Context Proper Generalized Decomposition Further improvements (linear models) Application to the NS equation PGD: algorithms and applications to several stochastic PDEs Olivier Le Maître 1 , 2 , 3 with A. Nouy, L. Tamellini, A. Ern, . . . 1- Duke University, Durham, North Carolina 2- KAUST, Saudi-Arabia 3- LIMSI CNRS, Orsay, France Numerical Methods for HighDim Pbs, Ecole des Ponts O. Le Maître PGD for stochastic PDEs
Context Proper Generalized Decomposition Further improvements (linear models) Application to the NS equation Content : 1 Context Parametric Uncertainty Galerkin formulation Proper Generalized Decomposition 2 Definition Algorithms An example Further improvements (linear models) 3 Hierarchical Decomposition (Damped) Wave equation Application to the NS equation 4 PGD for the Stochastic NS eq. Example O. Le Maître PGD for stochastic PDEs
Context Proper Generalized Decomposition Parametric Uncertainty Further improvements (linear models) Galerkin formulation Application to the NS equation 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 O. Le Maître PGD for stochastic PDEs
Context Proper Generalized Decomposition Parametric Uncertainty Further improvements (linear models) Galerkin formulation Application to the NS equation An example 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) . O. Le Maître PGD for stochastic PDEs
Context Proper Generalized Decomposition Parametric Uncertainty Further improvements (linear models) Galerkin formulation Application to the NS equation 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 )] . O. Le Maître PGD for stochastic PDEs
Context Proper Generalized Decomposition Parametric Uncertainty Further improvements (linear models) Galerkin formulation Application to the NS equation 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 Galerkin problem : (truncated) 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 . Large system of coupled linear problem, globally SPD. O. Le Maître PGD for stochastic PDEs
Context Proper Generalized Decomposition Parametric Uncertainty Further improvements (linear models) Galerkin formulation Application to the NS equation 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 O. Le Maître PGD for stochastic PDEs
Context Proper Generalized Decomposition Parametric Uncertainty Further improvements (linear models) Galerkin formulation Application to the NS equation Stochastic Galerkin solution U ( x , ξ ) ≈ � P 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 O. Le Maître PGD for stochastic PDEs
Context Proper Generalized Decomposition Parametric Uncertainty Further improvements (linear models) Galerkin formulation Application to the NS equation Content : 1 Context Parametric Uncertainty Galerkin formulation Proper Generalized Decomposition 2 Definition Algorithms An example Further improvements (linear models) 3 Hierarchical Decomposition (Damped) Wave equation Application to the NS equation 4 PGD for the Stochastic NS eq. Example O. Le Maître PGD for stochastic PDEs
Context Definition Proper Generalized Decomposition Algorithms Further improvements (linear models) An example Application to the NS equation Separated representation The rank- m 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 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 ? O. Le Maître PGD for stochastic PDEs
Context Definition Proper Generalized Decomposition Algorithms Further improvements (linear models) An example Application to the NS equation 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 ! Solve the Galerkin problem in V h ⊗ S P ′ < P to construct { u α } , and then � λ α ∈ S P � solve for the . Solve the Galerkin problem in V H ⊗ S P to construct { λ α } , and then solve with dim V H ≪ dim V h . � u α ∈ V h � for the See works by groups of Ghanem and Matthies. O. Le Maître PGD for stochastic PDEs
Context Definition Proper Generalized Decomposition Algorithms Further improvements (linear models) An example Application to the NS equation 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 ) O. Le Maître PGD for stochastic PDEs
Context Definition Proper Generalized Decomposition Algorithms Further improvements (linear models) An example Application to the NS equation Sequential construction : For i = 1 , 2 , 3 . . . i − 1 � β v + U i − 1 � � = J ( λ i u i ) = v ∈ V ,β ∈ S P J min β v + λ j u j v ∈ V ,β ∈ S P J min j = 1 The optimal couple ( λ i , u i ) solves simultaneously u i = D ( λ i , U i − 1 ) a) deterministic problem � � U i − 1 , λ i v A ( λ i u i , λ i v ) = B ( λ i v ) − A , ∀ v ∈ V λ i = S ( u i , U i − 1 ) b) stochastic problem � � U i − 1 , β u i ∀ β ∈ S P A ( λ i u i , β u i ) = B ( β u i ) − A , O. Le Maître PGD for stochastic PDEs
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