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Oliver Pajonk, Bojana Rosic, Alexander Litvinenko, Hermann G. Matthies

ISUME 2011, Prag, 2011-05-03

A Deterministic Filter for non-Gaussian State Estimation

Institute ofScientific Computing

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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 byPolynomial 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 byPolynomial 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

1 2 … 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: 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