Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Oliver Pajonk, Bojana Rosic, Alexander Litvinenko, Hermann G. Matthies ISUME 2011, Prag, A Deterministic Filter for non-Gaussian State Estimation Institute of Scientific Computing Picture: smokeonit (via Flickr.com)

3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation  Motivation / Problem Statement  State inference for dynamic system from measurements  Proposed Solution  Hilbert space of random variables (RVs) + representation of RVs by PCE  a recursive, PCE-based, minimum variance estimator  Examples  Method applied to: a bi-modal truth; the Lorenz-96 model  Conclusions Outline

3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation Motivation  Estimate state of a dynamic system from measurements  Lots of uncertainties and errors  Bayesian approach: Model “state of knowledge” by probabilities  New data should change/improve “state of knowledge”  Methods:  Bayes’ formula (expensive) or simplifications (approximations)  Common: Gaussianity, linearity  Kalman-filter-like methods  KF, EKF, UKF, Gaussian-Mixture, … popular: EnKF  All: Minimum variance estimates in Hilbert space  Question: What if we “go back there”? [Tarantola, 2004] [Evensen, 2009]

3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation  Motivation / Problem Statement  State inference for dynamic system from measurements  Proposed Solution  Hilbert space of random variables (RVs) + representation of RVs by PCE  a recursive, PCE-based, minimum variance estimator  Examples  Method applied to: a bi-modal truth; the Lorenz-96 model  Conclusions Outline

Tool 1: Hilbert Space of Random Variables 3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation *Under usual assumptions of uncorrelated errors! [Luenberger, 1969]

Tool 2: Representation of RVs by Polynomial Chaos Expansion (1/2) 3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation * Of course, there are still more representations – we skip them for brevity. [e.g. Holden, 1996]

Tool 2: Representation of RVs by Polynomial Chaos Expansion (2/2) 3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation [Pajonk et al, 2011] “min-var-update”:

3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation  Motivation / Problem Statement  State inference for dynamic system from measurements  Proposed Solution  Hilbert space of random variables (RVs) + representation of RVs by PCE  a recursive, PCE-based, minimum variance estimator  Examples  Method applied to: a bi-modal truth; the Lorenz-96 model  Conclusions Outline

Example 1: Bi-modal Identification 3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation 12 … 10

Example 2: Lorenz-84 Model 3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation [Lorenz, 1984]

Example 2: Lorenz-84 – Application of PCE-based updating 3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation  PCE  “Proper” uncertainty quantification  Updates  Variance reduction and shift of mean at update points  Skewed structure clearly visible, preserved by updates

Example 2: Lorenz-84 – Comparison with EnKF 3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation

Example 2: Lorenz-84 – Variance Estimates – PCE-based upd. 3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation

Example 2: Lorenz-84 – Variance Estimates – EnKF 3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation

Example 2: Lorenz-84 – Non-Gaussian Identification 3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation (a) PCE-based (b) EnKF

Conclusions & Outlook 3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation  Recursive, deterministic, non-Gaussian minimum variance estimation method  Skewed & bi-modal identification possible  Appealing mathematical properties: Rich mathematical structure of Hilbert spaces available  No closure assumptions besides truncation of PCE  Direct computation of update from PCE  efficient  Fully deterministic: Possible applications with security & real time requirements  Future: Scale it to more complex systems, e.g. geophysical applications  “Curse of dimensionality” (adaptivity, model reduction,…)  Development of algebra (numerical & mathematical)

References & Acknowledgements 3rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation  Pajonk, O.; Rosic, B. V.; Litvinenko, A. & Matthies, H. G., A Deterministic Filter for Non-Gaussian Bayesian Estimation, Physica D: Nonlinear Phenomena, 2011, Submitted for publication  Preprint:  The authors acknowledge the financial support from SPT Group for a research position at the Institute of Scientific Computing at the TU Braunschweig.  Lorenz, E. N., Irregularity: a fundamental property of the atmosphere, Tellus A, Blackwell Publishing Ltd, 1984, 36,  Evensen, G., The ensemble Kalman filter for combined state and parameter estimation, IEEE Control Systems Magazine, 2009, 29,  Tarantola, A., Inverse Problem Theory and Methods for Model Parameter Estimation, SIAM, Philadelphia, 2004  Luenberger, D. G., Optimization by Vector Space Methods, John Wiley & Sons, 1969  Holden, H.; Øksendal, B.; Ubøe, J. & Zhang, T.-S., Stochastic Partial Differential Equations, Birkhäuser Verlag, 1996