A survey of implicit particle filters for data assimilation

The implicit particle filter is a sequential Monte Carlo method for data assimilation. The idea is to focus the particles onto the high probability regions of the target probability density function (pdf) so that the number of particles required for a good approximation of this pdf remains manageable, even if the dimension of the state space is large. We explain how this idea is implemented, discuss special cases of practical importance, and work out the relations of the implicit particle filter with other data assimilation methods. We illustrate the theory with six examples.

[1]  Xuemin Tu,et al.  AN ITERATIVE IMPLEMENTATION OF THE IMPLICIT NONLINEAR FILTER , 2012 .

[2]  P. Bickel,et al.  Obstacles to High-Dimensional Particle Filtering , 2008 .

[3]  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.

[4]  Hedibert F. Lopes,et al.  Particle filters and Bayesian inference in financial econometrics , 2011 .

[5]  A. Chorin,et al.  Implicit sampling for particle filters , 2009, Proceedings of the National Academy of Sciences.

[6]  W. Reiher.  Hammersley, J. M., D. C. Handscomb: Monte Carlo Methods. Methuen & Co., London, and John Wiley & Sons, New York, 1964. VII + 178 S., Preis: 25 s , 1966 .

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

[8]  Kristine L. Bell,et al.  A Tutorial on Particle Filters for Online Nonlinear/NonGaussian Bayesian Tracking , 2007 .

[9]  R. E. Kalman,et al.  New Results in Linear Filtering and Prediction Theory , 1961 .

[10]  Stephen J. Wright,et al.  Numerical Optimization , 2018, Fundamental Statistical Inference.

[11]  R. E. Kalman,et al.  A New Approach to Linear Filtering and Prediction Problems , 2002 .

[12]  Alexandre Fournier,et al.  A case for variational geomagnetic data assimilation: insights from a one-dimensional, nonlinear, and sparsely observed MHD system , 2007, 0705.1777.

[13]  R. Fletcher.  Practical Methods of Optimization , 1988 .

[14]  Lance M. Leslie,et al.  Tropical Cyclone Prediction Using a Barotropic Model Initialized by a Generalized Inverse Method , 1993 .

[15]  Peter Jan,et al.  Particle Filtering in Geophysical Systems , 2009 .

[16]  Michael Ghil,et al.  Advanced data assimilation in strongly nonlinear dynamical systems , 1994 .

[17]  Toshio Tsuzmn,et al.  On the Formation of Dissipative Structures in Reaction-Diffusion Systems , 1975 .

[18]  J. Klauder,et al.  Numerical Integration of Multiplicative-Noise Stochastic Differential Equations , 1985 .

[19]  Philippe Courtier,et al.  Dual formulation of four‐dimensional variational assimilation , 1997 .

[20]  Jeffrey K. Uhlmann,et al.  New extension of the Kalman filter to nonlinear systems , 1997, Defense, Security, and Sensing.

[21]  G. Lord,et al.  A numerical scheme for stochastic PDEs with Gevrey regularity , 2004 .

[22]  B. Parlett.  The Symmetric Eigenvalue Problem , 1981 .

[23]  T. A. Zang,et al.  Spectral Methods: Fundamentals in Single Domains , 2010 .

[24]  Pierre Del Moral,et al.  Feynman-Kac formulae , 2004 .

[25]  A. Chorin,et al.  Implicit particle filtering for models with partial noise, and an application to geomagnetic data assimilation , 2011, 1109.3664.

[26]  A. Chorin,et al.  Stochastic Tools in Mathematics and Science , 2005 .

[27]  Nando de Freitas,et al.  Sequential Monte Carlo Methods in Practice , 2001, Statistics for Engineering and Information Science.

[28]  Olivier Talagrand,et al.  Assimilation of Observations, an Introduction (gtSpecial IssueltData Assimilation in Meteology and Oceanography: Theory and Practice) , 1997 .

[29]  P. Courtier,et al.  Variational Assimilation of Meteorological Observations With the Adjoint Vorticity Equation. I: Theory , 2007 .

[30]  M. Dowd.  A sequential Monte Carlo approach for marine ecological prediction , 2006 .

[31]  Rémi Bardenet,et al.  Monte Carlo Methods , 2013, Encyclopedia of Social Network Analysis and Mining. 2nd Ed..

[32]  Eric Vanden-Eijnden,et al.  DATA ASSIMILATION IN THE LOW NOISE, ACCURATE OBSERVATION REGIME WITH APPLICATION TO THE KUROSHIO CURRENT , 2012 .

[33]  L. Tierney,et al.  The validity of posterior expansions based on Laplace''s method , 1990 .

[34]  Youssef M. Marzouk,et al.  Bayesian inference with optimal maps , 2011, J. Comput. Phys..

[35]  Robert N. Miller,et al.  Representer‐based variational data assimilation in a nonlinear model of nearshore circulation , 2007 .

[36]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[37]  Robert N. Miller,et al.  Data assimilation into nonlinear stochastic models , 1999 .

[38]  D. Zupanski.  A General Weak Constraint Applicable to Operational 4DVAR Data Assimilation Systems , 1997 .

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

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

[41]  Neil J. Gordon,et al.  A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking , 2002, IEEE Trans. Signal Process..

[42]  P. Moral.  Measure-valued processes and interacting particle systems. Application to nonlinear filtering problems , 1998 .

[43]  Drew D. Creal.  A Survey of Sequential Monte Carlo Methods for Economics and Finance , 2012 .

[44]  M. Pitt,et al.  Filtering via Simulation: Auxiliary Particle Filters , 1999 .

[45]  G. Sivashinsky.  Nonlinear analysis of hydrodynamic instability in laminar flames—I. Derivation of basic equations , 1977 .

[46]  J. Hammersley,et al.  Monte Carlo Methods , 1965 .

[47]  Matthias Morzfeld,et al.  Implicit particle filters for data assimilation , 2010, 1005.4002.

[48]  Yoshiki Kuramoto,et al.  On the Formation of Dissipative Structures in Reaction-Diffusion Systems Reductive Perturbation Approach , 1975 .

[49]  Brad Weir,et al.  Implicit Estimation of Ecological Model Parameters , 2013, Bulletin of mathematical biology.

[50]  A. Chorin,et al.  Implicit Particle Methods and Their Connection with Variational Data Assimilation , 2012, 1205.1830.

[51]  P. Courtier,et al.  Variational Assimilation of Meteorological Observations With the Adjoint Vorticity Equation. Ii: Numerical Results , 2007 .

[52]  P. Courtier,et al.  A strategy for operational implementation of 4D‐Var, using an incremental approach , 1994 .

[53]  Joel Ferziger,et al.  Higher Order Methods for Incompressible Fluid Flow: by Deville, Fischer and Mund, Cambridge University Press, 499 pp. , 2003 .

[54]  Matthias Morzfeld,et al.  A random map implementation of implicit filters , 2011, J. Comput. Phys..

[55]  Milija Zupanski,et al.  Comparison of sequential data assimilation methods for the Kuramoto–Sivashinsky equation , 2009 .

[56]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[57]  Paul Krause,et al.  Dimensional reduction for a Bayesian filter. , 2004, Proceedings of the National Academy of Sciences of the United States of America.

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

[59]  N. Gordon,et al.  Novel approach to nonlinear/non-Gaussian Bayesian state estimation , 1993 .

[60]  D. Nychka.  Data Assimilation” , 2006 .

[61]  Arthur Gelb,et al.  Applied Optimal Estimation , 1974 .

[62]  Y. Trémolet.  Accounting for an imperfect model in 4D‐Var , 2006 .