On a Data Assimilation Method coupling Kalman Filtering, MCRE Concept and PGD Model Reduction for Real-Time Updating of Structural Mechanics Model

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1 On a Data Assimilation Method coupling, 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

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

3 In this work Objectives : Identification process for time dependent systems/parameters fast resolution robust even if highly corrupted data SIAM UQ Marchand et al April 5-8, / 30

4 In this work Objectives : Identification process for time dependent systems/parameters fast resolution 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 Marchand et al April 5-8, / 30

5 Outline SIAM UQ Marchand et al April 5-8, / 30

6 Data assimilation Dynamical system : { u (k+1) s (k) = M (k) u (k) (k) + e u = H (k) u (k) (k) + e s SIAM UQ Marchand et al April 5-8, / 30

7 Data assimilation Dynamical system : { u (k+1) Bayes theorem : s (k) = M (k) u (k) (k) + e u = H (k) u (k) (k) + e s ( π u (k) s (k)) = π ( s (k) u (k)) π ( u (k) s (0:k 1)) π ( s (k) s (0:k 1)) under the following hypothesis : State u (k) is a Markov process, Observations s (k) are statistically independent of state history SIAM UQ Marchand et al April 5-8, / 30

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

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

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

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

12 Inverse problems formulation Kalman filter is a very well-known method to solve inverse problems 3 3. Kaipio and Somersalo, Statistical and Computational Inverse Problems, 2006 SIAM UQ Marchand et al April 5-8, / 30

13 Inverse problems formulation Kalman filter is a very well-known method to solve inverse problems 3 Principle : Introduce model parameters vector ξ R np no a priori knowledge stationarity hypothesis : ξ t 0 ξ(k+1) = ξ (k) + e (k) ξ 3. Kaipio and Somersalo, Statistical and Computational Inverse Problems, 2006 SIAM UQ Marchand et al April 5-8, / 30

14 Inverse problems formulation Kalman filter is a very well-known method to solve inverse problems 3 Principle : Introduce model parameters vector ξ R np no a priori knowledge stationarity hypothesis : ξ t 0 ξ(k+1) = ξ (k) + e (k) ξ 3. Kaipio and Somersalo, Statistical and Computational Inverse Problems, 2006 SIAM UQ Marchand et al April 5-8, / 30

15 Inverse problems formulation Kalman filter is a very well-known method to solve inverse problems 3 Principle : Introduce model parameters vector ξ R np no a priori knowledge stationarity hypothesis : Joint Kalman Filter { ū (k+1) = ξ t 0 ξ(k+1) = ξ (k) + e (k) ξ M (k) ū (k) + ē (k) M s (k) = H (k) ū (k) + e (k) s Two formulations Dual Kalman filter { ξ (k+1) = ξ (k) + e (k) ξ s (k) = H (k) u (k) (ξ (k) ) + e (k) s 3. Kaipio and Somersalo, Statistical and Computational Inverse Problems, 2006 SIAM UQ Marchand et al April 5-8, / 30

16 Inverse problems formulation Kalman filter is a very well-known method to solve inverse problems 3 Principle : Introduce model parameters vector ξ R np no a priori knowledge stationarity hypothesis : [ ] u (k) ξ (k) Joint Kalman Filter { ū (k+1) = ξ t 0 ξ(k+1) = ξ (k) + e (k) ξ M (k) ū (k) + ē (k) M s (k) = H (k) ū (k) + e (k) s Two formulations Dual Kalman filter { ξ (k+1) = ξ (k) + e (k) ξ s (k) = H (k) u (k) (ξ (k) ) + e (k) s 3. Kaipio and Somersalo, Statistical and Computational Inverse Problems, 2006 SIAM UQ Marchand et al April 5-8, / 30

17 Inverse problems formulation Kalman filter is a very well-known method to solve inverse problems 3 Principle : Introduce model parameters vector ξ R np no a priori knowledge stationarity hypothesis : [ ] u (k) ξ (k) Joint Kalman Filter { ū (k+1) = ξ t 0 ξ(k+1) = ξ (k) + e (k) ξ M (k) ū (k) + ē (k) M s (k) = H (k) ū (k) + e (k) s Two formulations 3. Kaipio and Somersalo, Statistical and Computational Inverse Problems, 2006 Dual Kalman filter { ξ (k+1) = ξ (k) + e (k) ξ s (k) = H (k) u (k) (ξ (k) ) + e (k) s computed with another Kalman filter SIAM UQ Marchand et al April 5-8, / 30

18 Resolution schemes : UKF vs EKF The problem : Gaussian N ( x, C x ) Nonlinear operator A Gaussian N (ȳ, C y ) Two main approaches in Kalman filtering context SIAM UQ Marchand et al April 5-8, / 30

19 Resolution schemes : UKF vs EKF The problem : Gaussian N ( x, C x ) Nonlinear operator A Gaussian N (ȳ, C y ) 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 Marchand et al April 5-8, / 30

20 Resolution schemes : UKF vs EKF The problem : Gaussian N ( x, C x ) Nonlinear operator A Gaussian N (ȳ, C y ) Two main approaches in Kalman filtering context First order linearization, Extended Kalman filter 4 Deterministic Monte-Carlo like method, Unscented Transform, Unscented Kalman filter 5 4. Sorenson and Stubberud, Non-linear Filtering by Approximation of the a posteriori Density, Julier and Uhlmann, A new extension of the kalman filter to nonlinear systems, 1997 SIAM UQ Marchand et al April 5-8, / 30

21 Linearization vs Unscented Transform First Order Linearization Prior 11 Linearization A = x A 10 ȳ = A( x) C y = AC x A T Posterior 1 0 SIAM UQ Marchand et al April 5-8, / 30

22 Linearization vs Unscented Transform First Order Linearization Prior 11 Linearization A = x A 10 ȳ = A( x) C y = AC x A T Posterior 1 0 Unscented Transform Prior 11 σ-points propagation 10 {x i } i=1,..,2n+1 {y i } = A ({x i }) Posterior SIAM UQ Marchand et al April 5-8, / 30

23 Linearization vs Unscented Transform First Order Linearization Prior 11 Linearization A = x A 10 ȳ = A( x) C y = AC x A T Posterior 1 0 Unscented Transform Prior 11 σ-points propagation 10 {x i } i=1,..,2n+1 {y i } = A ({x i }) Posterior For the same computational cost SIAM UQ Marchand et al April 5-8, / 30

24 Why another approach? Kalman Filter based methods well-adapted for evolution problems and DDDAS paradigm SIAM UQ Marchand et al April 5-8, / 30

25 Why another approach? Kalman Filter based methods well-adapted for evolution problems and DDDAS paradigm But : methods very costly if degrees of freedom/parameters increase SIAM UQ Marchand et al April 5-8, / 30

26 Why another approach? Kalman Filter based methods well-adapted for evolution problems and DDDAS paradigm But : methods very costly if degrees of freedom/parameters increase Identification quality strongly depends on measurement noise SIAM UQ Marchand et al April 5-8, / 30

27 Outline SIAM UQ Marchand et al April 5-8, / 30

28 Principle of the method Keep the dual formulation { ξ (k+1) = ξ (k) + e (k) ξ s (k) = H (k) u (k) (ξ (k) ) + e (k) s Classically computed using a Kalman Filter SIAM UQ Marchand et al April 5-8, / 30

29 Principle of the method Keep the dual formulation { ξ (k+1) = ξ (k) + e (k) ξ s (k) = H (k) u (k) (ξ (k) ) + e (k) s Classically computed using a Kalman Filter But use another observation operator { ξ (k+1) = ξ (k) + e (k) ξ s (k) = H m (k) (ξ (k) ; s (k 1:k) ) + e (k) s SIAM UQ Marchand et al April 5-8, / 30

operator Defined from the modified Constitutive Relation Error functional { ξ (k+1) = ξ (k) + e

30 Principle of the method Keep the dual formulation { ξ (k+1) = ξ (k) + e (k) ξ s (k) = H (k) u (k) (ξ (k) ) + e (k) s Classically computed using a Kalman Filter But use another observation operator Defined from the modified Constitutive Relation Error functional { ξ (k+1) = ξ (k) + e (k) ξ s (k) = H m (k) (ξ (k) ; s (k 1:k) ) + e (k) s SIAM UQ Marchand et al April 5-8, / 30

31 MCRE framework The idea 6 : Weight the classical Constitutive Relation Error 7 by a measurements error term 6. Ladevèze et al, Updating of finite element models using vibration tests, Ladevèze and Leguillon, Error estimate procedure in the finite element method and application, 1983 SIAM UQ Marchand et al April 5-8, / 30

32 MCRE framework The idea 6 : Principle : Weight the classical Constitutive Relation Error 7 by a measurements error term Primal-dual formulation based on Legendre-Fenchel inequality applied to Helmholtz free energy 6. Ladevèze et al, Updating of finite element models using vibration tests, Ladevèze and Leguillon, Error estimate procedure in the finite element method and application, 1983 SIAM UQ Marchand et al April 5-8, / 30

33 MCRE framework The idea 6 : Principle : Weight the classical Constitutive Relation Error 7 by a measurements error term Primal-dual formulation based on Legendre-Fenchel inequality applied to Helmholtz free energy mcre functional for unsteady thermal problems : E m(u, q; ξ) = 1 2 (q K u) K 1 (q K u) dxdt + δ 2 Πu s 2 dt I t Ω I t U = { u H 1 (Ω) L 2 (I t) \ u = u d on Ω u, u = u 0 at t = t 0 } S(u) = { q [L 2 (Ω) L 2 (I t)] d \ q n = q d on Ω q, tu + q = f } 6. Ladevèze et al, Updating of finite element models using vibration tests, Ladevèze and Leguillon, Error estimate procedure in the finite element method and application, 1983 SIAM UQ Marchand et al April 5-8, / 30

34 mcre inverse problems Solution is defined by : p = argmin ξ P ad min E m (u, q; ξ) (u,q) U ad S ad SIAM UQ Marchand et al April 5-8, / 30

35 mcre inverse problems Solution is defined by : p = argmin ξ P ad min E m (u, q; ξ) (u,q) U ad S ad Admissible fields Constrained minimization SIAM UQ Marchand et al April 5-8, / 30

36 mcre inverse problems Solution is defined by : p = argmin ξ P ad min E m (u, q; ξ) (u,q) U ad S ad Parameters minimization Gradient based methods Admissible fields Constrained minimization SIAM UQ Marchand et al April 5-8, / 30

37 mcre inverse problems Solution is defined by : p = argmin ξ P ad min E m (u, q; ξ) (u,q) U ad S ad Parameters minimization Gradient based methods Fixed point Admissible fields Constrained minimization SIAM UQ Marchand et al April 5-8, / 30

38 mcre inverse problems Solution is defined by : p = argmin ξ P ad min E m (u, q; ξ) (u,q) U ad S ad Parameters minimization Gradient based methods Fixed point Admissible fields Constrained minimization Interest (i) Robustness of the method with highly corrupted data (ii) Strong mechanical content (iii) Model reduction integration SIAM UQ Marchand et al April 5-8, / 30

39 The Modified Kalman Filter { ξ (k+1) = ξ (k) ( + e (k) ξ s (k) = H m (k) ξ (k), s (k 1:k)) (k) + e s SIAM UQ Marchand et al April 5-8, / 30

40 The Modified Kalman Filter { ξ (k+1) = ξ (k) ( + e (k) ξ s (k) = H m (k) ξ (k), s (k 1:k)) (k) + e s Two steps for H m (ξ (k) (k), s (k 1:k)) SIAM UQ Marchand et al April 5-8, / 30

41 The Modified Kalman Filter { ξ (k+1) = ξ (k) ( + e (k) ξ s (k) = H m (k) ξ (k), s (k 1:k)) (k) + e s Two steps for H m (ξ (k) (k), s (k 1:k)) Step 1 : admissible fields computation u (k) = G mcre (ξ (k), s (k 1:k) ) SIAM UQ Marchand et al April 5-8, / 30

42 The Modified Kalman Filter { ξ (k+1) = ξ (k) ( + e (k) ξ s (k) = H m (k) ξ (k), s (k 1:k)) (k) + e s Two steps for H m (ξ (k) (k), s (k 1:k)) Step 1 : admissible fields computation u (k) = G mcre (ξ (k), s (k 1:k) ) Step 2 : projection Typically using boolean matrix H := Π H m (ξ (k), s (k) ) = H G mcre (ξ (k), s (k 1:k) ) SIAM UQ Marchand et al April 5-8, / 30

43 Optimization point of view Dual Kalman filter based identification can be seen as the minimization of n t ( T J(ξ) = s (k) H (k) u (k) (ξ )) (k) (k) 1 ( ) Cs s (k) H (k) u (k) (ξ (k) ) k=0 SIAM UQ Marchand et al April 5-8, / 30

44 Optimization point of view Dual Kalman filter based identification can be seen as the minimization of n t ( T J(ξ) = s (k) H (k) u (k) (ξ )) (k) (k) 1 ( ) Cs s (k) H (k) u (k) (ξ (k) ) k=0 min U Classical s (k) H (k) u (k) Cs (k) 1 SIAM UQ Marchand et al April 5-8, / 30

45 Optimization point of view Dual Kalman filter based identification can be seen as the minimization of n t ( T J(ξ) = s (k) H (k) u (k) (ξ )) (k) (k) 1 ( ) Cs s (k) H (k) u (k) (ξ (k) ) k=0 min U Classical s (k) H (k) u (k) Cs (k) 1 mcre based min q u + δ U S K 1,I (k) t 2 Πu s I (k) t SIAM UQ Marchand et al April 5-8, / 30

46 Optimization point of view Dual Kalman filter based identification can be seen as the minimization of n t ( T J(ξ) = s (k) H (k) u (k) (ξ )) (k) (k) 1 ( ) Cs s (k) H (k) u (k) (ξ (k) ) k=0 min U Classical s (k) H (k) u (k) Cs (k) 1 mcre based min q u + δ U S K 1,I (k) t 2 Πu s I (k) t Observations data strongly imposed Observations data weakly imposed SIAM UQ Marchand et al April 5-8, / 30

47 Technical points State estimation Admissible fields : (u ad, q ad ) = argmin (u,q) U ad S ad E m (u, q; ξ (k) ) SIAM UQ Marchand et al April 5-8, / 30

48 Technical points State estimation Admissible fields : (u ad, q ad ) = argmin (u,q) U ad S ad E m (u, q; ξ (k) ) t (0) τ (0) k Kalman time scale t (k 1) I (k) t t (k) mcre time scale τ (i 1) k τ (i) k τ (ns 1) k t (nt 1) SIAM UQ Marchand et al April 5-8, / 30

49 Technical points State estimation Admissible fields : (u ad, q ad ) = argmin (u,q) U ad S ad E m (u, q; ξ (k) ) t (0) τ (0) k Kalman time scale t (k 1) I (k) t t (k) mcre time scale τ (i 1) k τ (i) k λ lagrange multiplier field and stationarity conditions τ (ns 1) k t (nt 1) SIAM UQ Marchand et al April 5-8, / 30

50 Technical points State estimation Admissible fields : (u ad, q ad ) = argmin (u,q) U ad S ad E m (u, q; ξ (k) ) t (0) τ (0) k Kalman time scale t (k 1) I (k) t t (k) mcre time scale τ (i 1) k τ (i) k λ lagrange multiplier field and stationarity conditions After FE discretization : [ [ ] [ ] [ C 0 K K u + 0 C] u λ δπ T = Π K λ] with u(τ (0) (ns 1) [ ] Fext δπ T s k ) = u (k 1) and λ(τ k ) = 0 t τ (ns 1) k t (nt 1) SIAM UQ Marchand et al April 5-8, / 30

51 Technical points State estimation Admissible fields : (u ad, q ad ) = argmin (u,q) U ad S ad E m (u, q; ξ (k) ) t (0) τ (0) k Kalman time scale t (k 1) I (k) t t (k) mcre time scale τ (i 1) k τ (i) k λ lagrange multiplier field and stationarity conditions After FE discretization : [ [ ] [ ] [ C 0 K K u + 0 C] u λ δπ T = Π K λ] with u(τ (0) (ns 1) [ ] Fext δπ T s k ) = u (k 1) and λ(τ k ) = 0 t τ (ns 1) k t (nt 1) Coupled forward-backward problem in time SIAM UQ Marchand et al April 5-8, / 30

52 PGD based model reduction Find u X = X 1 X D such that B(u, v) = L(v) v X 8. Nouy, A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial differential equations, 2010 SIAM UQ Marchand et al April 5-8, / 30

53 PGD based model reduction Find u X = X 1 X D such that B(u, v) = L(v) v X Principle : Low-rank tensor approximation u u m = m w 1 i w 2 i w D i i=1 ; u m X m X 8. Nouy, A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial differential equations, 2010 SIAM UQ Marchand et al April 5-8, / 30

54 PGD based model reduction Find u X = X 1 X D such that B(u, v) = L(v) v X Principle : Low-rank tensor approximation u u m = m w 1 i w 2 i w D i i=1 Construction : many strategies 8 ; u m X m X ; progressive Galerkin approach 8. Nouy, A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial differential equations, 2010 SIAM UQ Marchand et al April 5-8, / 30

i=1 Construction : many strategies 8 u M 1 known ; u m X m X ;

B D (w D, w ) = L(w ) B D (u M 1, w ) Orthogonalization and update u M

Nouy, A priori model reduction through Proper Generalized Decomposition

55 PGD based model reduction Find u X = X 1 X D such that B(u, v) = L(v) v X Principle : Low-rank tensor approximation u u m = m w 1 i w 2 i w D i i=1 Construction : many strategies 8 u M 1 known ; u m X m X ; progressive Galerkin approach B 1 (w 1, w ) = L(w ) B 1 (u M 1, w ). B D (w D, w ) = L(w ) B D (u M 1, w ) Orthogonalization and update u M = u M 1 + w 1 w D Fixed point Greedy 8. Nouy, A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial differential equations, 2010 SIAM UQ Marchand et al April 5-8, / 30

56 PGD-mCRE Two fields problem : u and λ Two PGD decompositions simultaneously computed SIAM UQ Marchand et al April 5-8, / 30

57 PGD-mCRE Two fields problem : u and λ Two PGD decompositions simultaneously computed Many parameters to consider as extra-coordinates SIAM UQ Marchand et al April 5-8, / 30

58 PGD-mCRE Two fields problem : u and λ Two PGD decompositions simultaneously computed Many parameters to consider as extra-coordinates space, time parameters to identify ξ observations data initial condition SIAM UQ Marchand et al April 5-8, / 30

59 PGD-mCRE Two fields problem : u and λ space, time Two PGD decompositions simultaneously computed Many parameters to consider as extra-coordinates parameters to identify ξ observations data initial condition Projection into a reduced basis n init u (k) 0 = α i ψ i (x) i=0 SIAM UQ Marchand et al April 5-8, / 30

60 PGD-mCRE Two fields problem : u and λ space, time Two PGD decompositions simultaneously computed Many parameters to consider as extra-coordinates parameters to identify ξ observations data initial condition m u PGD = λ PGD = i=1 m i=1 φ u i φ λ i ψ u i ψ λ i Projection into a reduced basis n init u (k) 0 = α i ψ i (x) i=0 n p n obs n obs n init χ u j,i θ u k,i η u m,i ϕ u q,i j=1 k=1 m=1 q=1 n p n obs n obs n init χ λ j,i θ λ k,i η λ m,i ϕ λ q,i j=1 k=1 m=1 q=1 SIAM UQ Marchand et al April 5-8, / 30

Projection into a reduced basis n init u (k) 0 = α i ψ i (x) i=0 n p n obs n obs n init χ u j,i θ u k,i η u m,i ϕ u q,i j=1 k=1 m=1 q=1

61 PGD-mCRE Two fields problem : u and λ space, time Two PGD decompositions simultaneously computed Many parameters to consider as extra-coordinates parameters to identify ξ observations data initial condition m u PGD = λ PGD = i=1 m i=1 φ u i φ λ i ψ u i ψ λ i Projection into a reduced basis n init u (k) 0 = α i ψ i (x) i=0 n p n obs n obs n init χ u j,i θ u k,i η u m,i ϕ u q,i j=1 k=1 m=1 q=1 n p n obs n obs n init χ λ j,i θ λ k,i η λ m,i ϕ λ q,i j=1 k=1 m=1 q=1 n p + 2 n obs + n init 20 SIAM UQ Marchand et al April 5-8, / 30

62 Synthesis 8. Marchand et al, Real-time updating of structural mechanics models using Kalman filtering, modified Constitutive Relation Error and Proper Generalized Decomposition, 2016 SIAM UQ Marchand et al April 5-8, / 30

63 Synthesis offline Reduced basis computation for initial condition projection PGD admissible fields computation 8. Marchand et al, Real-time updating of structural mechanics models using Kalman filtering, modified Constitutive Relation Error and Proper Generalized Decomposition, 2016 SIAM UQ Marchand et al April 5-8, / 30

64 Synthesis offline Reduced basis computation for initial condition projection PGD admissible fields computation online at each time step Project current initial condition in reduced basis Evaluate PGD parametric solution for set of σ-points Project state into observation space Kalman parameters update 8. Marchand et al, Real-time updating of structural mechanics models using Kalman filtering, modified Constitutive Relation Error and Proper Generalized Decomposition, 2016 SIAM UQ Marchand et al April 5-8, / 30

65 Outline SIAM UQ Marchand et al April 5-8, / 30

66 Example 1 Problem setting q d (t) =? ρc, κ sensor location Time stepping for observation : 1000 Time stepping for identification : 100 Noise level : 20% u = ud PGD modes SIAM UQ Marchand et al April 5-8, / 30

67 Exemple 1 : Neumann B.C. identification 10 Joint Unscented Kalman Filter 10 Modified Kalman Filter time step time step Better accuracy exact mean variance Tuning parameters impact ε MKF = ξ true E [ξ MKF ] L 2 (I t ) ξ true L 2 (I t ) εmkf c s c ξ SIAM UQ Marchand et al April 5-8, / 30

68 Example 2 Problem setting κ 1? κ 2? κ 3? κ 4? u = u d sensor location Space modes Time stepping for observation : 1000 Time stepping for identification : 100 Noise level : 10% SIAM UQ Marchand et al April 5-8, / 30

69 Exemple 2 : conductivity identification Joint Unscented Kalman Filter 2 2 Modified Kalman Filter 1.5 κ1 κ ref Better 1.5 κ2 κ ref accuracy κ3 κ ref κ4 κ ref and robustness exact mean variance SIAM UQ Marchand et al April 5-8, / 30

70 Example 3 Problem setting Thermal source : f (x; x c ) = sinc 2 (π x x c (t) ) u = 0 f(x, x c) u = 0 sensor location To include x c as PGD s extra-coordinate f (x; x c ) N F i (x) G i (x c ) i=1 Using SVD SIAM UQ Marchand et al April 5-8, / 30

71 Exemple 3 : source localization Time stepping for observation : 1000 Time stepping for identification : 100 Noise level : 10% Modified Kalman Filter 8 xc identification 2 yc identification exact mean variance not compared to UKF since this problem requires to solve 5000 problems at each time step with the UKF approach SIAM UQ Marchand et al April 5-8, / 30

72 Exemple 3 : source localization Limits of PGD here PGD limits Space modes Source center modes Solution is relatively singular involves Initial condition should be project on many modes ninit 1 but np + 2 nobs + ninit / 20 SIAM UQ Marchand et al April 5-8, / 30

73 Outline SIAM UQ Marchand et al April 5-8, / 30

74 and future works Unscented Kalman Filter Modified Kalman Filter modified CRE Implementation Cost Robustness Implementation Robustness Cost Robustness Cost 2 Minimizations Proper Generalized Decomposition SIAM UQ Marchand et al April 5-8, / 30

75 and future works Unscented Kalman Filter Modified Kalman Filter modified CRE Implementation Cost Robustness Implementation Robustness Cost Robustness Cost 2 Minimizations Proper Generalized Decomposition Number of parameters significantly increases Extension to field identification split state and parameters meshes adaptive strategy 9 SIAM UQ Marchand et al April 5-8, / 30

Nonlinear and/or Non-normal Filtering. Jesús Fernández-Villaverde University of Pennsylvania

Nonlinear and/or Non-normal Filtering Jesús Fernández-Villaverde University of Pennsylvania 1 Motivation Nonlinear and/or non-gaussian filtering, smoothing, and forecasting (NLGF) problems are pervasive