Parameter identification and state estimation for linear parabolic - PowerPoint PPT Presentation

Parameter identification and state estimation for linear parabolic equations Sergiy Zhuk IBM Research - Ireland Joint work with J.Frank (Utrecht University), I.Herlin (INRIA), R.Shorten (IBM) and S.McKenna (IBM) Stochastic Modelling of Multiscale Systems NDNS+ workshop, Eindhoven Multiscale Institute December 5, 2013

Outline Minimax projection method Projection coefficients as a solution of DAE Bounding set for the projection error Ellipsoid containing the projection coefficients State estimation for a linear transport equation Parameter identification for linear Darcy equation 1 / 56 Estimation and identification for parabolic PDEs (Sergiy Zhuk) NDNS+ workshop, EMI

Problem statement Assume a > 0 and I ( · , t ) ∈ H 1 0 (Ω) satisfies for almost all t ∈ (0 , T ) the following equation: ∂ t I + M · ∇ I − a ∆ I = f , I ( x , 0) = f 0 ( x ) , where • x ∈ Ω ⊂ R n , n ≥ 2, Ω is an open bounded convex set; • M ( x , t ) = ( M 1 ( x , t ) . . . M n ( x , t )) ′ with M i ∈ L ∞ (0 , T , H 1 0 (Ω)) for all i = 1 , . . . , n ; • f ∈ L 2 (0 , T , L 2 (Ω)) and f 0 ∈ H 2 (Ω) ∩ H 1 0 (Ω). 2 / 56 Estimation and identification for parabolic PDEs (Sergiy Zhuk) NDNS+ workshop, EMI

Galerkin projection We expand the solution I into the following series: � I ( x , t ) = a i ( t ) ϕ i ( x ) , a i ( t ) := � I ( · , t ) , ϕ i � L 2 (Ω) , (1) i ∈ N where { ϕ k } k ∈ N is the orthonormal set of eigenfunctions of − ∆: ϕ k ∈ C ∞ (Ω) ∩ H 1 − ∆ ϕ k = λ k ϕ k , 0 (Ω) , ϕ k = 0 on ∂ Ω . 3 / 56 Estimation and identification for parabolic PDEs (Sergiy Zhuk) NDNS+ workshop, EMI

Galerkin projection We expand the solution I into the following series: � I ( x , t ) = a i ( t ) ϕ i ( x ) , a i ( t ) := � I ( · , t ) , ϕ i � L 2 (Ω) , (1) i ∈ N where { ϕ k } k ∈ N is the orthonormal set of eigenfunctions of − ∆: ϕ k ∈ C ∞ (Ω) ∩ H 1 − ∆ ϕ k = λ k ϕ k , 0 (Ω) , ϕ k = 0 on ∂ Ω . Define projection operator: P N I ( · , t ) = a ( t ) := ( a 1 ( t ) . . . a N ( t )) ′ , and reconstruction operator: N � P + N a ( t ) = a i ( t ) ϕ i i =1 3 / 56 Estimation and identification for parabolic PDEs (Sergiy Zhuk) NDNS+ workshop, EMI

Galerkin projection We expand the solution I into the following series: � I ( x , t ) = a i ( t ) ϕ i ( x ) , a i ( t ) := � I ( · , t ) , ϕ i � L 2 (Ω) , (1) i ∈ N where { ϕ k } k ∈ N is the orthonormal set of eigenfunctions of − ∆: ϕ k ∈ C ∞ (Ω) ∩ H 1 − ∆ ϕ k = λ k ϕ k , 0 (Ω) , ϕ k = 0 on ∂ Ω . Define projection operator: P N I ( · , t ) = a ( t ) := ( a 1 ( t ) . . . a N ( t )) ′ , and reconstruction operator: N � P + N a ( t ) = a i ( t ) ϕ i i =1 and the vector of the exact projection coefficients: a true := P N I . N 3 / 56 Estimation and identification for parabolic PDEs (Sergiy Zhuk) NDNS+ workshop, EMI

DAE for the projection coefficents Define a differential operator A ϕ = M · ∇ ϕ − a ∆ ϕ and a commutation error: e ( x , t ) := A P + N P N I ( x , t ) − P + N P N AI ( x , t ) . (2) 4 / 56 Estimation and identification for parabolic PDEs (Sergiy Zhuk) NDNS+ workshop, EMI

DAE for the projection coefficents Define a differential operator A ϕ = M · ∇ ϕ − a ∆ ϕ and a commutation error: e ( x , t ) := A P + N P N I ( x , t ) − P + N P N AI ( x , t ) . (2) Since a true ( t ) = P N I ( · , t ) it follows that a true solves: N N ∂ t P + N a = P + N P N ∂ t I = − A P + N a + e + P + N P N f . (3) 4 / 56 Estimation and identification for parabolic PDEs (Sergiy Zhuk) NDNS+ workshop, EMI

DAE for the projection coefficents Define a differential operator A ϕ = M · ∇ ϕ − a ∆ ϕ and a commutation error: e ( x , t ) := A P + N P N I ( x , t ) − P + N P N AI ( x , t ) . (2) Since a true ( t ) = P N I ( · , t ) it follows that a true solves: N N ∂ t P + N a = P + N P N ∂ t I = − A P + N a + e + P + N P N f . (3) As P N P + N = I , we get, multiplying (3) by P N , that a true solves N d a dt = − P N A P + N a + P N e + P N f 4 / 56 Estimation and identification for parabolic PDEs (Sergiy Zhuk) NDNS+ workshop, EMI

DAE for the projection coefficents On the other hand, ∂ t P + N a = P + N P N ∂ t I = − A P + N a + e + P + N P N f (4) has a solution if and only if − A P + N a + e is in the range of P + N . 5 / 56 Estimation and identification for parabolic PDEs (Sergiy Zhuk) NDNS+ workshop, EMI

DAE for the projection coefficents On the other hand, ∂ t P + N a = P + N P N ∂ t I = − A P + N a + e + P + N P N f (4) has a solution if and only if − A P + N a + e is in the range of P + N . This holds true, in turn, if ( I − P + N P N ) A P + N a = ( I − P + N P N ) e . 5 / 56 Estimation and identification for parabolic PDEs (Sergiy Zhuk) NDNS+ workshop, EMI

DAE for the projection coefficents On the other hand, ∂ t P + N a = P + N P N ∂ t I = − A P + N a + e + P + N P N f (4) has a solution if and only if − A P + N a + e is in the range of P + N . This holds true, in turn, if ( I − P + N P N ) A P + N a = ( I − P + N P N ) e . Noting that ( I − P + N P N ) e ( t ) = ( I − P + N P N ) A P + N a true N N ) ′ = P N , we compute: and, recalling that ( P + � ( I− P + N P N ) A P + � 2 � 2 N a true N A N ) a true · a true = � H N a true L 2 (Ω) = ( S N − A ′ R N , N N N N i , j =1 , A N = P N A P + where S N = {� A ϕ i , A ϕ j �} N N and 1 H N := ( S N − A ′ 2 . N A N ) 5 / 56 Estimation and identification for parabolic PDEs (Sergiy Zhuk) NDNS+ workshop, EMI

DAE for the projection coefficents On the other hand, ∂ t P + N a = P + N P N ∂ t I = − A P + N a + e + P + N P N f (4) has a solution if and only if − A P + N a + e is in the range of P + N . This holds true, in turn, if ( I − P + N P N ) A P + N a = ( I − P + N P N ) e . Noting that ( I − P + N P N ) e ( t ) = ( I − P + N P N ) A P + N a true N N ) ′ = P N , we compute: and, recalling that ( P + � ( I− P + N P N ) A P + � 2 � 2 N a true N A N ) a true · a true = � H N a true L 2 (Ω) = ( S N − A ′ R N , N N N N i , j =1 , A N = P N A P + where S N = {� A ϕ i , A ϕ j �} N N and 1 H N := ( S N − A ′ 2 . N A N ) Thus a true solves the algebraic equation: N 0 = H N a + e o for e o = − H N a true . N 5 / 56 Estimation and identification for parabolic PDEs (Sergiy Zhuk) NDNS+ workshop, EMI

DAE for the projection coefficients = P + N P + Finally we find that if ∂ t I + AI = f , I (0) = f 0 then a true N I N solves the following DAE: d a N a + e m + P N f , dt = − P N A P + 0 = H N a + e o , a (0) = P N f 0 , (5) e m = P N e = P N A ( P + e o = − H N P N I N P N I − I ) 1 where S N = {� A ϕ i , A ϕ j �} N 2 . i , j =1 and H N := ( S N − A ′ N A N ) 6 / 56 Estimation and identification for parabolic PDEs (Sergiy Zhuk) NDNS+ workshop, EMI

Outline Minimax projection method Projection coefficients as a solution of DAE Bounding set for the projection error Ellipsoid containing the projection coefficients State estimation for a linear transport equation Parameter identification for linear Darcy equation 7 / 56 Estimation and identification for parabolic PDEs (Sergiy Zhuk) NDNS+ workshop, EMI

A priori estimates For A ϕ = M · ∇ ϕ − a ∆ ϕ we get an estimate: N e m · e m = � P N A ( P + N P N I − I ) � 2 � � ϕ k , M · ∇ ( P + N P N I − I ) � 2 R N = L 2 (Ω) k =1 ≤ � ρ 1 ( · , t ) � L ∞ (Ω) �∇ ( P + N P N I − I ) � 2 L 2 (Ω) ≤ � ρ 1 ( · , t ) � L ∞ (Ω) λ − 1 N +1 � ∆ I ( · , t ) � 2 L 2 (Ω) where ρ 1 ( x , t ) := � M ( x , t ) � 2 R n . 8 / 56 Estimation and identification for parabolic PDEs (Sergiy Zhuk) NDNS+ workshop, EMI

A priori estimates For A ϕ = M · ∇ ϕ − a ∆ ϕ and I N := P + N P N I we get an estimate: e o · e o = � H N a true � 2 R N = � ( I − P + N P N ) A P + N P N I ( · , t ) � L 2 (Ω) N � � ϕ k , A P + N P N I � 2 � � ϕ k , M · ∇ P + N P N I � 2 = L 2 (Ω) = L 2 (Ω) k > N k > N � λ − 2 k �− ∆ ϕ k , M · ∇ P + N P N I � 2 = L 2 (Ω) k > N ≤ 2 λ − 1 N +1 � λ − 1 1 ρ 2 ( · , t ) + ρ 1 ( · , t ) � L ∞ (Ω) � ∆ I ( · , t ) � 2 L 2 (Ω) where ρ 1 ( x , t ) := � M ( x , t ) � 2 R n , ρ 2 ( x , t ) := � J M ( x , t ) � 2 , J M is the Jacobian of M . 9 / 56 Estimation and identification for parabolic PDEs (Sergiy Zhuk) NDNS+ workshop, EMI

Outline Minimax projection method Projection coefficients as a solution of DAE Bounding set for the projection error Ellipsoid containing the projection coefficients State estimation for a linear transport equation Parameter identification for linear Darcy equation 10 / 56 Estimation and identification for parabolic PDEs (Sergiy Zhuk) NDNS+ workshop, EMI

Uncertain DAE for the projection coefficents = P + N P + Finally we find that if ∂ t I + AI = f , I (0) = f 0 then a true N I N solves the following DAE: d a N a + e m + P N f , dt = − P N A P + (6) 0 = H N a + e o , a (0) = P N f 0 , e m = P N e = P N A ( P + e o = − H N P N I N P N I − I ) and � T � T � e m � 2 R N + � e o � 2 � ∆ I ( · , t ) � 2 R N dt ≤ C λ N +1 L 2 (Ω) dt 0 0 � T ≤ C 1 ( �∇ f 0 � 2 � f ( x , t ) � 2 L 2 (Ω) + L 2 (Ω) dt ) 0 ≤ C ∗ where C ∗ = C ∗ ( M , f 0 , f ) is a constant. 11 / 56 Estimation and identification for parabolic PDEs (Sergiy Zhuk) NDNS+ workshop, EMI