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Sequential implicit sampling methods for Bayesian inverse problems. (English) Zbl 1401.65012

Summary: The solutionto the inverse problems under the Bayesian framework is given by a posterior probability density. For large-scale problems, sampling the posterior can be an extremely challenging task. Markov chain Monte Carlo provides a general way for sampling but it can be computationally expensive. Gaussian type methods, such as the ensemble Kalman filter (EnKF), make Gaussian assumptions at some point(s) in the algorithms, even for the possible non-Gaussian densities, which may lead to inaccuracy. In this paper, the implicit sampling method, one of the importance sampling methods, and the newly proposed sequential implicit sampling method are investigated for the inverse problem involving time-dependent partial differential equations. The sequential implicit sampling method combines the idea of the EnKF and implicit sampling and it is particularly suitable for time-dependent problems. Moreover, the new method is capable of reducing the computational cost in the optimization, which is a necessary and the most expensive step in the implicit sampling method. The sequential implicit sampling method has been tested on a seismic wave inversion. The numerical experiments show its accuracy and efficiency by comparing it with some popular Gaussian approximation methods and importance sampling methods.

MSC:

65C05 Monte Carlo methods
35R30 Inverse problems for PDEs
62F15 Bayesian inference

Software:

EnKF
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. L. Anderson and S. L. Anderson, {\it A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts}, Monthly Weather Rev., 127 (1999), pp. 2741-2758.
[2] M. S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, {\it A tutorial on particle filters for online nonlinear/nongaussian Bayesian tracking}, IEEE Trans. Signal Process., 50 (2002), pp. 174-188.
[3] E. Atkins, M. Morzfeld, and A. J. Chorin, {\it Implicit particle methods and their connection with variational data assimilation}, Monthly Weather Rev., 141 (2013), pp. 1786-1803.
[4] A. Beskos, A. Jasra, E. Muzaffer, and A. Stuart, {\it Sequential Monte Carlo methods for Bayesian elliptic inverse problems}, Statist. Comput., 25 (2015), pp. 727-737. · Zbl 1331.65012
[5] P. Bickel, B. Li, and T. Bengtsson, {\it Sharp failure rates for the bootstrap particle filter in high dimensions}, in Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh, Inst. Math. Stat. Collect. 3, Institute of Mathematical Statistics, Beachwood, OH, 2008, pp. 318-329. · Zbl 1159.62004
[6] T. Bui-Thanh, O. Ghattas, J. Martin, and G. Stadler, {\it A computational framework for infinite-dimensional Bayesian inverse problems Part I: The linearized case, with application to global seismic inversion}, SIAM J. Sci. Comput., 35 (2013), pp. A2494-A2523. · Zbl 1287.35087
[7] G. Burgers, P. J. van Leeuwen, and G. Evensen, {\it Analysis scheme in the ensemble Kalman filter}, Monthly Weather Rev., 126 (2013), pp. 1719-1724.
[8] N. Chopin, {\it A sequential particle filter method for static models}, Biometrika, 89 (2002), pp. 539-551. · Zbl 1036.62062
[9] A. J. Chorin and O. H. Hald, {\it Stochastic Tools in Mathematics and Science}, Springer, New York, 2008. · Zbl 1086.60001
[10] A. J. Chorin and M. Morzfeld, {\it Conditions for successful data assimilation}, J. Geophys. Res. Atmospheres, 118 (2013), pp. 11522-11533.
[11] A. J. Chorin, M. Morzfeld, and X. Tu, {\it Implicit particle filters for data assimilation}, Commun. Appl. Math. Comput. Sci., 5 (2010), pp. 221-240. · Zbl 1229.60047
[12] A. J. Chorin, M. Morzfeld, and X. Tu, {\it Implicit sampling, with application to data assimilation}, Chin. Ann. Math. Ser. B, 34 (2013), pp. 89-98. · Zbl 1261.62084
[13] A. J. Chorin and X. Tu, {\it Implicit sampling for particle filters}, Proc. Natl. Acad. Sci. USA, 106 (2009), pp. 17249-17254.
[14] A. J. Chorin and X. Tu, {\it Interpolation and iteration for nonlinear filters}, M2AN Math. Model. Numer. Anal., 46 (2012), pp. 535-543. · Zbl 1395.62290
[15] G. Evensen, {\it Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics}, J. Geophys. Res., 99 (1994), pp. 10143-10162.
[16] G. Evensen, {\it Data Assimilation: The Ensemble Kalman Filter}, Springer, New York, 2006. · Zbl 1157.86001
[17] E. J. Fertig, J. Harlim, and B. R. Hunt, {\it A comparative study of 4D-Var and a 4D ensemble Kalman filter: Perfect model simulations with Lorenz-96}, Tellus, 59A (2007), pp. 96-100.
[18] J. Goodman and J. Weare, {\it Ensemble samplers with affine invariance}, Commun. Appl. Math. Comput. Sci., 5 (2010), pp. 65-80. · Zbl 1189.65014
[19] T. M. Hamill, J. S. Whitaker, and C. Snyder, {\it Distance-dependent filtering of background error covariance estimates in an ensemble Kalman filter}, Monthly Weather Rev., 129 (2001), pp. 2776-2790.
[20] P. L. Houtekamer and H. L. Mitchell, {\it Data assimilation using an ensemble Kalman filter technique}, Monthly Weather Rev., 126 (1998), pp. 796-811.
[21] B. R. Hunt, E. Kalnay, E. J. Kostelich, E. Ott, D. J. Patil, T. Sauer, I. Szunyogh, J. A. Yorke, and A. V. Zimin, {\it Four-dimensional ensemble Kalman filtering}, Tellus, 56A (2004), pp. 273-277.
[22] M. Iglesias, K. Law, and A. M. Stuart, {\it Evalutaion of Gaussian approximations for data assimilation in reservoir models}, Comput. Geosci., 17 (2013), pp. 851-885. · Zbl 1393.86020
[23] N. Kantas, A. Beskos, and A. Jasra, {\it Sequential Monte Carlo methods for high-dimensional inverse problems: A case study for the Navier-Stokes equations}, SIAM/ASA J. Uncertain. Quantif., 2 (2014), pp. 464-489. · Zbl 1308.65010
[24] C. Liu, Q. Xiao, and B. Wang, {\it An ensemble-based four-dimensional variational data assimilation scheme. Part I: Technical formulation and preliminary test}, Monthly Weather Rev., 136 (2008), pp. 3363-3373.
[25] J. S. Liu, {\it Monte Carlo Strategies for Scientific Computing}, Springer, New York, 2013.
[26] J. Martin, L. C. Wilcox, C. Burstedde, and O. Ghattas, {\it A stochastic Newton MCMC method for large-scale statistical inverse problems with application to seismic inversion}, SIAM J. Sci. Comput., 34 (2012), pp. A1460-A1487. · Zbl 1250.65011
[27] M. Morzfeld and A. J. Chorin, {\it Implicit particle filtering for models with partial noise, and an application to geomagnetic data assimilation}, Nonlinear Processes Geophys., 19 (2012), pp. 365-382.
[28] M. Morzfeld, X. Tu, E. Atkins, and A. J. Chorin, {\it A random map implementation of implicit filters}, J. Comput. Phys., 231 (2012), pp. 2049-2066. · Zbl 1242.65012
[29] M. Morzfeld, X. Tu, J. Wilkening, and A. J. Chorin, {\it Parameter estimation by implicit sampling}, Commun. Appl. Math. Comput. Sci., 10 (2015), pp. 205-225. · Zbl 1328.86002
[30] J. Nocedal and S. T. Wright, {\it Numerical Optimization}, 2nd ed., Springer, New York, 2006. · Zbl 1104.65059
[31] D. Oliver, A. Reynolds, and N. Liu, {\it Inverse Theory for Petroleum Reservoir Characterization and History Matching}, Cambridge University Press, Cambridge, UK, 2008.
[32] E. Ott, B. R. Hunt, I. Szunyogh, A. V. Zimin, E. J. Kostelich, M. Corazza, E. Kalnay, D. J. Patil, and J. A. Yorke, {\it A local ensemble kalman filter for atmospheric data assimilation}, Tellus, 56A (2004), pp. 415-428.
[33] N. Petra, J. Martin, G. Stadler, and O. Ghattas, {\it A computational framework for infinite-dimensional Bayesian inverse problems, Part II: Stochastic Newton MCMC with application to ice sheet flow inverse problems}, SIAM J. Sci. Comput., 36 (2014), pp. A1525-A1555. · Zbl 1303.35110
[34] C. Snyder, T. Bengtsson, P. Bickel, and J. Anderson, {\it Obstacles to high-dimensional particle filtering}, Monthly Weather Rev., 136 (2008), pp. 4629-4640.
[35] A. M. Stuart, {\it Inverse problems: A Bayesian perspective}, Acta Numer., 19 (2010), pp. 451-559. · Zbl 1242.65142
[36] O. Talagrand, {\it A study of the dynamics of four-dimensional data assimilation}, Tellus, 33 (1981), pp. 43-60.
[37] O. Talagrand and P. Courtier, {\it Variational assimilation of meteorological observations with the adjoint vorticity equation. I: Theory}, Q. J. Roy. Meteor. Soc., 113 (1987), pp. 1311-1328.
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