Abstract
There are many computational tasks, in which it is necessary to sample a given probability density function (or pdf for short), i.e., to use a computer to construct a sequence of independent random vectors x i (i = 1, 2, …), whose histogram converges to the given pdf. This can be difficult because the sample space can be huge, and more importantly, because the portion of the space, where the density is significant, can be very small, so that one may miss it by an ill-designed sampling scheme. Indeed, Markovchain Monte Carlo, the most widely used sampling scheme, can be thought of as a search algorithm, where one starts at an arbitrary point and one advances step-by-step towards the high probability region of the space. This can be expensive, in particular because one is typically interested in independent samples, while the chain has a memory. The authors present an alternative, in which samples are found by solving an algebraic equation with a random right-hand side rather than by following a chain; each sample is independent of the previous samples. The construction in the context of numerical integration is explained, and then it is applied to data assimilation.
Similar content being viewed by others
References
Doucet, A., de Freitas, N. and Gordon, N., Sequential Monte Carlo Methods in Practice, Springer-Verlag, New York, 2001.
Chorin, A. J. and Hald, O. H. Stochastic Tools in Mathematics and Science, 2nd edition, Springer-Verlag, New York, 2009.
Morzfeld, M., Tu, X., Atkins, E. and Chorin, A. J., A random map implementation of implicit filters, J. Comput. Phys., 231, 2012, 2049–2066.
Morzfeld, M. and Chorin, A. J., Implicit particle filtering for models with partial noise, and an application to geomagnetic data assimilation, Nonlin. Processes Geophys., 19, 2012, 365–382.
Kloeden, P. E. and Platen, E., Numerical Solution of Stochastic Differential Equations, 3rd edition, Springer-Verlag, New York, 1999.
Chorin, A. J. and Tu, X., Implicit sampling for particle filters, Proc. Nat. Acad. Sc. USA, 106, 2009, 17249–17254.
Chorin, A. J., Morzfeld, M. and Tu, X., Implicit particle filters for data assimilation, Commun. Appl. Math. Comput. Sci., 5(2), 2010, 221–240.
Arulampalam, M. S., Maskell, S., Gordon, N. and Clapp, T., A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking, IEEE Trans. Signal Process, 10, 2002, 197–208.
Bickel, P., Li, B. and Bengtsson, T., Sharp failure rates for the bootstrap particle filter in high dimensions, Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh, 2008, 318–329.
Snyder, C. C., Bengtsson, T., Bickel, P. and Anderson, J., Obstacles to high-dimensional particle filtering, Mon. Wea. Rev., 136, 2008, 4629–4640.
Gordon, N. J., Salmon, D. J. and Smith, A. F. M., Novel approach to nonlinear/non-Gaussian Bayesian state estimation, IEEE Proceedings F on Radar and Signal Processing, 140, 1993, 107–113.
Doucet, A., Godsill, S. and Andrieu, C., On sequential Monte Carlo sampling methods for Bayesian filtering, Statistics and Computing, 50, 2000, 174–188.
Del Moral, P., Feynman-Kac Formulae, Springer-Verlag, New York, 2004.
Del Moral, P., Measure-valued processes and interacting particle systems. Application to nonlinear filtering problems, Annals of Applied Probability, 8(2), 1998, 438–495.
Zaritskii, V. S. and Shimelevich, L. I., Monte Carlo technique in problems of optimal data processing, Automation and Remote Control, 12, 1975, 95–103.
Kalman, R. E., A new approach to linear filtering and prediction theory, Trans. ASME, Ser. D, 82, 1960, 35–48.
Kalman, R. E. and Bucy, R. S., New results in linear filtering and prediction theory, Trans. ASME, Ser. D, 83, 1961, 95–108.
Evensen, G., Data Assimilation, Springer-Verlag, New York, 2007.
Zakai, M., On the optimal filtering of diffusion processes, Zeit. Wahrsch., 11, 1969, 230–243.
Talagrand, O. and Courtier, P., Variational assimilation of meteorological observations with the adjoint vorticity equation. I: Theory, Q. J. R. Meteorol. Soc., 113, 1987, 1311–1328.
Bennet, A. F., Leslie, L. M., Hagelberg, C. R. and Powers, P. E., A cyclone prediction using a barotropic model initialized by a general inverse method, Mon. Weather Rev., 121, 1993, 1714–1728.
Courtier, P., Thepaut, J. N. and Hollingsworth, A., A strategy for operational implementation of 4D-var, using an incremental appoach, Q. J. R. Meteorol. Soc., 120, 1994, 1367–1387.
Courtier, P., Dual formulation of four-dimensional variational assimilation, Q. J. R. Meteorol. Soc., 123, 1997, 2449–2461.
Talagrand, O., Assimilation of observations, an introduction, J. R. Meteorol. Soc. of Japan, 75(1), 1997, 191–209.
Tremolet, Y., Accounting for an imperfect model in 4D-var, Q. J. R. Meteorol. Soc., 621(132), 2006, 2483–2504.
Atkins, E., Morzfeld, M. and Chorin, A. J., Implicit particle methods and their connection to variational data assimilation, Mon. Weather Rev., in press.
Kuramoto, Y. and Tsuzuki, T., On the formation of dissipative structures in reaction-diffusion systems, Progr. Theoret. Phys., 54, 1975, 687–699.
Sivashinsky, G., Nonlinear analysis of hydrodynamic instability in laminar flames, Part I, Derivation of basic equations, Acta Astronaut., 4, 1977, 1177–1206.
Chorin, A. J. and Krause, P., Dimensional reduction for a Bayesian filter, PNAS, 101, 2004, 15013–15017.
Jardak, M., Navon, I. M. and Zupanski, M., Comparison of sequential data assimilation methods for the Kuramoto-Sivashinsky equation, Int. J. Numer. Methods Fluids, 62, 2009, 374–402.
Lord, G. J. and Rougemont, J., A numerical scheme for stochastic PDEs with Gevrey regularity, IMA Journal of Numerical Analysis, 24, 2004, 587–604.
Jentzen, A. and Kloeden, P. E., Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space-time noise, Proc. R. Soc. A, 465, 2009, 649–667.
Fletcher, R., Practical Methods of Optimization, Wiley, New York, 1987.
Nocedal, J. and Wright, S. T., Numerical Optimization, 2nd edition, Springer-Verlag, New York, 2006.
Author information
Authors and Affiliations
Corresponding author
Additional information
In honor of the scientific heritage of Jacques-Louis Lions
Project supported by the Director, Office of Science, Computational and Technology Research, U. S. Department of Energy (No.DE-AC02-05CH11231) and the National Science Foundation (Nos.DMS-0705910, OCE-0934298).
Rights and permissions
About this article
Cite this article
Chorin, A.J., Morzfeld, M. & Tu, X. Implicit Sampling, with Application to Data Assimilation. Chin. Ann. Math. Ser. B 34, 89–98 (2013). https://doi.org/10.1007/s11401-012-0757-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11401-012-0757-5