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Institute of Scientific Computing Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen A Deterministic Filter for non-Gaussian State Estimation Oliver Pajonk, Bojana Rosic, Alexander Litvinenko, Hermann G. Matthies ISUME 2011, Prag, 2011 -05 -03 Picture: smokeonit (via Flickr. com)

Outline § 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 3 rd 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” [Tarantola, 2004] - Methods: - Bayes’ formula (expensive) or simplifications (approximations) - Common: Gaussianity, linearity Kalman-filter-like methods [Evensen, 2009] - KF, EKF, UKF, Gaussian-Mixture, … popular: En. KF - All: Minimum variance estimates in Hilbert space Question: What if we “go back there”? 3 rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation

Outline § 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 3 rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation

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

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

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

Outline § 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 3 rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation

Example 1: Bi-modal Identification 1 2 … 10 3 rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation

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

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

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

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

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

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

Conclusions & Outlook § 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) 3 rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation

References & Acknowledgements § 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: http: //www. digibib. tu-bs. de/? docid=00038994 § 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, 98 -110 § Evensen, G. , The ensemble Kalman filter for combined state and parameter estimation, IEEE Control Systems Magazine, 2009, 29, 82 -104 § 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 3 rd May 2011 | Oliver Pajonk, ISUME 2011 | A Filter for non-Gaussian State Estimation