Multilevel ensemble Kalman filtering for spatio-temporal processes

We design and analyse the performance of a multilevel ensemble Kalman filter method (MLEnKF) for filtering settings where the underlying state-space model is an infinite-dimensional spatio-temporal process. We consider underlying models that needs to be simulated by numerical methods, with discretization in both space and time. The multilevel Monte Carlo (MLMC) sampling strategy, achieving variance reduction through pairwise coupling of ensemble particles on neighboring resolutions, is used in the sample-moment step of MLEnKF to produce an efficient hierarchical filtering method for spatio-temporal models. Under sufficient regularity, MLEnKF is proven to be more efficient for weak approximations than EnKF, asymptotically in the large-ensemble and fine-numerical-resolution limit. Numerical examples support our theoretical findings.

[1]  Michael B. Giles,et al.  Multilevel Monte Carlo Path Simulation , 2008, Oper. Res..

[2]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[3]  Kody J. H. Law,et al.  Multilevel ensemble Kalman filtering , 2015, SIAM J. Numer. Anal..

[4]  G. Evensen,et al.  Analysis Scheme in the Ensemble Kalman Filter , 1998 .

[5]  R. Handel,et al.  Can local particle filters beat the curse of dimensionality , 2013, 1301.6585.

[6]  Edriss S. Titi,et al.  Determining Modes for Continuous Data Assimilation in 2D Turbulence , 2003 .

[7]  Yan Zhou,et al.  Multilevel Sequential Monte Carlo Samplers for Normalizing Constants , 2016, ACM Trans. Model. Comput. Simul..

[8]  Edriss S. Titi,et al.  Discrete data assimilation in the Lorenz and 2D Navier–Stokes equations , 2010, 1010.6105.

[9]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[10]  A. Stuart,et al.  Well-posedness and accuracy of the ensemble Kalman filter in discrete and continuous time , 2013, 1310.3167.

[11]  T. Tarn,et al.  Observers for nonlinear stochastic systems , 1975, 1975 IEEE Conference on Decision and Control including the 14th Symposium on Adaptive Processes.

[12]  A. Stuart,et al.  Data Assimilation: A Mathematical Introduction , 2015, 1506.07825.

[13]  Jan Mandel,et al.  Convergence of the Square Root Ensemble Kalman Filter in the Large Ensemble Limit , 2014, SIAM/ASA J. Uncertain. Quantification.

[14]  Kai Lai Chung,et al.  A Course in Probability Theory , 1949 .

[15]  T. Başar,et al.  A New Approach to Linear Filtering and Prediction Problems , 2001 .

[16]  T. J. Dodwell,et al.  A Hierarchical Multilevel Markov Chain Monte Carlo Algorithm with Applications to Uncertainty Quantification in Subsurface Flow , 2013, SIAM/ASA J. Uncertain. Quantification.

[17]  D. Luenberger.  Optimization by Vector Space Methods , 1968 .

[18]  D. Crisan,et al.  Fundamentals of Stochastic Filtering , 2008 .

[19]  Karsten Urban.  Wavelets in Numerical Simulation - Problem Adapted Construction and Applications , 2002, Lecture Notes in Computational Science and Engineering.

[20]  Arnulf Jentzen,et al.  Analysis and Numerical Approximations , 2015 .

[21]  A. Beskos,et al.  On the stability of sequential Monte Carlo methods in high dimensions , 2011, 1103.3965.

[22]  D. S. McCormick,et al.  Accuracy and stability of filters for dissipative PDEs , 2012, 1203.5845.

[23]  J. Mandel,et al.  On the convergence of the ensemble Kalman filter , 2009, Applications of mathematics.

[24]  F. Gland,et al.  Large sample asymptotics for the ensemble Kalman filter , 2009 .

[25]  A. Beskos,et al.  Multilevel sequential Monte Carlo samplers , 2015, 1503.07259.

[26]  Stefan Heinrich,et al.  Multilevel Monte Carlo Methods , 2001, LSSC.

[27]  A. Jazwinski.  Stochastic Processes and Filtering Theory , 1970 .

[28]  P. Bickel,et al.  Sharp failure rates for the bootstrap particle filter in high dimensions , 2008, 0805.3287.

[29]  P. Moral.  Feynman-Kac Formulae: Genealogical and Interacting Particle Systems with Applications , 2004 .

[30]  A. Stuart,et al.  Analysis of the 3DVAR filter for the partially observed Lorenz'63 model , 2012, 1212.4923.

[31]  Simon J. Godsill,et al.  On sequential Monte Carlo sampling methods for Bayesian filtering , 2000, Stat. Comput..

[32]  Jonathan Weare,et al.  Particle filtering with path sampling and an application to a bimodal ocean current model , 2009, J. Comput. Phys..

[33]  Yan Zhou,et al.  Multilevel Particle Filters , 2015, SIAM J. Numer. Anal..

[34]  Xin T. Tong,et al.  Nonlinear stability and ergodicity of ensemble based Kalman filters , 2015, 1507.08307.

[35]  Alexandros Beskos,et al.  Sequential Monte Carlo Methods for High-Dimensional Inverse Problems: A Case Study for the Navier-Stokes Equations , 2013, SIAM/ASA J. Uncertain. Quantification.

[36]  Pierre Del Moral,et al.  On the stability and the uniform propagation of chaos properties of Ensemble Kalman-Bucy filters , 2016, 1605.09329.

[37]  Edriss S. Titi,et al.  Data Assimilation algorithm for 3D B\'enard convection in porous media employing only temperature measurements , 2015, 1506.08678.

[38]  Pierre Del Moral,et al.  On the Stability and the Uniform Propagation of Chaos of a Class of Extended Ensemble Kalman-Bucy Filters , 2016, SIAM J. Control. Optim..

[39]  Ajay Jasra,et al.  Forward and Inverse Uncertainty Quantification using Multilevel Monte Carlo Algorithms for an Elliptic Nonlocal Equation , 2016, 1603.06381.

[40]  Andrew J Majda,et al.  Concrete ensemble Kalman filters with rigorous catastrophic filter divergence , 2015, Proceedings of the National Academy of Sciences.

[41]  A. M. Stuart,et al.  Accuracy and stability of the continuous-time 3DVAR filter for the Navier–Stokes equation , 2012, 1210.1594.

[42]  P. Diaconis,et al.  The sample size required in importance sampling , 2015, 1511.01437.

[43]  P. Leeuwen,et al.  Nonlinear data assimilation in geosciences: an extremely efficient particle filter , 2010 .

[44]  M. Giles,et al.  Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation , 2012, 1202.6283.

[45]  Alexandros Beskos,et al.  Particle Filtering for Stochastic Navier-Stokes Signal Observed with Linear Additive Noise , 2017, SIAM J. Sci. Comput..

[46]  Andrew M. Stuart,et al.  Complexity analysis of accelerated MCMC methods for Bayesian inversion , 2012, 1207.2411.

[47]  Hamidou Tembine,et al.  Deterministic Mean-Field Ensemble Kalman Filtering , 2014, SIAM J. Sci. Comput..

[48]  LEXANDER,et al.  A Deterministic Filter for non-Gaussian Bayesian Estimation , 2011 .

[49]  P. Kloeden,et al.  Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space–time noise , 2009, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[50]  G. Evensen.  Sequential data assimilation with a nonlinear quasi‐geostrophic model using Monte Carlo methods to forecast error statistics , 1994 .

[51]  A. Beskos,et al.  A stable particle filter for a class of high-dimensional state-space models , 2017, Advances in Applied Probability.

[52]  P. Bickel,et al.  Curse-of-dimensionality revisited : Collapse of importance sampling in very large scale systems , 2005 .

[53]  A. GREGORY,et al.  Multilevel Ensemble Transform Particle Filtering , 2015, SIAM J. Sci. Comput..