Linear-quadratic optimal control for the Oseen equations with stabilized finite elements M. Braack 1 , B. Tews 1 1 Mathematical Seminar, University of Kiel, Germany Funded by the DFG Priority Program 1253 (Opt. PDE) Workshop on Numerical Analysis of Singularly Perturbed Problems TU Dresden, November 16-18, 2011 1 / 18
Outline Optimal control problem with Oseen system 1 A priori analysis for SUPG+PSPG and LPS 2 Numerical results 3 Summary 4 2 / 18
1. Optimal control problem Variables: v = velocity , p = pressure , u = control Spaces: 0 (Ω) d × L 2 u ∈ Q ⊂ L 2 (Ω) d (subspace) y := ( v , p ) ∈ X := H 1 0 (Ω) , Equation of state: − µ ∆ v + ( b · ∇ ) v + σ v + ∇ p + u = f in Ω ∇ · v = 0 in Ω v = 0 on ∂ Ω , Parameters: µ > 0 , σ ≥ 0 , div b = 0 Target functional: ( α ≥ 0) 1 0 + α 2 � v − v d � 2 2 � u � 2 J ( v , u ) := → min . 0 3 / 18
Variational formulation Bilinear forms: A ( y , ϕ ) := ( ∇ · v , ξ ) + ( σ v , φ ) + (( b · ∇ ) v , φ ) + ( µ ∇ v , ∇ φ ) − ( p , ∇ · φ ) B ( u , φ ) := ( u , φ ) . Oseen state equation: A ( y , ϕ ) + B ( u , φ ) = � f , φ � ∀ ϕ = ( φ , ξ ) ∈ X . Lagrangian functional: L : X × Q × X → R with Lagrange multiplier z = ( z v , z p ) ∈ X , J ( y , u ) − A ( y , z ) − B ( u , z v ) + � f , z v � . L ( y , u , z ) := Necessary and sufficient condition for optimality: ∇L ( y , u , z ) = 0 4 / 18
Karush-Kuhn-Tucker system State equation: ( ∂ z L ( y , u , z )( ψ ) ≡ 0) A ( y , ϕ ) + B ( u , φ ) = � f , φ � ∀ ϕ = ( φ , ξ ) ∈ X . Adjoint equation: ( ∂ y L ( y , u , z )( ψ ) ≡ 0) ( v − v d , ψ v ) ∀ ψ = ( ψ v , ψ p ) ∈ X A ( ψ , z ) = Gradient equation: ( ∂ u L ( y , u , z )( λ ) ≡ 0) B ( λ , z v ) α ( u , λ ) = ∀ λ ∈ Q . 5 / 18
OD or DO ? OD = first optimize, than discretize DO = first discretize, than optimize We know: For non-symmetric stabilization (e.g., SUPG, PSPG): DO � = OD For symmetric stabilization (e.g., LPS, EOS): DO = OD Theoretical result for symmetric stabilization for Q r elements ( y , z ) and Q m elements ( u ) (Br. 2009): � u − u h � 0 � C ( α, σ, µ ) ( ε ( y ) + ε ( z ) + ε ( u )) with “ ” ε ( y ) 2 X L K h 2 r K � v � 2 r +1 , K + h 2 r +1 � p � 2 := r +1 , K K K ∈T h ε ( u ) 2 X h 2 m +2 � u � 2 := K m +1 , K K ∈T h µ + σ h 2 L K := K + h K � b � ∞ , K + h K 6 / 18
2. A priori analysis We want to know: Practical accuracy of symmetric stabilization Theoretical and practical accuracy of SUPG+PSPG for DO and OD Methods to be analyzed: SUPG+PSPG optimize-discretize 1 SUPG+PSPG discretize-optimize 2 LPS discretize-optimize=optimize-discretize 3 Mesh- and method-depending (semi-)norms: | 2 µ | v | 2 1 + σ � v � 2 0 + ρ � p � 2 | | | y | | := sd 0 � δ K � ( b · ∇ ) v + ∇ p � 2 0 , K + γ K �∇ · v � 2 + 0 , K K ∈T h | 2 µ | v | 2 1 + σ � v � 2 0 + S lps | | | y | | := h ( y , y ) lps Interpolation error: If y ∈ H r +1 (Ω) d +1 , | | | y − I h y | | | lps + | | | y − I h y | | | sd � ε ( y ) 7 / 18
SUPG+PSPG optimize-discretize Theorem For enough regularity, it holds 1 + η 2 ε ( u ) + η 3 / 2 α ε ( y ( u h )) + η 1 / 2 � � � � � u − u h � 0 � α ε ( z ( y ( u h ))) α � − 1 � σ + µ η := . c Ω Proof. Using reduced functional j h ( u ) := J ( v h ( u ) , u ): α � u − u h � 2 j ′′ ≤ h ( u − u h , u − u h ) 0 j ′ h ( u )( u − u h ) − j ′ ( u h )( u − u h ) = j ′ h ( u )( u − u h ) − j ′ ( u )( u − u h ) = Using Gradient eq. α � u − u h � 2 � z v h − z v ( u h ) � 0 � I h u − u � 0 + � z v ( u h ) − z v � 0 � I h u − u � 0 � 0 + � z v ( u h ) − z v h � 0 � u h − u � 0 + α � u h − u � 0 � I h u − u � 0 Use now proper bounds for � z v ( u h ) − z v h � 0 . Important: Stable discretization of adjoint eq. 8 / 18
SUPG+PSPG discretize-optimize Theorem For enough regularity, it holds λε ( z ) + λ (1 + η ) ε ( y ) + (1 + λη 3 / 2 ) | � u − u h � 0 � | u − I h u | | � 1 / 2 � � +( λ + α − 1 ) D 2 K ( z ) K ∈T h K � ( b · ∇ ) z v + ∇ z p � 0 , K , and λ := 1 with D K ( z ) := δ 1 / 2 α (1 + η ) 1 / 2 . Reason. Projection error of adjoint ω := I h z − z h : A ( ϕ , ω ) + S sd h , u =0 ( ϕ , ω ) 1 | 2 | | | ω | | ≤ β sup sd | | | ϕ | | | sd ϕ ∈ X h By perturbed Galerkin orthogonality in adjoint eq.: A ( ϕ , ω ) + S sd h , u =0 ( ϕ , ω ) = ( v − v h ( I h u ) , ϕ v ) − A ( ϕ , z − I h z ) + S sd h , u =0 ( ϕ , I h z ) 9 / 18
Momentum residual: − µ ∆ ϕ v + ( b · ∇ ) ϕ v + σ ϕ v + ∇ ϕ p R ( ϕ ) := Stabilization term at zero control: R ( ϕ ) , ( b · ∇ ) I h z v + ∇ I h z p � � S sd ∇· ϕ v , ∇· I h z v � � � � � h , u =0 ( ϕ , I h z ) = δ K K + γ K K K ∈T h �� 1 / 2 � � � D 2 K ( I h z ) + h 2 l +1 | z v | | 2 ≤ | | | ϕ | | | sd | K l +1 , K K ∈T h Spurious term D 2 K ( I h z ) results from inconsistent discrete adjoint eq. K � ( b · ∇ ) z v + ∇ z p � 0 , K D K ( z ) := δ 1 / 2 10 / 18
LPS LPS is symmetric ⇒ DO=OD Theorem For enough regularity, it holds � u − u h � 2 α − 2 η ( ε ( y ) + ε ( z )) + (1 + α − 2 ) ε ( u ) . � 0 11 / 18
Qualitative comparison η ≤ 1 and λ � α − 1 / 2 σ ≥ 1 ⇒ ⇒ all stated a priori estimates become independent of If σ, α ≥ 1: SUPG discretize-optimize: � 1 / 2 � � D 2 � u − u h � 0 � ε ( y ) + ε ( z ) + ε ( u ) + K ( z ) K ∈T h SUPG optimize-discretize: � u − u h � 0 ε ( y ( u h )) + ε ( z ( y ( u h ))) + ε ( u ) � LPS: � u − u h � 0 � ε ( y ) + ε ( z ) + ε ( u ) 12 / 18
3. Numerical results Boundary conditions on unit square: v 1 = 0; n ) − pn 2 = 0 µ ( ∇ v 2 ,� on Γ S ∪ Γ N v 2 = 0; n ) − pn 1 = 0 µ ( ∇ v 1 ,� on Γ E ∪ Γ W . Exact solution y = ( v , p ): v 1 ( x , y ) = g ( y ) , v 2 ( x , y ) = g ( x ) , p = 0 , with viscosity-depending function (exponential layer, µ = 7 . 5 · 10 − 3 ) g ( x ) := x − 1 − e x /µ 1 − e 1 /µ . Adjoint state z = ( z v , z p ) and control u : z p = 0 z v , 1 ( x , y ) = g (1 − y ) , z v , 2 ( x , y ) = g (1 − x ) , u = − z v . SUPG+PSPG parameters: γ 0 = δ 0 = 0 . 2 13 / 18
Errors and convergence orders with SUPG/PSPG Q1: h = � y − y h � � z − z h � � u − u h � j ( u ) − j h ( u h ) 2 − l � · � 0 order | | | · | | | sd order � · � 0 order | | | · | | | sd order � · � 0 order value order SUPG/PSPG Q1 optimize-discretize 3 2.65e-1 9.58e-1 2.68e-1 9.76e-1 1.89e-1 2.91e-2 4 1.50e-1 0.82 7.77e-1 0.30 1.50e-1 0.83 7.77e-1 0.33 1.06e-1 0.83 2.06e-2 0.50 5 6.62e-2 1.18 7.44e-1 0.06 6.62e-2 1.18 7.43e-1 0.06 4.68e-2 1.18 1.03e-2 1.00 6 2.42e-2 1.45 5.30e-1 0.49 2.42e-2 1.45 5.29e-1 0.49 1.71e-2 1.45 4.24e-3 1.28 7 8.15e-3 1.57 2.96e-1 0.84 8.15e-3 1.57 2.96e-1 0.84 5.76e-3 1.57 1.59e-3 1.42 SUPG/PSPG Q1 discretize-optimize 3 2.67e-1 9.55e-1 2.70e-1 9.70e-1 1.52e-1 4.33e-2 4 1.50e-1 0.83 7.77e-1 0.30 1.51e-1 0.84 7.78e-1 0.32 7.85e-2 0.95 2.53e-2 0.78 5 6.61e-2 1.19 7.44e-1 0.06 6.65e-2 1.19 7.44e-1 0.06 2.86e-2 1.46 1.17e-2 1.12 6 2.42e-2 1.45 5.30e-1 0.49 2.43e-2 1.45 5.30e-1 0.49 6.62e-3 2.11 4.58e-3 1.35 7 8.11e-3 1.58 2.96e-1 0.84 8.16e-3 1.57 2.96e-1 0.84 1.52e-3 2.12 1.65e-3 1.47 14 / 18
Errors and convergence orders with SUPG/PSPG Q2: h = � y − y h � � z − z h � � u − u h � j ( u ) − j h ( u h ) 2 − l � · � 0 order | | | · | | | sd order � · � 0 order | | | · | | | sd order � · � 0 order value order SUPG/PSPG Q2 optimize-discretize 3 1.91e-1 7.52e-1 1.89e-1 7.47e-1 1.34e-1 6.59e-2 4 9.86e-2 0.96 9.67e-1 -0.36 9.77e-2 0.95 9.66e-1 -0.37 6.91e-2 0.95 3.22e-2 1.03 5 4.05e-2 1.29 6.27e-1 0.63 4.00e-2 1.29 6.26e-1 0.63 2.83e-2 1.29 1.17e-2 1.26 6 1.06e-2 1.93 2.32e-1 1.44 1.03e-2 1.95 2.31e-1 1.44 7.32e-3 1.95 2.62e-3 2.16 7 1.69e-3 2.65 5.68e-2 2.03 1.55e-3 2.74 5.68e-2 2.03 1.09e-3 2.75 2.54e-4 3.37 SUPG/PSPG Q2 discretize-optimize 3 1.89e-1 7.50e-1 2.20e-1 1.68e+0 9.01e-2 8.05e-2 4 9.74e-2 0.95 9.66e-1 -0.37 1.29e-1 0.77 1.62e+0 0.05 4.49e-2 1.01 3.83e-2 1.07 5 4.01e-2 1.28 6.26e-1 0.63 7.54e-2 0.78 1.52e+0 0.09 2.59e-2 0.80 1.35e-2 1.50 6 1.05e-2 1.93 2.32e-1 1.44 3.76e-2 1.00 1.20e+0 0.34 1.31e-2 0.98 3.00e-3 2.17 7 1.66e-3 2.66 5.58e-2 2.03 1.48e-2 1.35 7.86e-1 0.62 5.27e-3 1.32 3.18e-4 3.24 15 / 18
Recommend
More recommend
