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Sampling, feasibility, and priors in data assimilation
1. | Department of Mathematics, University of California, Berkeley, and Lawrence Berkeley National Laboratory, Berkeley, CA 94720, United States |
2. | College of Earth, Ocean and Atmospheric Science, Oregon State University, Corvallis, OR 97331, United States |
3. | Department of Mathematics, University of Arizona, Tucson, AZ 85721, United States |
4. | Department of Mathematics, University of Kansas, Lawrence, KS 66045, United States |
References:
[1] |
M. Ades and P. van Leeuwen, An exploration of the equivalent weights particle filter, Quarterly Journal of the Royal Meteorological Society, 139 (2013), 820-840.
doi: 10.1002/qj.1995. |
[2] |
M. Arulampalam, S. Maskell, N. Gordon and T. Clapp, A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking, IEEE Transaction on Signal Processing, 50 (2002), 174-188.
doi: 10.1109/78.978374. |
[3] |
E. Atkins, M. Morzfeld and A. J. Chorin, Implicit particle methods and their connection with variational data assimilation, Monthly Weather Review, 141 (2013), 1786-1803.
doi: 10.1175/MWR-D-12-00145.1. |
[4] |
T. Bengtsson, P. Bickel and B. Li, Curse of dimensionality revisited: The collapse of importance sampling in very large scale systems, IMS Collections: Probability and Statistics: Essays in Honor of David A. Freedman, 2 (2008), 316-334.
doi: 10.1214/193940307000000518. |
[5] |
T. Berry and J. Harlim, Linear theory for filtering nonlinear multiscale systems with model error, Proc. R. Soc. A, 470 (2014), 20140168, 25pp.
doi: 10.1098/rspa.2014.0168. |
[6] |
P. Bickel, T. Bengtsson and J. Anderson, Sharp failure rates for the bootstrap particle filter in high dimensions, Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh, 3 (2008), 318-329.
doi: 10.1214/074921708000000228. |
[7] |
M. Bocquet, C. Pires and L. Wu, Beyond Gaussian statistical modeling in geophysical data assimilation, Monthly Weather Review, 138 (2010), 2997-3023.
doi: 10.1175/2010MWR3164.1. |
[8] |
N. H. Chan, J. P. Kadane, R. N. Miller and W. Palma, Estimation of tropical sea level anomaly by an improved Kalman filter, Journal of Physical Oceanography, 26 (1996), 1286-1303.
doi: 10.1175/1520-0485(1996)026<1286:EOTSLA>2.0.CO;2. |
[9] |
A. J. Chorin and O. H. Hald, Stochastic Tools in Mathematics and Science, 3rd edition, Springer, 2013.
doi: 10.1007/978-1-4614-6980-3. |
[10] |
A. J. Chorin and O. H. Hald, Estimating the uncertainty in underresolved nonlinear dynamics, Mathematics and Mechanics of Solids, 19 (2014), 28-38.
doi: 10.1177/1081286513505465. |
[11] |
A. J. Chorin and F. Lu, Discrete approach to stochastic parametrization and dimension reduction in nonlinear dynamics, Proceedings of the National Academy of Sciences, 112 (2015), 9804-9809.
doi: 10.1073/pnas.1512080112. |
[12] |
A. J. Chorin and M. Morzfeld, Conditions for successful data assimilation, Journal of Geophysical Research - Atmospheres, 118 (2013), p11,522-11,533.
doi: 10.1002/2013JD019838. |
[13] |
A. J. Chorin, M. Morzfeld and X. Tu, Implicit particle filters for data assimilation, Communications in Applied Mathematics and Computational Science, 5 (2010), 221-240.
doi: 10.2140/camcos.2010.5.221. |
[14] |
A. J. Chorin and X. Tu, Implicit sampling for particle filters, Proceedings of the National Academy of Sciences, 106 (2009), 17249-17254.
doi: 10.1073/pnas.0909196106. |
[15] |
D. Crommelin and E. Vanden-Eijnden, Subgrid-scale parameterization with conditional Markov chains, Journal of the Atmospheric Sciences, 65 (2008), 2661-2675.
doi: 10.1175/2008JAS2566.1. |
[16] |
A. Doucet, N. de Freitas and N. Gordon (eds.), Sequential Monte Carlo Methods in Practice, Springer, 2001.
doi: 10.1007/978-1-4757-3437-9. |
[17] |
A. Doucet, S. Godsill and C. Andrieu, On sequential Monte Carlo sampling methods for Bayesian filtering, Statistics and Computing, 10 (2000), 197-208. |
[18] |
G. Evensen, Data Assimilation: The Ensemble Kalman Filter, Second edition. Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-642-03711-5. |
[19] |
I. Fatkulin and E. Vanden-Eijnden, A computational strategy for multi-scale systems with applications to Lorenz'96 model, Journal of Computational Physics, 200 (2004), 605-638.
doi: 10.1016/j.jcp.2004.04.013. |
[20] |
G. E. Forsythe, M. A. Malcom and C. Mohler, Computer Methods for Mathematical Computations, Prentice-Hall, 1977. |
[21] |
A. Fournier, G. Hulot, D. Jault, W. Kuang, W. Tangborn, N. Gillet, E. Canet, J. Aubert and F. Lhuillier, An introduction to data assimilation and predictability in geomagnetism, Space Science Review, 155 (2010), 247-291. |
[22] |
J. Goodman, K. Lin and M. Morzfeld, Small-noise analysis and symmetrization of implicit Monte Carlo samplers, Communications on Pure and Applied Mathematics, (2015).
doi: 10.1002/cpa.21592. |
[23] |
J. Goodman and J. Weare, Ensemble samplers with affine invariance, Communications in Applied Mathematics and Computational Sciences, 5 (2010), 65-80.
doi: 10.2140/camcos.2010.5.65. |
[24] |
J. Harlim, Data assimilation with model error from unresolved scales,, , ().
|
[25] |
R. Kalman, A new approach to linear filtering and prediction theory, Transactions of the ASME-Journal of Basic Engineering, 82 (1960), 35-48. |
[26] |
M. Kalos and P. Whitlock, Monte Carlo Methods, Second edition. Wiley-Blackwell, Weinheim, 2008.
doi: 10.1002/9783527626212. |
[27] |
P. Lancaster and L. Rodman, Algebraic Riccati Equations, Oxford University Press, 1995. |
[28] |
J. Liu and R. Chen, Blind deconvolution via sequential imputations, Journal of the American Statistical Association, 90 (1995), 567-576.
doi: 10.1080/01621459.1995.10476549. |
[29] |
E. N. Lorenz, Deterministic non-periodic flow, Journal of Atmospheric Science, 20 (1963), 130-141. |
[30] |
E. Lorenz, Predictability: A problem partly solved, Proc. Seminar on Predictability, (2009), 40-58.
doi: 10.1017/CBO9780511617652.004. |
[31] |
A. Majda and J. Harlim, Physics constrained nonlinear regression models for time series, Nonlinearity, 26 (2013), 201-217.
doi: 10.1088/0951-7715/26/1/201. |
[32] |
R. N. Miller and L. Ehret, Application of the implicit particle filter to a model of nearshore circulation, Journal of Geophysical Research, 119 (2014), 2363-2385.
doi: 10.1002/2013JC009440. |
[33] |
R. N. Miller, M. Ghil and F. Gauthiez, Advanced data assimilation in strongly nonlinear systems, Journal of Atmospheric Science, 51 (1994), 1037-1056.
doi: 10.1175/1520-0469(1994)051<1037:ADAISN>2.0.CO;2. |
[34] |
M. Morzfeld and H. Godinez, Estimation and prediction for an orbital propagation model using data assimilation, Advances in the Astronautical Sciences Spaceflight Mechanics, 152 (2015), 2003-2021. |
[35] |
M. Morzfeld and D. Hodyss, Analysis of the ensemble Kalman filter for marginal and joint posteriors,, submitted for publication., ().
|
[36] |
M. Morzfeld, X. Tu, E. Atkins and A. J. Chorin, A random map implementation of implicit filters, Journal of Computational Physics, 231 (2012), 2049-2066.
doi: 10.1016/j.jcp.2011.11.022. |
[37] |
M. Morzfeld, X. Tu, J. Wilkening and A. J. Chorin, Parameter estimation by implicit sampling, Communications in Applied Mathematics and Computational Science, 10 (2015), 205-225.
doi: 10.2140/camcos.2015.10.205. |
[38] |
T. Moselhy and Y. Marzouk, Bayesian inference with optimal maps, Journal of Computational Physics, 231 (2012), 7815-7850.
doi: 10.1016/j.jcp.2012.07.022. |
[39] |
C. Nicolis, Probabilistic aspects of error growth in atmospheric dynamics, Quarterly Jourmal of the Royal Meteorological Society, 118 (1992), 553-567. |
[40] |
A. B. Owen, Monte Carlo Theory, Methods and Examples, 2013., URL , ().
|
[41] |
Y. Pokern, A. Stuart and P. Wiberg, Parameter estimation for partially observed hypoelliptic diffusions, Journal of the Royal Statistical Society: Series B, 71 (2009), 49-73.
doi: 10.1111/j.1467-9868.2008.00689.x. |
[42] |
C. Rasmussen and C. Williams, Gaussian Processes for Machine Learning, MIT Press, 2006. |
[43] |
N. Recca, A New Methodology for Importance Sampling, Masters Thesis, Courant Institute of Mathematical Sciences, New York University, 2011. |
[44] |
A. Samson and M. Thieullen, A contrast estimator for completely or partially observed hypoelliptic diffusion, Stochastic Processes and their Applications, 122 (2012), 2521-2552.
doi: 10.1016/j.spa.2012.04.006. |
[45] |
C. Snyder, Particle filters, the "optimal'' proposal and high-dimensional systems, Proceedings of the ECMWF Seminar on Data Assimilation for Atmosphere and Ocean, (2011). |
[46] |
C. Snyder, T. Bengtsson, P. Bickel and J. Anderson, Obstacles to high-dimensional particle filtering, Monthly Weather Review, 136 (2008), 4629-4640. |
[47] |
C. Snyder, T. Bengtsson and M. Morzfeld, Performance bounds for particle filters using the optimal proposal, Monthly Weather Review, 143 (2015), 4750-4761. |
[48] |
A. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica, 19 (2010), 451-559.
doi: 10.1017/S0962492910000061. |
[49] |
M. Tippet, J. Anderson, C. Bishop, T. Hamil and J. Whitaker, Ensemble square root filters, Monthly Weather Review, 131 (2003), 1485-1490. |
[50] |
P. van Leeuwen, Particle filtering in geophysical systems, Monthly Weather Review, 137 (2009), 4089-4114. |
[51] |
P. van Leeuwen, Nonlinear data assimilation in geosciences: an extremely efficient particle filter, Quarterly Journal of the Royal Meteorological Society, 136 (2010), 1991-1999. |
[52] |
E. Vanden-Eijnden and J. Weare, Rare event simulation and small noise diffusions, Communications on Pure and Applied Mathematics, 65 (2012), 1770-1803.
doi: 10.1002/cpa.21428. |
[53] |
E. Vanden-Eijnden and J. Weare, Data assimilation in the low noise, accurate observation regime with application to the Kuroshio current, Monthly Weather Review, 141 (2013), 1822-1841. |
[54] |
B. Weir, R. N. Miller and Y. Spitz, Implicit estimation of ecological model parameters, Bulletin of Mathematical Biology, 75 (2013), 223-257.
doi: 10.1007/s11538-012-9801-6. |
[55] |
D. Wilks, Effects of stochastic parameterizations in the Lorenz '96 model, Quarterly Journal of the Royal Meteorological Society, 131 (2005), 389-407. |
[56] |
V. Zaritskii and L. Shimelevich, Monte Carlo technique in problems of optimal data processing, Automation and Remote Control, 36 (1975), 2015-2022. |
show all references
References:
[1] |
M. Ades and P. van Leeuwen, An exploration of the equivalent weights particle filter, Quarterly Journal of the Royal Meteorological Society, 139 (2013), 820-840.
doi: 10.1002/qj.1995. |
[2] |
M. Arulampalam, S. Maskell, N. Gordon and T. Clapp, A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking, IEEE Transaction on Signal Processing, 50 (2002), 174-188.
doi: 10.1109/78.978374. |
[3] |
E. Atkins, M. Morzfeld and A. J. Chorin, Implicit particle methods and their connection with variational data assimilation, Monthly Weather Review, 141 (2013), 1786-1803.
doi: 10.1175/MWR-D-12-00145.1. |
[4] |
T. Bengtsson, P. Bickel and B. Li, Curse of dimensionality revisited: The collapse of importance sampling in very large scale systems, IMS Collections: Probability and Statistics: Essays in Honor of David A. Freedman, 2 (2008), 316-334.
doi: 10.1214/193940307000000518. |
[5] |
T. Berry and J. Harlim, Linear theory for filtering nonlinear multiscale systems with model error, Proc. R. Soc. A, 470 (2014), 20140168, 25pp.
doi: 10.1098/rspa.2014.0168. |
[6] |
P. Bickel, T. Bengtsson and J. Anderson, Sharp failure rates for the bootstrap particle filter in high dimensions, Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh, 3 (2008), 318-329.
doi: 10.1214/074921708000000228. |
[7] |
M. Bocquet, C. Pires and L. Wu, Beyond Gaussian statistical modeling in geophysical data assimilation, Monthly Weather Review, 138 (2010), 2997-3023.
doi: 10.1175/2010MWR3164.1. |
[8] |
N. H. Chan, J. P. Kadane, R. N. Miller and W. Palma, Estimation of tropical sea level anomaly by an improved Kalman filter, Journal of Physical Oceanography, 26 (1996), 1286-1303.
doi: 10.1175/1520-0485(1996)026<1286:EOTSLA>2.0.CO;2. |
[9] |
A. J. Chorin and O. H. Hald, Stochastic Tools in Mathematics and Science, 3rd edition, Springer, 2013.
doi: 10.1007/978-1-4614-6980-3. |
[10] |
A. J. Chorin and O. H. Hald, Estimating the uncertainty in underresolved nonlinear dynamics, Mathematics and Mechanics of Solids, 19 (2014), 28-38.
doi: 10.1177/1081286513505465. |
[11] |
A. J. Chorin and F. Lu, Discrete approach to stochastic parametrization and dimension reduction in nonlinear dynamics, Proceedings of the National Academy of Sciences, 112 (2015), 9804-9809.
doi: 10.1073/pnas.1512080112. |
[12] |
A. J. Chorin and M. Morzfeld, Conditions for successful data assimilation, Journal of Geophysical Research - Atmospheres, 118 (2013), p11,522-11,533.
doi: 10.1002/2013JD019838. |
[13] |
A. J. Chorin, M. Morzfeld and X. Tu, Implicit particle filters for data assimilation, Communications in Applied Mathematics and Computational Science, 5 (2010), 221-240.
doi: 10.2140/camcos.2010.5.221. |
[14] |
A. J. Chorin and X. Tu, Implicit sampling for particle filters, Proceedings of the National Academy of Sciences, 106 (2009), 17249-17254.
doi: 10.1073/pnas.0909196106. |
[15] |
D. Crommelin and E. Vanden-Eijnden, Subgrid-scale parameterization with conditional Markov chains, Journal of the Atmospheric Sciences, 65 (2008), 2661-2675.
doi: 10.1175/2008JAS2566.1. |
[16] |
A. Doucet, N. de Freitas and N. Gordon (eds.), Sequential Monte Carlo Methods in Practice, Springer, 2001.
doi: 10.1007/978-1-4757-3437-9. |
[17] |
A. Doucet, S. Godsill and C. Andrieu, On sequential Monte Carlo sampling methods for Bayesian filtering, Statistics and Computing, 10 (2000), 197-208. |
[18] |
G. Evensen, Data Assimilation: The Ensemble Kalman Filter, Second edition. Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-642-03711-5. |
[19] |
I. Fatkulin and E. Vanden-Eijnden, A computational strategy for multi-scale systems with applications to Lorenz'96 model, Journal of Computational Physics, 200 (2004), 605-638.
doi: 10.1016/j.jcp.2004.04.013. |
[20] |
G. E. Forsythe, M. A. Malcom and C. Mohler, Computer Methods for Mathematical Computations, Prentice-Hall, 1977. |
[21] |
A. Fournier, G. Hulot, D. Jault, W. Kuang, W. Tangborn, N. Gillet, E. Canet, J. Aubert and F. Lhuillier, An introduction to data assimilation and predictability in geomagnetism, Space Science Review, 155 (2010), 247-291. |
[22] |
J. Goodman, K. Lin and M. Morzfeld, Small-noise analysis and symmetrization of implicit Monte Carlo samplers, Communications on Pure and Applied Mathematics, (2015).
doi: 10.1002/cpa.21592. |
[23] |
J. Goodman and J. Weare, Ensemble samplers with affine invariance, Communications in Applied Mathematics and Computational Sciences, 5 (2010), 65-80.
doi: 10.2140/camcos.2010.5.65. |
[24] |
J. Harlim, Data assimilation with model error from unresolved scales,, , ().
|
[25] |
R. Kalman, A new approach to linear filtering and prediction theory, Transactions of the ASME-Journal of Basic Engineering, 82 (1960), 35-48. |
[26] |
M. Kalos and P. Whitlock, Monte Carlo Methods, Second edition. Wiley-Blackwell, Weinheim, 2008.
doi: 10.1002/9783527626212. |
[27] |
P. Lancaster and L. Rodman, Algebraic Riccati Equations, Oxford University Press, 1995. |
[28] |
J. Liu and R. Chen, Blind deconvolution via sequential imputations, Journal of the American Statistical Association, 90 (1995), 567-576.
doi: 10.1080/01621459.1995.10476549. |
[29] |
E. N. Lorenz, Deterministic non-periodic flow, Journal of Atmospheric Science, 20 (1963), 130-141. |
[30] |
E. Lorenz, Predictability: A problem partly solved, Proc. Seminar on Predictability, (2009), 40-58.
doi: 10.1017/CBO9780511617652.004. |
[31] |
A. Majda and J. Harlim, Physics constrained nonlinear regression models for time series, Nonlinearity, 26 (2013), 201-217.
doi: 10.1088/0951-7715/26/1/201. |
[32] |
R. N. Miller and L. Ehret, Application of the implicit particle filter to a model of nearshore circulation, Journal of Geophysical Research, 119 (2014), 2363-2385.
doi: 10.1002/2013JC009440. |
[33] |
R. N. Miller, M. Ghil and F. Gauthiez, Advanced data assimilation in strongly nonlinear systems, Journal of Atmospheric Science, 51 (1994), 1037-1056.
doi: 10.1175/1520-0469(1994)051<1037:ADAISN>2.0.CO;2. |
[34] |
M. Morzfeld and H. Godinez, Estimation and prediction for an orbital propagation model using data assimilation, Advances in the Astronautical Sciences Spaceflight Mechanics, 152 (2015), 2003-2021. |
[35] |
M. Morzfeld and D. Hodyss, Analysis of the ensemble Kalman filter for marginal and joint posteriors,, submitted for publication., ().
|
[36] |
M. Morzfeld, X. Tu, E. Atkins and A. J. Chorin, A random map implementation of implicit filters, Journal of Computational Physics, 231 (2012), 2049-2066.
doi: 10.1016/j.jcp.2011.11.022. |
[37] |
M. Morzfeld, X. Tu, J. Wilkening and A. J. Chorin, Parameter estimation by implicit sampling, Communications in Applied Mathematics and Computational Science, 10 (2015), 205-225.
doi: 10.2140/camcos.2015.10.205. |
[38] |
T. Moselhy and Y. Marzouk, Bayesian inference with optimal maps, Journal of Computational Physics, 231 (2012), 7815-7850.
doi: 10.1016/j.jcp.2012.07.022. |
[39] |
C. Nicolis, Probabilistic aspects of error growth in atmospheric dynamics, Quarterly Jourmal of the Royal Meteorological Society, 118 (1992), 553-567. |
[40] |
A. B. Owen, Monte Carlo Theory, Methods and Examples, 2013., URL , ().
|
[41] |
Y. Pokern, A. Stuart and P. Wiberg, Parameter estimation for partially observed hypoelliptic diffusions, Journal of the Royal Statistical Society: Series B, 71 (2009), 49-73.
doi: 10.1111/j.1467-9868.2008.00689.x. |
[42] |
C. Rasmussen and C. Williams, Gaussian Processes for Machine Learning, MIT Press, 2006. |
[43] |
N. Recca, A New Methodology for Importance Sampling, Masters Thesis, Courant Institute of Mathematical Sciences, New York University, 2011. |
[44] |
A. Samson and M. Thieullen, A contrast estimator for completely or partially observed hypoelliptic diffusion, Stochastic Processes and their Applications, 122 (2012), 2521-2552.
doi: 10.1016/j.spa.2012.04.006. |
[45] |
C. Snyder, Particle filters, the "optimal'' proposal and high-dimensional systems, Proceedings of the ECMWF Seminar on Data Assimilation for Atmosphere and Ocean, (2011). |
[46] |
C. Snyder, T. Bengtsson, P. Bickel and J. Anderson, Obstacles to high-dimensional particle filtering, Monthly Weather Review, 136 (2008), 4629-4640. |
[47] |
C. Snyder, T. Bengtsson and M. Morzfeld, Performance bounds for particle filters using the optimal proposal, Monthly Weather Review, 143 (2015), 4750-4761. |
[48] |
A. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica, 19 (2010), 451-559.
doi: 10.1017/S0962492910000061. |
[49] |
M. Tippet, J. Anderson, C. Bishop, T. Hamil and J. Whitaker, Ensemble square root filters, Monthly Weather Review, 131 (2003), 1485-1490. |
[50] |
P. van Leeuwen, Particle filtering in geophysical systems, Monthly Weather Review, 137 (2009), 4089-4114. |
[51] |
P. van Leeuwen, Nonlinear data assimilation in geosciences: an extremely efficient particle filter, Quarterly Journal of the Royal Meteorological Society, 136 (2010), 1991-1999. |
[52] |
E. Vanden-Eijnden and J. Weare, Rare event simulation and small noise diffusions, Communications on Pure and Applied Mathematics, 65 (2012), 1770-1803.
doi: 10.1002/cpa.21428. |
[53] |
E. Vanden-Eijnden and J. Weare, Data assimilation in the low noise, accurate observation regime with application to the Kuroshio current, Monthly Weather Review, 141 (2013), 1822-1841. |
[54] |
B. Weir, R. N. Miller and Y. Spitz, Implicit estimation of ecological model parameters, Bulletin of Mathematical Biology, 75 (2013), 223-257.
doi: 10.1007/s11538-012-9801-6. |
[55] |
D. Wilks, Effects of stochastic parameterizations in the Lorenz '96 model, Quarterly Journal of the Royal Meteorological Society, 131 (2005), 389-407. |
[56] |
V. Zaritskii and L. Shimelevich, Monte Carlo technique in problems of optimal data processing, Automation and Remote Control, 36 (1975), 2015-2022. |
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Yuan Pei. Continuous data assimilation for the 3D primitive equations of the ocean. Communications on Pure and Applied Analysis, 2019, 18 (2) : 643-661. doi: 10.3934/cpaa.2019032 |
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