图书馆订阅: Guest
Begell Digital Portal Begell 数字图书馆 电子图书 期刊 参考文献及会议录 研究收集
国际不确定性的量化期刊
影响因子: 4.911 5年影响因子: 3.179 SJR: 1.008 SNIP: 0.983 CiteScore™: 5.2

ISSN 打印: 2152-5080
ISSN 在线: 2152-5099

Open Access

国际不确定性的量化期刊

DOI: 10.1615/Int.J.UncertaintyQuantification.2019027245
pages 321-330

A MULTILEVEL APPROACH FOR SEQUENTIAL INFERENCE ON PARTIALLY OBSERVED DETERMINISTIC SYSTEMS

Ajay Jasra
Department of Statistics & Applied Probability, National University of Singapore, Singapore
Kody J.H. Law
School of Mathematics, University of Manchester, Manchester, M139PL, UK
Yi Xu
Department of Statistics & Applied Probability, National University of Singapore, Singapore

ABSTRACT

In this article we consider sequential inference on partially observed deterministic systems. Examples include: inference on the expected position of a dynamical system, with random initial position, or Bayesian static parameter inference for unobserved partial differential equations (PDEs), both associated to sequentially observed real data. Such statistical models are found in a wide variety of real applications, including weather prediction. In many practical scenarios one must discretize the system, but even under such discretization, it is not possible to compute the associated expected value (integral) required for inference. Such quantities are then approximated by Monte Carlo methods, and the associated cost to achieve a given level of error in this context can substantially be reduced by using multilevel Monte Carlo (MLMC). MLMC relies upon exact sampling of the model of interest, which is not always possible. We devise a sequential Monte Carlo (SMC) method, which does not require exact sampling, to leverage the MLMC method. We prove that for some models with n data points, that to achieve a mean square error (MSE) in estimation of O( 2) (for some 0 < < 1) our MLSMC method has a cost of O(n 2 -2) versus an SMC method that just approximates the most precise discretiztion of O(n 2 -3). This is illustrated on two numerical examples.

REFERENCES

  1. Law, K., Stuart, A., and Zygalakis, K., Data Assimilation, Cham, Switzerland: Springer, 2015.

  2. Kantas, N., Beskos, A., and Jasra, A., Sequential Monte Carlo Methods for High-Dimensional Inverse Problems: A Case Study for the Navier-Stokes Equations, SIAM/ASA J. Uncertain. Quantific., 2(1):464-489, 2014.

  3. Beskos, A., Crisan, D., and Jasra, A., On the Stability of Sequential Monte Carlo Methods in High Dimensions, Annals Appl. Prob., 24(4):1396-1445,2014.

  4. Oliver, D.S., Reynolds, A.C., and Liu, N., Inverse Theory for Petroleum Reservoir Characterization and History Matching, Cambridge: Cambridge University Press, 2008.

  5. Carrassi, A., Bocquet, M., Bertino, L., and Evensen, G., Data Assimilation in the Geosciences: An Overview of Methods, Issues, and Perspectives, Wiley Interdisc. Rev.: Climate Change, 9(5):e535,2018.

  6. Paulin, D., Jasra, A., Crisan, D., and Beskos, A., On Concentration Properties of Partially Observed Chaotic Systems, Adv. Appl. Prob, 50(2):440-479,2018.

  7. Paulin, D., Jasra, A., Crisan, D., and Beskos, A., Optimization-Based Methods for Partially Observed Chaotic Systems, Foun-dations Comput. Mathemat, pp. 1-75,2017. DOI: s10208-018-9388-x.

  8. Giles, M.B., Multilevel Monte Ccarlo Path Simulation, Oper. Res., 56(3):607-617, 2008.

  9. Giles, M.B., Multilevel Monte Carlo Methods, in Monte Carlo and Quasi-Monte Carlo Methods 2012, Springer, pp. 83-103, 2013.

  10. Heinrich, S., Multilevel Monte Carlo Methods, in Large-Scale Scientific Computing Methods, S. Margenov, J. Wasniewski, and P. Yalamov, Eds., Berlin: Springer, 2001.

  11. Beskos, A., Jasra, A., Law, K., Tempone, R., and Zhou, Y., Multilevel Sequential Monte Carlo Samplers, Stochastic Proc. Appl., 127(5):1417-1440, 2017.

  12. Hoang, V.H., Schwab, C., and Stuart, A.M., Complexity Analysis of Accelerated MCMC Methods for Bayesian Inversion, Inverse Probl., 29(8):085010,2013.

  13. Robert, C. and Casella, G., Monte Carlo Statistical Methods, Berlin: Springer Science & Business Media, 2013.

  14. Chopin, N., A Sequential Particle Filter Method for Static Models, Biometrika, 89(3):539-552, 2002.

  15. Del Moral, P., Doucet, A., and Jasra, A., Sequential Monte Carlo Samplers, J. Royal Stat. Soc.: Ser. B, 68(3):411-436, 2006.

  16. Del Moral, P., Feynman-Kac Formulae, in Feynman-Kac Formulae, Berlin: Springer, pp. 47-93, 2004.


Articles with similar content:

A MULTI-INDEX MARKOV CHAIN MONTE CARLO METHOD
International Journal for Uncertainty Quantification, Vol.8, 2018, issue 1
Ajay Jasra, Yan Zhou, Kengo Kamatani, Kody J.H. Law
AN OPTIMAL SAMPLING RULE FOR NONINTRUSIVE POLYNOMIAL CHAOS EXPANSIONS OF EXPENSIVE MODELS
International Journal for Uncertainty Quantification, Vol.5, 2015, issue 3
Michael Sinsbeck, Wolfgang Nowak
Method of Summary Representations for Solving Problems of Mathematical Safe on Graphs
Journal of Automation and Information Sciences, Vol.51, 2019, issue 12
Artem L. Gurin
FORWARD AND INVERSE UNCERTAINTY QUANTIFICATION USING MULTILEVEL MONTE CARLO ALGORITHMS FOR AN ELLIPTIC NONLOCAL EQUATION
International Journal for Uncertainty Quantification, Vol.6, 2016, issue 6
Ajay Jasra, Yan Zhou, Kody J.H. Law
APPLICATIONS OF NEUTROSOPHIC CUBIC SETS IN MULTI-CRITERIA DECISION-MAKING
International Journal for Uncertainty Quantification, Vol.7, 2017, issue 5
Ahmed Ali, Muhammad Gulistan, Jianming Zhan, Madad Khan