On a Data Assimilation Method coupling Kalman Filtering, MCRE - PowerPoint PPT Presentation

On a Data Assimilation Method coupling Kalman Filtering, MCRE Concept and PGD Model Reduction for Real-Time Updating of Structural Mechanics Model 2016 SIAM Conference on Uncertainty Quantification Basile Marchand 1 , Ludovic Chamoin 1 , Christian Rey 2 1 LMT/ENS Cachan/CNRS/Paris-Saclay University, France 2 SAFRAN, Research and Technology Center, France April 5-8, 2016

DDDAS Paradigm DDDAS 1 paradigm : a continuous exchange between Introduction the physical system Basics on Kalman Filtering and Proposed its numerical model Approach Numerical Results Conclusion ξ Numerical identification u model control s S Real system observation ξ c 1. Darema, Dynamica Data Driven Applications Systems : A New Paradigm for Application Simulations and Measurements , 2003 SIAM UQ 2016 - Marchand et al April 5-8, 2016 2 / 30

In this work Objectives : Introduction Basics on Identification process Kalman Filtering Proposed Approach ◮ for time dependent systems/parameters Numerical Results ◮ fast resolution Conclusion ◮ robust even if highly corrupted data SIAM UQ 2016 - Marchand et al April 5-8, 2016 3 / 30

In this work Objectives : Introduction Basics on Identification process Kalman Filtering Proposed Approach ◮ for time dependent systems/parameters Numerical Results ◮ fast resolution Conclusion ◮ robust even if highly corrupted data Tools : Kalman filter for evolution aspect modified Constitutive Relation Error for robustness offline/online process based on Proper Generalized Decomposition SIAM UQ 2016 - Marchand et al April 5-8, 2016 3 / 30

Outline Introduction Basics on Kalman Filtering Basics on Kalman Filtering Proposed Approach Numerical Results Proposed Approach Conclusion Numerical Results Conclusion SIAM UQ 2016 - Marchand et al April 5-8, 2016 4 / 30

Data assimilation Dynamical system : Introduction Basics on Kalman Filtering = M ( k ) u ( k ) + e u � u ( k + 1 ) ( k ) Proposed = H ( k ) u ( k ) + e s Approach s ( k ) ( k ) Numerical Results Conclusion SIAM UQ 2016 - Marchand et al April 5-8, 2016 5 / 30

Data assimilation Dynamical system : Introduction Basics on Kalman Filtering = M ( k ) u ( k ) + e u � u ( k + 1 ) ( k ) Proposed = H ( k ) u ( k ) + e s Approach s ( k ) ( k ) Numerical Results Bayes theorem : Conclusion s ( k ) | u ( k ) � u ( k ) | s ( 0 : k − 1 ) � � � = π π � u ( k ) | s ( k ) � π � s ( k ) | s ( 0 : k − 1 ) � π under the following hypothesis : ◮ State u ( k ) is a Markov process, ◮ Observations s ( k ) are statistically independent of state history SIAM UQ 2016 - Marchand et al April 5-8, 2016 5 / 30

Linear Kalman Filter Principle Kalman filter 2 is a bayesian filter combined with Maximum a Posteriori Introduction method in the case of Gaussian probability density functions. Basics on Kalman Filtering Proposed Approach Numerical Results Conclusion 2. Kalman, A new approach to linear filtering and prediction problems , 1960 SIAM UQ 2016 - Marchand et al April 5-8, 2016 6 / 30

Linear Kalman Filter Principle Kalman filter 2 is a bayesian filter combined with Maximum a Posteriori Introduction method in the case of Gaussian probability density functions. Basics on Kalman Filtering Two main steps : Proposed Approach Numerical Results Conclusion u u ( • + 1 2 ) u ( • ) a s ( • ) t t ( k − 1 ) t ( k ) t ( k + 1 ) t ( k + 2 ) t ( k + 3 ) t ( k + 4 ) 2. Kalman, A new approach to linear filtering and prediction problems , 1960 SIAM UQ 2016 - Marchand et al April 5-8, 2016 6 / 30

Linear Kalman Filter Principle Kalman filter 2 is a bayesian filter combined with Maximum a Posteriori Introduction method in the case of Gaussian probability density functions. Basics on Kalman Filtering Two main steps : Proposed Approach 2 ) of state (a) Prediction step where is realized a priori estimation u ( k + 1 Numerical Results system Conclusion u u ( • + 1 2 ) u ( • ) a s ( • ) t t ( k − 1 ) t ( k ) t ( k + 1 ) t ( k + 2 ) t ( k + 3 ) t ( k + 4 ) 2. Kalman, A new approach to linear filtering and prediction problems , 1960 SIAM UQ 2016 - Marchand et al April 5-8, 2016 6 / 30

Linear Kalman Filter Principle Kalman filter 2 is a bayesian filter combined with Maximum a Posteriori Introduction method in the case of Gaussian probability density functions. Basics on Kalman Filtering Two main steps : Proposed Approach 2 ) of state (a) Prediction step where is realized a priori estimation u ( k + 1 Numerical Results system Conclusion (b) Assimilation step where is realized a posteriori estimation u a using observations data u u ( • + 1 2 ) u ( • ) a s ( • ) t t ( k − 1 ) t ( k ) t ( k + 1 ) t ( k + 2 ) t ( k + 3 ) t ( k + 4 ) 2. Kalman, A new approach to linear filtering and prediction problems , 1960 SIAM UQ 2016 - Marchand et al April 5-8, 2016 6 / 30

Inverse problems formulation Kalman filter is a very well-known method to solve inverse problems 3 Introduction Basics on Kalman Filtering Proposed Approach Numerical Results Conclusion 3. Kaipio and Somersalo, Statistical and Computational Inverse Problems , 2006 SIAM UQ 2016 - Marchand et al April 5-8, 2016 7 / 30

Inverse problems formulation Kalman filter is a very well-known method to solve inverse problems 3 Introduction Introduce model parameters vector ξ ∈ R n p Principle : Basics on Kalman Filtering Proposed no a priori knowledge → stationarity hypothesis : Approach Numerical ∂ ξ Results ξ ( k + 1 ) = ξ ( k ) + e ( k ) ∂ t ≃ 0 ⇒ Conclusion ξ 3. Kaipio and Somersalo, Statistical and Computational Inverse Problems , 2006 SIAM UQ 2016 - Marchand et al April 5-8, 2016 7 / 30

Inverse problems formulation Kalman filter is a very well-known method to solve inverse problems 3 Introduction Introduce model parameters vector ξ ∈ R n p Principle : Basics on Kalman Filtering Proposed no a priori knowledge → stationarity hypothesis : Approach Numerical ∂ ξ Results ξ ( k + 1 ) = ξ ( k ) + e ( k ) ∂ t ≃ 0 ⇒ Conclusion ξ 3. Kaipio and Somersalo, Statistical and Computational Inverse Problems , 2006 SIAM UQ 2016 - Marchand et al April 5-8, 2016 7 / 30

Inverse problems formulation Kalman filter is a very well-known method to solve inverse problems 3 Introduction Introduce model parameters vector ξ ∈ R n p Principle : Basics on Kalman Filtering Proposed no a priori knowledge → stationarity hypothesis : Approach Numerical ∂ ξ Results ξ ( k + 1 ) = ξ ( k ) + e ( k ) ∂ t ≃ 0 ⇒ Conclusion ξ Two formulations Joint Kalman Filter Dual Kalman filter u ( k + 1 ) = ¯ u ( k ) + ¯ ξ ( k + 1 ) = ξ ( k ) + e ( k ) � � e ( k ) M ( k ) ¯ ¯ M ξ s ( k ) = ¯ u ( k ) + e ( k ) s ( k ) = H ( k ) u ( k ) ( ξ ( k ) ) + e ( k ) H ( k ) ¯ s s 3. Kaipio and Somersalo, Statistical and Computational Inverse Problems , 2006 SIAM UQ 2016 - Marchand et al April 5-8, 2016 7 / 30

Inverse problems formulation Kalman filter is a very well-known method to solve inverse problems 3 Introduction Introduce model parameters vector ξ ∈ R n p Principle : Basics on Kalman Filtering Proposed no a priori knowledge → stationarity hypothesis : Approach Numerical ∂ ξ Results ξ ( k + 1 ) = ξ ( k ) + e ( k ) ∂ t ≃ 0 ⇒ Conclusion ξ Two formulations Joint Kalman Filter Dual Kalman filter u ( k + 1 ) = ¯ u ( k ) + ¯ ξ ( k + 1 ) = ξ ( k ) + e ( k ) � � e ( k ) M ( k ) ¯ ¯ M ξ s ( k ) = ¯ u ( k ) + e ( k ) s ( k ) = H ( k ) u ( k ) ( ξ ( k ) ) + e ( k ) H ( k ) ¯ s s � u ( k ) � ξ ( k ) 3. Kaipio and Somersalo, Statistical and Computational Inverse Problems , 2006 SIAM UQ 2016 - Marchand et al April 5-8, 2016 7 / 30

Inverse problems formulation Kalman filter is a very well-known method to solve inverse problems 3 Introduction Introduce model parameters vector ξ ∈ R n p Principle : Basics on Kalman Filtering Proposed no a priori knowledge → stationarity hypothesis : Approach Numerical ∂ ξ Results ξ ( k + 1 ) = ξ ( k ) + e ( k ) ∂ t ≃ 0 ⇒ Conclusion ξ Two formulations Joint Kalman Filter Dual Kalman filter u ( k + 1 ) = ¯ u ( k ) + ¯ ξ ( k + 1 ) = ξ ( k ) + e ( k ) � � e ( k ) M ( k ) ¯ ¯ M ξ s ( k ) = ¯ u ( k ) + e ( k ) s ( k ) = H ( k ) u ( k ) ( ξ ( k ) ) + e ( k ) H ( k ) ¯ s s � u ( k ) � computed with another ξ ( k ) Kalman filter 3. Kaipio and Somersalo, Statistical and Computational Inverse Problems , 2006 SIAM UQ 2016 - Marchand et al April 5-8, 2016 7 / 30

Resolution schemes : UKF vs EKF The problem : Introduction Basics on Gaussian Gaussian Kalman Filtering Nonlinear operator N ( ¯ x , C x ) N ( ¯ y , C y ) A Proposed Approach Numerical Results Conclusion Two main approaches in Kalman filtering context SIAM UQ 2016 - Marchand et al April 5-8, 2016 8 / 30

Resolution schemes : UKF vs EKF The problem : Introduction Basics on Gaussian Gaussian Kalman Filtering Nonlinear operator N ( ¯ x , C x ) N ( ¯ y , C y ) A Proposed Approach Numerical Results Conclusion Two main approaches in Kalman filtering context ◮ First order linearization, Extended Kalman filter 4 4. Sorenson and Stubberud, Non-linear Filtering by Approximation of the a posteriori Density , 1968 SIAM UQ 2016 - Marchand et al April 5-8, 2016 8 / 30