Amezcua, J., Kalnay, E., Ide, K. & Reich, S. (2014), ‘Ensemble transform Kalman–Bucy filters’, Q.J.R. Meteor. Soc. 140, 995–1004.CrossRefGoogle Scholar

Anderson, J. (2007), ‘An adaptive covariance inflation error correction algorithm for ensemble filters’, Tellus 59, 210–224.Google Scholar

Anderson, J. (2010), ‘A non-Gaussian ensemble filter update for data assimilation’, Mon. Wea. Rev. 138, 4186–4198.CrossRefGoogle Scholar

Anderson, J. (2012), ‘Localization and sampling error correction in ensemble Kalman filter data assimilation’, Mon. Wea. Rev. 140, 2359–2371.CrossRefGoogle Scholar

Andrieu, C. & Roberts, G. (2009), ‘The pseudo-marginal approach for efficient Monte-Carlo computations’, Ann. Statist. 37, 697–725.CrossRefGoogle Scholar

Arnold, V. (1989), Mathematical Methods of Classical Mechanics, 2nd edn, Springer-Verlag, New York.

Arulampalam, M., Maskell, S., Gordon, N. & Clapp, T. (2002), ‘A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking’, IEEE Trans. Sign. Process. 50, 174–188.CrossRefGoogle Scholar

Ascher, U. (2008), Numerical Methods for Evolutionary Differential Equations, SIAM, Philadelphia, PA.CrossRefGoogle Scholar

Bain, A. & Crisan, D. (2009), Fundamentals of Stochastic Filtering, Vol. 60 of Stochastic Modelling and Applied Probability, Springer-Verlag, New York.CrossRefGoogle Scholar

Beaumont, M. (2003), ‘Estimation of population growth or decline in genetically monitored populations’, Genetics 164, 1139–1160.Google ScholarPubMed

Benamou, J. & Brenier, Y. (2000), ‘A computational fluid mechanics solution to the Monge–Kantorovitch mass transfer problem’, Numer. Math. 84, 375–393.CrossRefGoogle Scholar

Bengtsson, T., Bickel, P. & Li, B. (2008), ‘Curse of dimensionality revisited: Collapse of the particle filter in very large scale systems’. In IMS Collections, Probability and Statistics: Essays in Honor of David F. Freedman, 2, 316–334.Google Scholar

Bergemann, K. & Reich, S. (2010), ‘A mollified ensemble Kalman filter’, Q.J.R. Meteorolog. Soc. 136, 1636–1643.CrossRefGoogle Scholar

Bergemann, K. & Reich, S. (2012), ‘An ensemble Kalman–Bucy filter for continuous data assimilation’, Meteorolog. Zeitschrift 21, 213–219.CrossRefGoogle Scholar

Bloom, S., Takacs, L., Silva, A.D. & Ledvina, D. (1996), ‘Data assimilation using incremental analysis updates’, Q.J.R. Meteorolog. Soc. 124, 1256–1271.Google Scholar

Bocquet, M. & Sakov, P. (2012), ‘Combining inflation-free and iterative ensemble Kalman filters for strongly nonlinear systems’, Nonlin. Processes Geophys. 19, 383–399.CrossRefGoogle Scholar

Bocquet, M., Pires, C. & Wu, L. (2010), ‘Beyond Gaussian statistical modeling in geophysical data assimilaition’, Mon. Wea. Rev. 138, 2997–3022.CrossRefGoogle Scholar

Breźniak, Z. & Zastawniak, T. (1999), Basic Stochastic Processes, Springer-Verlag, London.CrossRefGoogle Scholar

Bröcker, J. (2012), ‘Evaluating raw ensembles with the continuous ranked probability score’, Q.J.R. Meteorolog. Soc. 138, 1611–1617.CrossRefGoogle Scholar

Bungartz, H.-J. & Griebel, M. (2004), ‘Sparse grids’, Acta Numerica 13, 147–269.CrossRefGoogle Scholar

Caflisch, R. (1998), ‘Monte Carlo and Quasi-Monte Carlo methods’, Acta Numerica 7, 1–49.CrossRefGoogle Scholar

Chen, Y. & Oliver, D. (2013), ‘Levenberg–Marquardt forms of the iterative ensemble smoother for efficient history matching and uncertainty quantification’, Computational Geoscience 17, 689–703.CrossRefGoogle Scholar

Cheng, Y. & Reich, S. (2013), ‘A McKean optimal transportation perspective on Feynman–Kac formulae with application to data assimilation’., arXiv:1311.6300. To appear in Frontiers in Applied Dynamical Systems, Vol. 1, Springer-Verlag, New York.Google Scholar

Chorin, A. & Hald, O. (2009), Stochastic Tools in Mathematics and Science, 2nd edn, Springer-Verlag, Berlin.CrossRefGoogle Scholar

Chorin, A., Morzfeld, M. & Tu, X. (2010), ‘Implicit filters for data assimilation’, Comm. Appl. Math. Comp. Sc. 5, 221–240.Google Scholar

Coles, S. (2001), An Introduction to Statistical Modeling of Extreme Values, Springer-Verlag, New York.CrossRefGoogle Scholar

Collet, P. & Eckmann, J.-P. (2006), Concepts and Results in Chaotic Dynamics, Springer-Verlag, Berlin.Google Scholar

Crisan, D. & Xiong, J. (2010), ‘Approximate McKean–Vlasov representation for a class of SPDEs’, Stochastics 82, 53–68.CrossRefGoogle Scholar

Daley, R. (1993), Atmospheric Data Analysis, Cambridge University Press, Cambridge.Google Scholar

Daum, F. & Huang, J. (2011), ‘Particle filter for nonlinear filters’. In Acoustics, Speech and Signal Processing (ICASSP), 2011 IEEE International Conference, IEEE, New York, 5920–5923.Google Scholar

del Moral, P. (2004), Feynman–Kac Formulae: Genealogical and Interacting Particle Systems with Applications, Springer-Verlag, New York.CrossRefGoogle Scholar

Doucet, A., de Freitas, N. & Gordon, N., editors (2001), Sequential Monte Carlo Methods in Practice, Springer-Verlag, Berlin.

Emerik, A. & Reynolds, A. (2012), ‘Ensemble smoother with multiple data assimilation’, Computers & Geosciences 55, 3–15.Google Scholar

Engl, H., Hanke, M. & Neubauer, A. (2000), Regularization of Inverse Problems, Kluwer Academic Publishers, Dordrecht.

Evans, L. (1998), Partial Differential Equations, American Mathematical Society, Providence, RI.

Evans, L. (2013), An Introduction to Stochastic Differential Equations, American Mathematical Society, Providence, RI.

Evensen, G. (2006), Data Assimilation. The Ensemble Kalman Filter, Springer-Verlag, New York.

Frei, M. & Küunsch, H. (2013), ‘Mixture ensemble Kalman filters’, Computational Statistics and Data Analysis 58, 127–138.CrossRefGoogle Scholar

Gardiner, C. (2004), Handbook of Stochastic Methods, 3rd edn, Springer-Verlag.

Gneiting, T., Balabdaoui, F. & Raftery, A. (2007), ‘Probabilistic forecasts, calibration and sharpness’, J. R. Statist. Soc. B 69, 243–268.CrossRefGoogle Scholar

Golub, G. & Loan, C.V. (1996), Matrix Computations, 3rd edn, Johns Hopkins University Press, Baltimore, MD.

González-Tokman, C. & Hunt, B. (2013), ‘Ensemble data assimilation for hyperbolic systems’, Physica D 243, 128–142.CrossRefGoogle Scholar

Hand, D. (2008), Statistics: A Very Short Introduction, Oxford University Press, Oxford.

Hastie, T., Tibshirani, R. & Friedman, J. (2009), The Elements of Statistical Learning, 2nd edn, Springer-Verlag, New York.

Higham, D. (2001), ‘An algorithmic introduction to numerical simulation of stochastic differential equations’, SIAM Review 43, 525–546.CrossRefGoogle Scholar

Hoke, J. & Anthes, R. (1976), ‘The initialization of numerical models by a dynamic relaxation technique’, Mon. Wea. Rev. 104, 1551–1556.2.0.CO;2>CrossRefGoogle Scholar

Holtz, M. (2011), Sparse Grid Quadrature in High Dimensions with Applications in Finance and Insurance, Springer-Verlag, Berlin.CrossRefGoogle Scholar

Hunt, B., Kostelich, E. & Szunyogh, I. (2007), ‘Efficient data assimilation for spatialtemporal chaos: A local ensemble transform Kalman filter’, Physica D 230, 112–137.CrossRefGoogle Scholar

Janjić, T., McLaughlin, D., Cohn, S. & Verlaan, M. (2014), ‘Convervation of mass and preservation of positivity with ensemble-type Kalman filter algorithms’, Mon. Wea. Rev. 142, 755–773.CrossRefGoogle Scholar

Jazwinski, A. (1970), Stochastic Processes and Filtering Theory, Academic Press, New York.Google Scholar

Julier, S.J. & Uhlmann, J.K. (1997), ‘A new extension of the Kalman filter to nonlinear systems’. In Proc. AeroSense: 11th Int. Symp. Aerospace/Defense Sensing, Simulation and Controls, pp. 182–193.Google Scholar

Kaipio, J. & Somersalo, E. (2005), Statistical and Computational Inverse Problems, Springer-Verlag, New York.Google Scholar

Kalnay, E. (2002), Atmospheric Modeling, Data Assimilation and Predictability, Cambridge University Press, Cambridge.CrossRefGoogle Scholar

Katok, A. & Hasselblatt, B. (1995), Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge.CrossRefGoogle Scholar

Kitanidis, P. (1995), ‘Quasi-linear geostatistical theory for inverting’, Water Resources Research 31, 2411–2419.CrossRefGoogle Scholar

Kloeden, P. & Platen, E. (1992), Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin.CrossRefGoogle Scholar

Küunsch, H. (2005), ‘Recursive Monte Carlo filter: Algorithms and theoretical analysis’, Ann. Statist. 33, 1983–2021.CrossRefGoogle Scholar

Lasota, A. & Mackey, M. (1994), Chaos, Fractals, and Noise, 2nd edn, Springer-Verlag, New York.CrossRefGoogle Scholar

Law, K., Shukia, A. & Stuart, A. (2014), ‘Analysis of the 3DVAR filter for the partially observed Lorenz ’63 model’, Discrete and Continuous Dynamical Systems A 34, 1061–1078.Google Scholar

Lei, J. & Bickel, P. (2011), ‘A moment matching ensemble filter for nonlinear and non-Gaussian data assimilation’, Mon. Wea. Rev. 139, 3964–3973.CrossRefGoogle Scholar

Lewis, J., Lakshmivarahan, S. & Dhall, S. (2006), Dynamic Data Assimilation: A Least Squares Approach, Cambridge University Press, Cambridge.CrossRefGoogle Scholar

Liu, J. (2001), Monte Carlo Strategies in Scientific Computing, Springer-Verlag, New York.Google Scholar

Livings, D., Dance, S. & Nichols, N. (2008), ‘Unbiased ensemble square root filters’, Physica D 237, 1021–1028.CrossRefGoogle Scholar

Lorenz, E. (1963), ‘Deterministic non-periodic flows’, J. Atmos. Sci. 20, 130–141.2.0.CO;2>CrossRefGoogle Scholar

Majda, A. & Harlim, J. (2012), Filtering Complex Turbulent Systems, Cambridge University Press, Cambridge.CrossRefGoogle Scholar

McCann, R. (1995), ‘Existence and uniqueness of monotone measure-preserving maps’, Duke Math. J. 80, 309–323.CrossRefGoogle Scholar

McKean, H. (1966), ‘A class of Markov processes associated with nonlinear parabolic equations’, Proc. Natl. Acad. Sci. USA 56, 1907–1911.CrossRefGoogle Scholar

Meyn, S. & Tweedie, R. (1993), Markov Chains and Stochastic Stability, Springer-Verlag, London.CrossRefGoogle Scholar

Miyoshi, T. (2011), ‘The Gaussian approach to adaptive covariance inflation and its implementation with the local ensemble transform Kalman filter’, Mon. Wea. Rev. 139, 1519–1535.CrossRefGoogle Scholar

Morzfeld, M. & Chorin, A. (2012), ‘Implicit particle filtering for models with partial noise and an application to geomagnetic data assimilation’, Nonlinear Processes in Geophysics 19, 365–382.CrossRefGoogle Scholar

Morzfeld, M., Tu, X., Atkins, E. & Chorin, A. (2012), ‘A random map implementation of implicit filters’, J. Comput. Phys. 231, 2049–2066.CrossRefGoogle Scholar

Moselhy, T.E. & Marzouk, Y. (2012), ‘Bayesian inference with optimal maps’, J. Comput. Phys. 231, 7815–7850.CrossRefGoogle Scholar

Moser, J. (1965), ‘On the volume elements on a manifold’, Trans. Amer. Math. Soc. 120, 286–294.CrossRefGoogle Scholar

Neal, R. (1996), Bayesian Learning for Neural Networks, Springer-Verlag, New York.CrossRefGoogle Scholar

Nerger, L., Pfander, T.J., Schröter, J. & Hiller, W. (2012), ‘A regulated localization scheme for ensemble-based Kalman filters’, Q.J.R. Meteorolog. Soc. 138, 802–812.CrossRefGoogle Scholar

Nocedal, J. & Wright, S. (2006), Numerical Optimization, 2nd edn, Springer-Verlag, New York.Google Scholar

Øksendal, B. (2000), Stochastic Differential Equations, 5th edn, Springer-Verlag, Berlin.Google Scholar

Oliver, D. (1996), ‘On conditional simulation to inaccurate data’, Math. Geology 28, 811–817.CrossRefGoogle Scholar

Oliver, D., He, N. & Reynolds, A. (1996), ‘Conditioning permeability fields on pressure data. Technical Report: presented at the 5th European Conference on the Mathematics of Oil Recovery, Leoben, Austria.Google Scholar

Olkin, I. & Pukelsheim, F. (1982), ‘The distance between two random vectors with given dispersion matrices’, Lin. Alg. and its Applns. 48, 257–263.Google Scholar

Ott, E., Hunt, B., Szunyogh, I., Zimin, A., Kostelich, E., Corazza, M., Kalnay, E., Patil, D. & Yorke, J.A. (2004), ‘A local ensemble Kalman filter for atmospheric data assimilation’, Tellus A 56, 415–428.Google Scholar

Otto, F. (2001), ‘The geometry of dissipative evolution equations: the porous medium equation’, Comm. Part. Diff. Eqs. 26, 101–174.CrossRefGoogle Scholar

Pavliotis, G. (2014), Stochastic Processes and Applications, Springer-Verlag, New York.CrossRefGoogle Scholar

Pele, O. & Werman, M. (2009), ‘Fast and robust earth mover's distances’. In 12th IEEE International Conference on Computer Vision, IEEENew York, pp. 460–467.Google Scholar

Raftery, A. (1995), ‘Bayesian model selection in social research’, Sociological Methodology 25, 111–163.CrossRefGoogle Scholar

Rebeschini, P. & van Handel, R. (2013), ‘Can local particle filters beat the curse of dimensionality?’, arXiv:1301.6585.

Reich, S. (2011), ‘A dynamical systems framework for intermittent data assimilation’, BIT Numer Math 51, 235–249.CrossRefGoogle Scholar

Reich, S. (2012), ‘A Gaussian mixture ensemble transform filter’, Q.J.R. Meteorolog. Soc. 138, 222–233.CrossRefGoogle Scholar

Reich, S. (2013a), ‘A guided sequential Monte Carlo method for the assimilation of data into stochastic dynamical systems’. In Recent Trends in Dynamical Systems, Proceedings in Mathematics and Statistics, 35, Springer-Verlag, Basel, pp. 205–220.CrossRefGoogle Scholar

Reich, S. (2013b), ‘A nonparametric ensemble transform method for Bayesian inference’, SIAM J. Sci. Comput. 35, A2013–A2024.CrossRefGoogle Scholar

Reich, S. & Cotter, C. (2013), ‘Ensemble filter techniques for intermittent data assimilation’. In Large Scale Inverse Problems. Computational Methods and Applications in the Earth Sciences, M. Cullen, M.A.Freitag, S. Kindermann & R.Scheichl, Scheichl, eds, Radon Ser. Comput. Appl. Math., 13, de Gruyter, Berlin, pp. 91–134.Google Scholar

Robert, C. (2001), The Bayesian Choice: From Decision-Theoretic Motivations to Computational Implementations, 2nd edn, Springer-Verlag, New York.Google Scholar

Robert, C. & Casella, G. (2004), Monte Carlo Statistical Methods, 2nd edn, Springer-Verlag, New York.CrossRefGoogle Scholar

Robinson, J. (2001), Infinite-Dimensional Dynamical Systems, Cambridge University Press, Cambridge.CrossRefGoogle Scholar

Sakov, P., Oliver, D. & Bertino, L. (2012), ‘An iterative EnKF for strongly nonlinear systems’, Mon. Wea. Rev. 140, 1988–2004.CrossRefGoogle Scholar

Särkkä, S. (2013), Bayesian Filtering and Smoothing, Cambridge University Press, Cambridge.CrossRefGoogle Scholar

Simon, D. (2006), Optimal State Estimation, Wiley, New York.CrossRefGoogle Scholar

Smith, K. (2007a), ‘Cluster ensemble Kalman filter’, Tellus 59A, 749–757.Google Scholar

Smith, L. (2007b), Chaos: A Very Short Introduction, Oxford University Press, Oxford.CrossRef

Smith, R. (2014), Uncertainty Quantification, SIAM, Philadelphia, PA.Google Scholar

Stordal, A., Karlsen, H., Nævdal, G., Skaug, H. & Vallés, B. (2011), ‘Bridging the ensemble Kalman filter and particle filters: the adaptive Gaussian mixture filter’, Comput. Geosci. 15, 293–305.CrossRefGoogle Scholar

Strang, G. (1986), Introduction to Applied Mathematics, 2nd edn, Wellesley-Cambridge Press, Wellesley, MA.Google Scholar

Stuart, A. (2010), ‘Inverse problems: a Bayesian perspective’, Acta Numerica 17, 451–559.Google Scholar

Süli, E. & Mayers, D. (2006), An Introduction to Numerical Analysis, Cambridge University Press, Cambridge.Google Scholar

Takens, F. (1981), ‘Detecting strange attractors in turbulence’. In Dynamical Systems and Turbulence. Lecture Notes in Mathematics, 898, Springer-Verlag, Berlin, pp. 366–381.Google Scholar

Tarantola, A. (2005), Inverse Problem Theory and Methods for Model Parameter Estimation, SIAM, Philadelphia, PA.CrossRefGoogle Scholar

Tijms, H. (2012), Understanding Probability, 3rd edn, Cambridge University Press, Cambridge.CrossRefGoogle Scholar

Tippett, M., Anderson, J., Bishop, G., Hamill, T. & Whitaker, J. (2003), ‘Ensemble square root filters’, Mon. Wea. Rev. 131, 1485–1490.2.0.CO;2>CrossRefGoogle Scholar

van Leeuwen, P. (2010), ‘Nonlinear data assimilation in the geosciences: an extremely efficient particle filter’, Q.J.R. Meteorolog. Soc. 136, 1991–1996.CrossRefGoogle Scholar

van Leeuwen, P. & Ades, M. (2013), ‘Efficient fully nonlinear data assimilation for geophysical fluid dynamics’, Computers & Geosciences 55, 16–27.CrossRefGoogle Scholar

Verhulst, F. (2000), Nonlinear Differential Equations and Dynamical Systems, 2nd edn, Springer-Verlag, Berlin.Google Scholar

Villani, C. (2003), Topics in Optimal Transportation, American Mathematical Society, Providence, RI.CrossRefGoogle Scholar

Villani, C. (2009), Optimal Transportation: Old and New, Springer-Verlag, Berlin.CrossRefGoogle Scholar

Wang, X., Bishop, C. & Julier, S. (2004), ‘Which is better, an ensemble of positivenegative pairs or a centered spherical simplex ensemble?’, Mon. Wea. Rev. 132, 1590–1505.2.0.CO;2>CrossRefGoogle Scholar

Yang, T., Mehta, P. & Meyn, S. (2013), ‘Feedback particle filter’, IEEE Trans. Automatic Control 58, 2465–2480.CrossRefGoogle Scholar