Weighted reduced basis for the approximation of viscous flows with - PowerPoint PPT Presentation

Weighted reduced basis for the approximation of viscous flows with random coefficients Peng Chen 1 Gianluigi Rozza 2 Alfio Quarteroni 1 in collaboration with 1 CMCS - MATHICSE - ´ Ecole Polytechnique F´ ed´ erale de Lausanne, Switzerland 2 SISSA MathLab - International School for Advanced Studies, Trieste, Italy Workshop Numerical Methods for High Dimensional Problems Ecole Nationale des Ponts, ParisTech, Paris, France Acknowledgements: Federico Negri (EPFL) Chen, Rozza (EPFL-SISSA) Stochastic optimal control with Stokes Constraints April 14-18, 2014 1 / 32

Outline Stochastic Stokes problem 1 Constrained optimal control, saddle point formulation 2 Numerical approximation 3 Numerical experiments 4 Conclusions and perspectives 5 Chen, Rozza (EPFL-SISSA) Stochastic optimal control with Stokes Constraints April 14-18, 2014 2 / 32

Stochastic Stokes problem Stochastic Stokes equations with random input data Let (Ω , F , P ) be a complete probability space, where Ω is a set of outcomes ω ∈ Ω , F is a σ -algebra of events and P is a probability measure defined as P : F → [ 0 , 1 ] with P (Ω) = 1 . We consider a stochastic Stokes equations in physical domain D ∈ R d − ν ( ω ) △ u ( · , ω ) + ∇ p ( · , ω ) = f ( · , ω ) in D ,     ∇ · u ( · , ω ) = 0 in D ,  Prob ( ω ) (1) u ( · , ω ) = 0 on ∂ D D ,     ν ( ω ) ∇ u ( · , ω ) · n − p ( · , ω ) n = h ( · , ω ) on ∂ D N , where the uncertainties ω arise from the viscosity ν , force term f and Neumann BC h . Finite dimensional noise assumption The uncertainties depend on N random variables y = ( y 1 , . . . , y N ) : Ω → R N : N � e.g. multicomponent fluid: ν ( y ( ω )) = ν 0 + ( ν n − ν 0 ) y n ( ω ); (2) n = 1 N � � e.g. truncated random fields: f ( x , y ( ω )) = E [ f ]( x ) + λ n f n ( x ) y n ( ω ) . (3) n = 1 Chen, Rozza (EPFL-SISSA) Stochastic optimal control with Stokes Constraints April 14-18, 2014 4 / 32

Stochastic Stokes problem Parametrization of the stochastic Stokes equations ... so that the stochastic problem Prob ( ω ) becomes a parametric problem − ν ( y ) △ u ( · , y ) + ∇ p ( · , y ) = f ( · , y ) in D ,     ∇ · u ( · , y ) = 0 in D ,  Prob ( y ) (4) u ( · , y ) = 0 on ∂ D D ,     ν ( y ) ∇ u ( · , y ) · n − p ( · , y ) n = h ( · , y ) on ∂ D N , Remark: Prob ( y ) stochastic/parametric problem with random/parameter vector n = 1 Γ n ⊂ R N and probability density function ρ := ⊗ N y : Ω → Γ := ⊗ N n = 1 ρ n : Γ → R . Stochastic Hilbert Spaces � � � � L 2 E [ v 2 ] := ( v ( y )) 2 ρ ( y ) dy < ∞ ρ (Γ) := v : Γ → R ; � � Γ G := ( L 2 ρ (Γ) ⊗ L 2 ( D )) d ; H := ( L 2 ρ (Γ) ⊗ L 2 ( ∂ D N )) d ; ρ (Γ) ⊗ H 1 ( D )) d : v = 0 on ∂ D D � � v ∈ ( L 2 V := ; � � � Q := L 2 q ∈ L 2 ( D ) : ρ (Γ) ⊗ Q ( D ); Q ( D ) := qdx = 0 . D Chen, Rozza (EPFL-SISSA) Stochastic optimal control with Stokes Constraints April 14-18, 2014 5 / 32

Stochastic Stokes problem Weak formulation of stochastic Stokes problem The weak formulation of Prob ( y ) reads: find { u , p } ∈ V × Q such that � a ( u , v ) + b ( v , p ) = ( f , v ) + ( h , v ) ∂ D N ∀ v ∈ V , (5) b ( u , q ) = 0 ∀ q ∈ Q , � � a ( w , v ) := ν ∇ w ⊗ ∇ v ρ ( y ) dxdy ∀ w , v ∈ V ; Γ D � � b ( v , q ) := − ∇ · v q ρ ( y ) dxdy ∀ v ∈ V , q ∈ Q ; Γ D � � ( f , v ) := f · v ρ ( y ) dxdy f ∈ G , v ∈ V ; Γ D � � ( h , v ) ∂ D N := h · v ρ ( y ) dxdy h ∈ H , v ∈ V . Γ ∂ D N Remark: d -dimensional deterministic integral and N -dimensional stochastic integral Assumption on the random input data P ( ω : ν min ≤ ν ( y ( ω )) ≤ ν max ) = 1 , 0 < ν min < ν max < ∞ ; || f || G < ∞ and || h || H < ∞ . Chen, Rozza (EPFL-SISSA) Stochastic optimal control with Stokes Constraints April 14-18, 2014 6 / 32

Stochastic Stokes problem Well-posedness of stochastic Stokes problem Under the assumption above, there exists a unique solution to the stochastic Stokes problem (5). Moreover, the following stability estimate holds (Brezzi, 1974) � C P || f || G + α a + γ a � || u || V ≤ 1 C T || h || H , (6) α a β b and �� 1 + γ a � C P || f || G + γ a ( α a + γ a ) � || p || Q ≤ 1 C T || h || H , (7) β b α a α a β b where the positive constants α a , γ a , β b , γ b are defined such that ∀ w , v ∈ V and a ( v , v ) ≥ α a || v || 2 a ( w , v ) ≤ γ a || w || V || v || V ∀ v ∈ V 0 , (8) V being V 0 the kernel of b given by V 0 := { v ∈ V : b ( v , q ) = 0 , ∀ q ∈ Q} , and b ( v , q ) q ∈Q sup inf || v || V || q || Q ≥ β b , and b ( v , q ) ≤ γ b || v || V || q || Q ∀ v ∈ V , ∀ q ∈ Q . (9) v ∈V The constants C P and C T are due to Poincar´ e inequality and trace theorem. Chen, Quarteroni, Rozza. Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations, submitted, 2013. Chen, Rozza (EPFL-SISSA) Stochastic optimal control with Stokes Constraints April 14-18, 2014 7 / 32

Constrained optimal control, saddle point formulation Stochastic optimal control problem with Stokes constraint Cost functional (tracking) A possible distributed cost functional is defined by discrepancy + regularization � 1 � � � ( p − p d ) 2 dx + α � ( u − u d ) 2 dx + 1 f 2 dx J ( u , p , f ) = E . (10) 2 2 2 D D D Remark: may not involve the second term of pressure or more general observation u d . Constrained optimal control problem Find an optimal solution { u ∗ , p ∗ , f ∗ } ∈ V × Q × G such that J ( u ∗ , p ∗ , f ∗ ) = { u , p , f }∈V×Q×G J ( u , p , f ) subject to that { u , p , f } solve Prob ( y ) . min (11) Theorem: existence of the stochastic optimal solution By Lions’ argument (Lions, 1971), we have that there exists a stochastic optimal solution { u ∗ , p ∗ , f ∗ } ∈ V × Q × G of the constrained optimal control problem (11). Chen, Rozza (EPFL-SISSA) Stochastic optimal control with Stokes Constraints April 14-18, 2014 9 / 32

Constrained optimal control, saddle point formulation Lagrangian formulation - the first order optimality system Define a compound bilinear form for the weak formulation of Stokes problem as B ( { u , p , f } , { v , q } ) = a ( u , v ) + b ( v , p ) + b ( u , q ) − ( f , v ) . (12) Associated with this bilinear form, we define the Lagrangian functional as L ( { u , p , f } , { u a , p a } ) = J ( u , p , f ) + B ( { u , p , f } , { u a , p a } ) − ( h , u a ) ∂ D N , (13) where { u a , p a } ∈ V × Q are the adjoint (or dual) variables of the Stokes problem. First order optimality system  ( { u , p } , { v a , q a } ) B ( { v a , q a , 0 } , { u a , p a } ) +   = ( u d , v a ) + ( p d , p a ) ∀{ v a , q a } ∈ V × Q ,  (14) α ( f , g ) − ( u a , g ) = 0 ∀ g ∈ G ,   B ( { u , p , f } , { v , q } ) = ( h , v ) ∂ D N ∀{ v , q } ∈ V × Q ,  ∀ v a ∈ V ,  ( u , v a ) + a ( u a , v a ) + b ( v a , p a ) = ( u d , v a )  ∀ q a ∈ Q ,  ( p , q a ) + b ( u a , q a ) = ( p d , q a )    − ( u a , g ) α ( f , g ) = 0 ∀ g ∈ G , a ( u , v ) + b ( v , p ) − ( f , v ) = ( h , v ) ∂ D N ∀ v ∈ V ,     b ( u , q ) = 0 ∀ q ∈ Q ,  (15) Chen, Rozza (EPFL-SISSA) Stochastic optimal control with Stokes Constraints April 14-18, 2014 10 / 32

Constrained optimal control, saddle point formulation An equivalent stochastic saddle point formulation (Gunzburger, Bochev, 2004) Let A : ( V × Q × G ) × ( V × Q × G ) → R be a compound bilinear form defined as A ( { u , p , f } , { v , q , g } ) = ( u , v ) + ( p , q ) + α ( f , g ) . (16) An equivalent saddle point formulation Find { u , p , f } ∈ V × Q × G and { u a , p a } ∈ V × Q such that  A ( { u , p , f } , { v a , q a , g } ) + B ( { v a , q a , g } , { u a , p a } )  = ( { u d , p d , 0 } , { v a , q a , g } ) ∀{ v a , q a , g } ∈ V × Q × G , (17) B ( { u , p , f } , { v , q } ) = ( h , v ) ∂ D N ∀{ v , q } ∈ V × Q .  Theorem: there exists a unique optimal solution. Moreover, the optimal solution { u , p , f } and the adjoint variables { u a , p a } satisfy the following stability estimates: ||{ u , p , f }|| V×Q×G ≤ α 1 ||{ u d , p d }|| L×Q + β 1 || h || H (18) and ||{ u a , p a }|| V×Q ≤ α 2 ||{ u d , p d }|| L×Q + β 2 || h || H (19) where the constants α 1 , β 1 , α 2 , β 2 depends on the data, see more details in Chen, Quarteroni, Rozza. Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations, submitted, 2013. Chen, Rozza (EPFL-SISSA) Stochastic optimal control with Stokes Constraints April 14-18, 2014 11 / 32