Proper Generalized Decomposition for Linear and Non-Linear - PowerPoint PPT Presentation

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Proper Generalized Decomposition for Linear and Non-Linear Stochastic Models Olivier Le Maître 1 Lorenzo Tamellini 2 and Anthony Nouy 3 1 LIMSI-CNRS, Orsay, France 2 MOX, Politecnico Milano, Italy 3 GeM, Ecole Centrale Nantes, France Radon Special Semester, WS on Multiscale Simulation & Analysis in Energy and the Environment, Linz - (12-16)/12/2011

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks 1 Motivation Linear elliptic problem Probabilistic framework Stochastic Galerkin problem Stochastic Discretization 2 Proper Generalized Decomposition Optimal Decomposition Algorithms An example Nonlinear problems : N-S 3 Stochastic Navier-Stokes equations Discretization Results

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Parametric model uncertainty : A model M involving uncertain input parameters D Treat uncertainty in a probabilistic framework : D ( θ ) ∈ (Θ , Σ , d µ ) Assume D = D ( ξ ( θ )) , where ξ ∈ R N with known probability law The model solution is stochastic and solves : M ( U ( ξ ); D ( ξ )) = 0 a.s. Uncertainty in the model solution : U ( ξ ) can be high-dimensional U ( ξ ) can be analyzed by sampling techniques, solving multiple deterministic problems ( e.g. MC) We would like to construct a functional approximation of U ( ξ ) � U ( ξ ) ≈ u k Ψ k ( ξ ) k

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Linear elliptic problem Consider the deterministic linear scalar elliptic problem (in Ω ) Find u ∈ V s.t. : a ( u , v ) = b ( v ) , ∀ v ∈ V where � a ( u , v ) ≡ k ( x ) ∇ u ( x ) · ∇ v ( x ) d x (bilinear form) Ω � b ( v ) ≡ f ( x ) v ( x ) d x (+ BC terms ) (linear form) Ω ǫ < k ( x ) and f ( x ) given (problem data) V (= H 1 0 (Ω)) deterministic space (vector space) .

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Probabilistic framework Stochastic elliptic problem : Conductivity k , source field f (and BCs) uncertain Considered as random : Probability space (Θ , Σ , d µ ) : � � h 2 � h ∈ L 2 (Θ , d µ ) = E [ h ] ≡ h ( θ ) d µ ( θ ) , ⇒ E < ∞ . Θ Assume 0 < ǫ 0 ≤ k a.e. in Θ × Ω , k ( x , · ) ∈ L 2 (Θ , d µ ) a.e. in Ω and f ∈ L 2 (Ω , Θ , d µ ) Find U ∈ V ⊗ L 2 (Θ , d µ ) s.t. Variational formulation : ∀ V ∈ V ⊗ L 2 (Θ , d µ ) , A ( U , V ) = B ( V ) where A ( U , V ) . = E [ a ( U , V )] and B ( V ) . = E [ b ( V )] .

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Stochastic Galerkin problem Stochastic expansion : Let { Ψ 0 , Ψ 1 , Ψ 2 , . . . } be an orthonormal basis of L 2 (Θ , d µ ) W ∈ V ⊗ L 2 (Θ , d µ ) has for expansion + ∞ � W ( x , θ ) = w α ( x )Ψ α ( θ ) , w α ( x ) ∈ V α = 0 Truncated expansion : span { Ψ 0 , . . . , Ψ P } = S P ⊂ L 2 (Θ , d µ ) Galerkin problem : Find U ∈ V ⊗ S P s.t. ∀ V ∈ V ⊗ S P , A ( U , V ) = B ( V ) with U = � P α = 0 u α Ψ α and V = � P α = 0 v α Ψ α

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Stochastic Galerkin problem Stochastic expansion : Let { Ψ 0 , Ψ 1 , Ψ 2 , . . . } be an orthonormal basis of L 2 (Θ , d µ ) W ∈ V ⊗ L 2 (Θ , d µ ) has for expansion + ∞ � W ( x , θ ) = w α ( x )Ψ α ( θ ) , w α ( x ) ∈ V α = 0 Truncated expansion : span { Ψ 0 , . . . , Ψ P } = S P ⊂ L 2 (Θ , d µ ) Galerkin problem : Find { u 0 , . . . , u P } s.t. for β = 0 , . . . , P � a α,β ( u α , v β ) = b β ( v β ) , ∀ v β ∈ V α � with a α,β ( u , v ) := Ω E [ k Ψ α Ψ β ] ∇ u · ∇ v d x , � b β ( v ) := Ω E [ f Ψ β ] v ( x ) d x .

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Stochastic Galerkin problem Stochastic expansion : Let { Ψ 0 , Ψ 1 , Ψ 2 , . . . } be an orthonormal basis of L 2 (Θ , d µ ) W ∈ V ⊗ L 2 (Θ , d µ ) has for expansion + ∞ � W ( x , θ ) = w α ( x )Ψ α ( θ ) , w α ( x ) ∈ V α = 0 Truncated expansion : span { Ψ 0 , . . . , Ψ P } = S P ⊂ L 2 (Θ , d µ ) Galerkin problem : Find { u 0 , . . . , u P } s.t. for β = 0 , . . . , P � a α,β ( u α , v β ) = b β ( v β ) , ∀ v β ∈ V α Large system of coupled linear problem, globally SDP.

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Stochastic Discretization Stochastic parametrization Parameterization using N independent R -valued r.v. ξ ( θ ) = ( ξ 1 · · · ξ N ) Let Ξ ⊆ R N be the range of ξ ( θ ) and p ξ its pdf The problem is solved in the image space (Ξ , B (Ξ) , p ξ ) U ( θ ) ≡ U ( ξ ( θ )) Stochastic basis : Ψ α ( ξ ) Spectral polynomials (Hermite, Legendre, Askey scheme, . . . ) [Ghanem and Spanos, 1991], [Xiu and Karniadakis 2001] Piecewise continuous polynomials (Stochastic elements, multiwavelets, . . . ) [Deb et al , 2001], [olm et al , 2004] Truncature w.r.t. polynomial order : advanced selection strategy [Nobile et al , 2010] Size of dim S P - Curse of dimensionality

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Stochastic Discretization U ( x , ξ ) ≈ � P Stochastic Galerkin solution α = 0 u α ( x )Ψ α ( ξ ) Find { u 0 , . . . u P } s.t. � α a α,β ( u α , v β ) = b β ( v β ) , ∀ v β = 0 ,... P ∈ V A priori selection of the subspace S P Is the truncature / selection of the basis well suited ? Size of the Galerkin problem scales with P + 1 : iterative solver Memory requirements may be an issue for large bases Paradigm : Decouple the modes computation (smaller size problems, complexity reduction) Use reduced basis representation : find important components in U (reduce complexity and memory requirements) Proper Generalized Decomposition ∗ ∗ Also GSD : Generalized Spectral Decomposition

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Stochastic Discretization 1 Motivation Linear elliptic problem Probabilistic framework Stochastic Galerkin problem Stochastic Discretization 2 Proper Generalized Decomposition Optimal Decomposition Algorithms An example Nonlinear problems : N-S 3 Stochastic Navier-Stokes equations Discretization Results

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Optimal Decomposition The m -terms PGD approximation of U is [Nouy, 2007, 2008, 2010] m < P � U ( x , θ ) ≈ U m ( x , θ ) = λ α ∈ S P , u α ∈ V . u α ( x ) λ α ( θ ) , α = 1 separated representation Interpretation : U is approximated on the stochastic reduced basis { λ 1 , . . . , λ m } of S P the deterministic reduced basis { u 1 , . . . , u m } of V none of which is selected a priori The questions are then : how to define the (deterministic or stochastic) reduced basis ? how to compute the reduced basis and the m -terms PGD of U ?

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Optimal Decomposition Optimal L 2 -spectral decomposition : POD, KL decomposition � � m � � U m − U � 2 U m ( x , θ ) = u α ( x ) λ α ( θ ) minimizes E L 2 (Ω) α = 1 The modes u α are the m dominant eigenvectors of the kernel E [ U ( x , · ) U ( y , · )] : � E [ U ( x , · ) U ( y , · )] u α ( y ) d y = β u α ( x ) , � u α � L 2 (Ω) = 1 . Ω The modes are orthonormal : � λ α ( θ ) = U ( x , θ ) u α ( x ) d x Ω However U ( x , θ ) , so E [ u ( x , · ) u ( y , · )] is not known !

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Optimal Decomposition Optimal L 2 -spectral decomposition : POD, KL decomposition � � � m � U m − U � 2 U m ( x , θ ) = u α ( x ) λ α ( θ ) minimizes E L 2 (Ω) α = 1 Solve the Galerkin problem in V h ⊗ S P ′ < P to construct { u α } , and � λ α ∈ S P � then solve for the . Solve the Galerkin problem in V H ⊗ S P to construct { λ α } , and � u α ∈ V h � with dim V H ≪ dim V h . then solve for the See works by groups of Ghanem and Matthies.

Motivation Proper Generalized Decomposition Nonlinear problems : N-S Conclusion & closing remarks Optimal Decomposition Alternative definition of optimality A ( · , · ) is symmetric positive definite, so U minimizes the energy functional J ( V ) ≡ 1 2 A ( V , V ) − B ( V ) We define U m through � m � � J ( U m ) = { u α } , { λ α } J min u α λ α . α = 1 Equivalent to minimizing a Rayleigh quotient Optimality w.r.t the A -norm (change of metric) : � V � 2 A = E [ a ( V , V )] = A ( V , V )