37 | 0.1699 | |
40.22 | 0.3093 | 43.57 | 0.2517 | |
139.72 | 0.3908 | 153.56 | 0.3305 | |
488.51 | 0.4639 | 541.71 | 0.4055 |
WEnKI | WEnSRF | |||
Moments | Est. | Re. Error | Est. | Re. Error |
3.82 | 0.0056 | 3.88 | 0.0098 | |
14.73 | 0.0114 | 15.19 | 0.0192 | |
57.19 | 0.0177 | 59.86 | 0.0281 | |
223.79 | 0.0243 | 237.75 | 0.0366 | |
882.83 | 0.0312 | 951.95 | 0.0447 | |
EnKI | EnSRF | |||
Moments | Est. | Re. Error | Est. | Re. Error |
3.69 | 0.0413 | 3.70 | 0.0391 | |
13.66 | 0.0833 | 13.73 | 0.0785 | |
50.90 | 0.1258 | 51.35 | 0.1181 | |
190.68 | 0.1687 | 193.24 | 0.1575 | |
718.31 | 0.2117 | 732.17 | 0.1965 | |
WEnKF | IS | |||
Moments | Est. | Re. Error | Est. | Re. Error |
3.40 | 0.1156 | 3.52 | 0.0858 | |
11.65 | 0.2181 | 12.37 | 0.1699 | |
40.22 | 0.3093 | 43.57 | 0.2517 | |
139.72 | 0.3908 | 153.56 | 0.3305 | |
488.51 | 0.4639 | 541.71 | 0.4055 |
Case | WEnKI | WEnSRF | EnKI | EnSRF |
Example 1 | 0.362s | 0.197s | 0.138s | 0.178s |
Example 2 | 50.041s | 41.739s | 26.564s | 18.518s |
Example 3 | 0.198s | 0.115s | 0.120s | 0.072s |
Case | WEnKI | WEnSRF | EnKI | EnSRF |
Example 1 | 0.362s | 0.197s | 0.138s | 0.178s |
Example 2 | 50.041s | 41.739s | 26.564s | 18.518s |
Example 3 | 0.198s | 0.115s | 0.120s | 0.072s |
WEnKI | WEnSRF | |||
Moments | Est. | Re. Error | Est. | Re. Error |
3.30 | 0.0055 | 3.32 | 0.0017 | |
10.99 | 0.0147 | 11.19 | 0.0023 | |
36.99 | 0.0279 | 38.12 | 0.0019 | |
125.53 | 0.0451 | 131.47 | 0.0001 | |
429.99 | 0.0664 | 459.16 | 0.0030 | |
EnKI | EnSRF | |||
Moments | Est. | Re. Error | Est. | Re. Error |
2.96 | 0.1084 | 3.28 | 0.0112 | |
9.07 | 0.1872 | 11.04 | 0.0111 | |
29.17 | 0.2332 | 38.25 | 0.0053 | |
100.32 | 0.2369 | 137.43 | 0.0455 | |
379.73 | 0.1755 | 516.22 | 0.1208 | |
WEnKF | IS | |||
Moments | Est. | Re. Error | Est. | Re. Error |
3.40 | 0.1658 | 3.24 | 0.0245 | |
7.72 | 0.3077 | 10.50 | 0.0592 | |
21.74 | 0.4287 | 34.10 | 0.1037 | |
61.62 | 0.5313 | 110.81 | 0.1571 | |
175.99 | 0.6179 | 360.27 | 0.2178 |
WEnKI | WEnSRF | |||
Moments | Est. | Re. Error | Est. | Re. Error |
3.30 | 0.0055 | 3.32 | 0.0017 | |
10.99 | 0.0147 | 11.19 | 0.0023 | |
36.99 | 0.0279 | 38.12 | 0.0019 | |
125.53 | 0.0451 | 131.47 | 0.0001 | |
429.99 | 0.0664 | 459.16 | 0.0030 | |
EnKI | EnSRF | |||
Moments | Est. | Re. Error | Est. | Re. Error |
2.96 | 0.1084 | 3.28 | 0.0112 | |
9.07 | 0.1872 | 11.04 | 0.0111 | |
29.17 | 0.2332 | 38.25 | 0.0053 | |
100.32 | 0.2369 | 137.43 | 0.0455 | |
379.73 | 0.1755 | 516.22 | 0.1208 | |
WEnKF | IS | |||
Moments | Est. | Re. Error | Est. | Re. Error |
3.40 | 0.1658 | 3.24 | 0.0245 | |
7.72 | 0.3077 | 10.50 | 0.0592 | |
21.74 | 0.4287 | 34.10 | 0.1037 | |
61.62 | 0.5313 | 110.81 | 0.1571 | |
175.99 | 0.6179 | 360.27 | 0.2178 |
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