Sampling-free linear Bayesian updating of model state and parameters using a square root approach

We present a sampling-free implementation of a linear Bayesian filter based on a square root formulation. It employs spectral series expansions of the involved random variables, one such example being Wiener's polynomial chaos. The method is compared to several related methods, as well as a full Bayesian update, on a simple scalar example. Additionally it is applied to a combined state and parameter estimation problem for a chaotic system, the well-known Lorenz-63 model. There, we compare it to the ensemble square root filter (EnSRF), which is essentially a probabilistic implementation of the same underlying estimator. The spectral method is found to be more robust than the probabilistic one, especially for variance estimation. This is to be expected due to the sampling-free implementation.

[1]  N. Wiener.  The Homogeneous Chaos , 1938 .

[2]  W. K. Hastings,et al.  Monte Carlo Sampling Methods Using Markov Chains and Their Applications , 1970 .

[3]  Hermann G. Matthies,et al.  Uncertainty Quantification with Stochastic Finite Elements , 2007 .

[4]  Anthony Nouy,et al.  Generalized spectral decomposition for stochastic nonlinear problems , 2009, J. Comput. Phys..

[5]  G. Evensen.  Data Assimilation: The Ensemble Kalman Filter , 2006 .

[6]  W. Hackbusch.  A Sparse Matrix Arithmetic Based on $\Cal H$-Matrices. Part I: Introduction to ${\Cal H}$-Matrices , 1999, Computing.

[7]  Ronald F. Boisvert,et al.  Uncertainty Quantification in Scientific Computing , 2012, IFIP Advances in Information and Communication Technology.

[8]  J. Hansen,et al.  Implications of Stochastic and Deterministic Filters as Ensemble-Based Data Assimilation Methods in Varying Regimes of Error Growth , 2004 .

[9]  Jeffrey L. Anderson,et al.  Comments on “Sigma-Point Kalman Filter Data Assimilation Methods for Strongly Nonlinear Systems” , 2009 .

[10]  Adam Bobrowski,et al.  Functional analysis for probability and stochastic processes , 2013 .

[11]  Paul Malliavin,et al.  Stochastic Analysis , 1997, Nature.

[12]  D. Nualart.  GAUSSIAN HILBERT SPACES (Cambridge Tracts in Mathematics 129) By SVANTE JANSON: 340 pp., £40.00, ISBN 0 521 56128 0 (Cambridge University Press, 1997) , 1998 .

[13]  Hermann G. Matthies,et al.  A deterministic filter for non-Gaussian Bayesian estimation— Applications to dynamical system estimation with noisy measurements , 2012 .

[14]  S. Janson.  Gaussian Hilbert Spaces , 1997 .

[15]  J. Whitaker,et al.  Ensemble Square Root Filters , 2003, Statistical Methods for Climate Scientists.

[16]  B. Øksendal.  Stochastic differential equations : an introduction with applications , 1987 .

[17]  B. Khattatov,et al.  Data assimilation : making sense of observations , 2010 .

[18]  Youmin Tang,et al.  Sigma-Point Kalman Filter Data Assimilation Methods for Strongly Nonlinear Systems , 2009 .

[19]  D. Berry,et al.  Statistics: Theory and Methods , 1990 .

[20]  Rainer Niekamp,et al.  Low Rank Approximation in Spectral Stochastic Finite Element method with Solution Space Adaption , 2010 .

[21]  Roger Ghanem,et al.  Ingredients for a general purpose stochastic finite elements implementation , 1999 .

[22]  P. Oke,et al.  Implications of the Form of the Ensemble Transformation in the Ensemble Square Root Filters , 2008 .

[23]  George Saad,et al.  Stochastic data assimilation with application to multi -phase flow and health monitoring problems , 2007 .

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

[25]  R. Fleming.  The dynamics of uncertainty: application to parameterization constants in climate models , 1993 .

[26]  Y. Bar-Shalom,et al.  The probabilistic data association filter , 2009, IEEE Control Systems.

[27]  T. Hida,et al.  Lectures on White Noise Functionals , 2008 .

[28]  O. L. Maître,et al.  Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics , 2010 .

[29]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

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

[31]  Hermann G. Matthies,et al.  Direct Bayesian update of polynomial chaos representations , 2011 .

[32]  P. Houtekamer,et al.  A Sequential Ensemble Kalman Filter for Atmospheric Data Assimilation , 2001 .

[33]  Adrian Sandu,et al.  A Polynomial Chaos-Based Kalman Filter Approach for Parameter Estimation of Mechanical Systems , 2010 .

[34]  Steven Finette,et al.  Polynomial Chaos Quantification of the Growth of Uncertainty Investigated with a Lorenz Model , 2010 .

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

[36]  A. N. Kolmogorov,et al.  Foundations of the theory of probability , 1960 .

[37]  Peter Whittle,et al.  Probability via expectation , 1992 .

[38]  Dongbin Xiu,et al.  A generalized polynomial chaos based ensemble Kalman filter with high accuracy , 2009, J. Comput. Phys..

[39]  G. Evensen.  Sampling strategies and square root analysis schemes for the EnKF , 2004 .

[40]  J. Butcher.  Numerical methods for ordinary differential equations , 2003 .

[41]  Frank M. Selten,et al.  Finding the effective parameter perturbations in atmospheric models: the LORENZ63 model as case study , 2004 .

[42]  Jeff Gill,et al.  What are Bayesian Methods , 2008 .

[43]  Benjamin L. Pence,et al.  A maximum likelihood approach to recursive polynomial chaos parameter estimation , 2010, Proceedings of the 2010 American Control Conference.

[44]  M. Allen,et al.  Sensitivity analysis of the climate of a chaotic system , 2000 .

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

[46]  G. Evensen.  The ensemble Kalman filter for combined state and parameter estimation , 2009, IEEE Control Systems.

[47]  Jürgen Potthoff,et al.  White Noise: An Infinite Dimensional Calculus , 1993 .

[48]  G. Kivman,et al.  Sequential parameter estimation for stochastic systems , 2003 .

[49]  Dirk P. Kroese,et al.  Kernel density estimation via diffusion , 2010, 1011.2602.

[50]  A. Andrews,et al.  A square root formulation of the Kalman covariance equations. , 1968 .

[51]  Tarek A. El-Moselhy,et al.  Field solver technologies for variation-aware interconnect parasitic extraction , 2010 .

[52]  J. Hartigan.  Linear Bayesian Methods , 1969 .

[53]  Albert Tarantola,et al.  Inverse problem theory - and methods for model parameter estimation , 2004 .

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

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

[56]  Bernt Øksendal,et al.  Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach , 1996 .

[57]  Peter R. Oke,et al.  A deterministic formulation of the ensemble Kalman filter: an alternative to ensemble square root filters , 2008 .

[58]  Nancy Nichols,et al.  Unbiased ensemble square root filters , 2007 .

[59]  D. Xiu.  Numerical Methods for Stochastic Computations: A Spectral Method Approach , 2010 .

[60]  J. Annan,et al.  Efficient parameter estimation for a highly chaotic system , 2004 .

[61]  Hermann G. Matthies,et al.  Parametric and Uncertainty Computations with Tensor Product Representations , 2011, WoCoUQ.

[62]  Lingzao Zeng,et al.  A stochastic collocation based Kalman filter for data assimilation , 2010 .