Lecture 2 Error Estimation and Control for Problems with Uncertain - PowerPoint PPT Presentation

Lecture 2 Error Estimation and Control for Problems with Uncertain Coefficients Serge Prudhomme D´ epartement de math´ ematiques et de g´ enie industriel Polytechnique Montr´ eal DCSE Fall School 2019 TU Delft, The Netherlands, November 4-8, 2019 S. Prudhomme (Polytechnique Montr´ eal) Problems with Uncertain Coefficients November 4-8, 2019 1 / 53

Outline Outline Error estimation for PDEs with uncertain coefficients. Adaptive scheme. Numerical examples. “. . . It is not possible to decide (a) between h or p refinement and (b) whether one should enrich the approximation space V h or S h . . . better approaches, yet to be conceived, are consequently needed.” Spectral Methods for Uncertainty Quantification, Le Maˆ ıtre & Knio 2010 A few words about validation. Application of GOEE to Bayesian Inference. Numerical examples. S. Prudhomme (Polytechnique Montr´ eal) Problems with Uncertain Coefficients November 4-8, 2019 2 / 53

Introduction Motivation A ( λ ; u ) = f ( λ ) → Q ( u ( λ )) A h ( λ ; u h ) = f h ( λ ) → Q ( u h ( λ )) � �� � � �� � M ( λ )= Q ( u ) M h ( λ )= Q ( u h ) Q h,N ( u ) Q N ( u ) λ 2 λ 2 λ λ 1 1 Surrogate model M h,N Surrogate model M N M h ≈ M h,N ( λ ) = Q ( u h,N ) M ≈ M N ( λ ) = Q ( u N ) S. Prudhomme (Polytechnique Montr´ eal) Problems with Uncertain Coefficients November 4-8, 2019 3 / 53

Introduction References Le Maˆ ıtre et al., 2007, 2010 Polynomial chaos, Stochastic Galerkin, Burger’s equation Almeida and Oden, 2010 convection-diffusion, sparse grid collocation Butler, Dawson, and Wildey, 2011 Stochastic Galerkin, PC representation of the discretization error (ignore truncation error) Butler, Constantine, and Wildey, 2012 Ignore physical discretization error, pseudo-spectral projection, improved linear functional . . . Bryant, Wildey, Prudhomme, SIAMJUQ, 2015 Pseudo-spectral projection method, adaptivity with respect to quantities of interest S. Prudhomme (Polytechnique Montr´ eal) Problems with Uncertain Coefficients November 4-8, 2019 4 / 53

Model Problem Model Problem and Discretization Model Problem: A ( λ ; u ) = f ( λ ) , ∀ x ∈ D where uncertain coefficients λ assumes the following representation: ∀ ξ ∈ Ω ∈ R n λ = λ ( ξ ) , (or Ξ ) Weak formulation for a given ξ : Find u ( ξ ) ∈ V such that B ξ ( u, v ) = F ξ ( v ) ∀ v ∈ V Finite element approximation: Find u h ( ξ ) ∈ V h ⊂ V such that ∀ v h ∈ V h B ξ ( u h , v h ) = F ξ ( v h ) S. Prudhomme (Polytechnique Montr´ eal) Problems with Uncertain Coefficients November 4-8, 2019 5 / 53

Model Problem Surrogate Approximation Assume that the surrogate approximation of u h is given by: N � u h,N ( x, ξ ) = u h i ( x )Ψ i ( ξ ) i =0 where: i ( x ) ∈ V h ⊂ V , ∀ i = 0 , . . . , N . u h Ψ i ( ξ ) is a basis function in a finite subpace of L 2 (Ω) . The space L 2 (Ω) is endowed with the norm: � � � 1 / 2 ( v ( ξ )) 2 ρ ( ξ ) dξ � v � L 2 (Ω) = Ω where ρ ( ξ ) is the probability density function of ξ . S. Prudhomme (Polytechnique Montr´ eal) Problems with Uncertain Coefficients November 4-8, 2019 6 / 53

Goal-oriented error estimation Goal-oriented error estimation (linear case) Quantity of interest (QoI): � Q ξ ( u ) = k ( x ) u ( x , ξ ) d x D Adjoint problem: Find p ( · , ξ ) ∈ V such that B ξ ( v, p ) = Q ξ ( v ) ∀ v ∈ V Error representation: Q ξ ( u ) − Q ξ ( u h ) = F ξ ( p ) − B ξ ( u h , p ) := R ξ ( u h ; p ) S. Prudhomme (Polytechnique Montr´ eal) Problems with Uncertain Coefficients November 4-8, 2019 7 / 53

Goal-oriented error estimation Goal-oriented error estimation Error estimator: Q ξ ( u ) − Q ξ ( u h ) = R ξ ( u h ; p ) ≈ η ( ξ ) Orthogonality property: If p h ∈ V h then R ξ ( u h ; p h ) = 0 Higher-order approximation of adjoint solution: V , V h ⊂ � p ( ξ ) ∈ � Compute ˜ V ⊂ V and η ( ξ ) = R ξ ( u h ; ˜ p ) Other choices πp h − p h ) Local interpolation: R ξ ( u h ; p ) ≈ R ξ ( u h ; ˜ Residual based: R ξ ( u h ; p ) = B ξ ( e u , e p ) ≈ � η u ( ξ ) η p ( ξ ) 1 Becker & Rannacher 2001, Oden & Prudhomme, 2001 S. Prudhomme (Polytechnique Montr´ eal) Problems with Uncertain Coefficients November 4-8, 2019 8 / 53

Goal-oriented error estimation Case with Uncertain Parameters Goal is to estimate: � u h,N �� � � � � � � Q ξ u − Q ξ � L 2 (Ω) Adjoint solution: M � p M ( x, ξ ) = p ≈ ˜ p i ( x )Ψ i ( ξ ) ˜ i =0 where: V , i = 0 , . . . , M , with V h ⊂ � p i ( x ) ∈ � ˜ V ⊂ V . Ψ i ( ξ ) are the basis functions in a finite subpace of L 2 (Ω) . However, we can choose M = N . S. Prudhomme (Polytechnique Montr´ eal) Problems with Uncertain Coefficients November 4-8, 2019 9 / 53

Goal-oriented error estimation Error Estimate � � � � � R ξ ( u h,N , ˜ p N ) η = � L 2 (Ω) where: R ξ ( u h,N , v ) = F ξ ( v ) − B ξ ( u h,N , v ) Such an estimate requires a large number of residual evaluations for the computation of the L 2 norm. Instead we compute: � � M � R ξ ( u h,N , ˜ p N ) ≈ R ξ ( u h,N , ˜ p N ) i Ψ i ( ξ ) := E ( ξ ) i =0 and consider estimate of total approximation error as: η ≈ �E ( ξ ) � L 2 (Ω) S. Prudhomme (Polytechnique Montr´ eal) Problems with Uncertain Coefficients November 4-8, 2019 10 / 53

Goal-oriented error estimation Error decomposition Decomposition: � � � u h,N � � � � u h � � u h � � u h,N � Q ξ u − Q ξ = Q ξ u − Q ξ + Q ξ − Q ξ � �� � � �� � error due to error due to approx physical discretization in parameter space Bound on total error: � u h,N �� � u h �� � � � � � � � � � � � Q ξ u − Q ξ L 2 (Ω) ≤ � Q ξ u − Q ξ � � L 2 (Ω) � u h,N �� � u h � � � � + � Q ξ − Q ξ � L 2 (Ω) S. Prudhomme (Polytechnique Montr´ eal) Problems with Uncertain Coefficients November 4-8, 2019 11 / 53

Goal-oriented error estimation Estimations of error constributions Physical space discretization error: � � � u h � � � u h ; ˜ Q ξ u − Q ξ ≈ R ξ p But: � �� N � � � � u h ; ˜ u h ; ˜ Ψ i ( ξ ) := E D ( ξ ) R ξ p ≈ R ξ p i i =0 with M = N . With the same expansion order for u h,N , ˜ p N , and E D , the coefficients of all three responses can be computed simultaneously. Then, by quadrature with m points: � u h �� � m � 1 / 2 � � � � � � � � � 2 ω k � E D ( ξ k ) � Q ξ u − Q ξ L 2 (Ω) ≈ � k =1 Parameter space discretization error: � � � u h � � u h,N � ≈ E ( ξ ) − E D ( ξ ) := E Ω ( ξ ) ⇒ � E Ω ( ξ ) � Q ξ − Q ξ L 2 (Ω) S. Prudhomme (Polytechnique Montr´ eal) Problems with Uncertain Coefficients November 4-8, 2019 12 / 53

Adaptive Strategy Adaptivity Strategy � � E D � � � E Ω � � � if L 2 (Ω) > L 2 (Ω) Refine physical approximation space V h ( h ← h 2 ) else Refine random approximation space S N ( N ← N + 1 ) end for a given physical mesh, refine approximation in Ω to the level of physical discretization error use error indicator to guide h refinement in parameter space anisotropic p -refinement in higher dimensions S. Prudhomme (Polytechnique Montr´ eal) Problems with Uncertain Coefficients November 4-8, 2019 13 / 53

non-Intrusive approach Pseudo-spectral projection method Model Problem: A ( λ ; u ) = f ( λ ) , ∀ x ∈ D where uncertain coefficients λ assumes the following representation: ∀ ξ ∈ Ω ∈ R n λ = λ ( ξ ) , (or Ξ ) Non-intrusive approach (“pseudo-spectral projection method”): N N � � u m k ( x )Ψ k ( ξ ) := u N ( x , ξ ) u ( x , ξ ) ≈ u k ( x )Ψ k ( ξ ) ≈ k =0 k =0 where � m ( N ) � u ( x , ξ j ) Ψ k ( ξ j ) w j := u m u k ( x ) := u ( x , ξ )Ψ k ( ξ ) ρ ( ξ ) d ξ ≈ k ( x ) Ω j =1 S. Prudhomme (Polytechnique Montr´ eal) Problems with Uncertain Coefficients November 4-8, 2019 14 / 53

non-Intrusive approach Model Problem and Discretization Gaussian quadrature: Select quadrature rule { ξ j , w j } m ( N ) according to probability density ρ . j =1 Parameterized discrete solution (the surrogate/reduced model): Solve for u h ( x , ξ j ) . Then: m ( N ) � u h,m u h ( x , ξ j ) Ψ k ( ξ j ) w j ( x ) = k j =1 and N � u h,m u h,N ( x , ξ ) = ( x )Ψ k ( ξ ) k k =0 S. Prudhomme (Polytechnique Montr´ eal) Problems with Uncertain Coefficients November 4-8, 2019 15 / 53

Numerical results Example 1: Smooth response surface in 2D Convection-diffusion problem in 2D: � π � � 10 sin � 2 ξ 1 in D = (0 , 1) 2 −∇ · (2 ∇ u ) + · ∇ u = f ( ξ ) 10 cos ( πξ 2 ) u = 0 on ∂D Loading f is chosen such that, with ξ 1 , ξ 2 ∼ U (0 , 1) : � ξ 1 ( x − ξ 1 ) 2 �� ξ 2 ( y − ξ 2 ) 2 � ξ 1 ( x − x 2 ) e − 20 ξ 2 ( y − y 2 ) e − 20 u ( x, y, ξ ) = 400 Quantity of interest: � 1 � 1 � Q ( u ( · , ξ )) = 1 u ( x, y, ξ ) dxdy ≈ q ( x, y ) u ( x, y, ξ ) dxdy 4 0 . 5 0 . 5 D S. Prudhomme (Polytechnique Montr´ eal) Problems with Uncertain Coefficients November 4-8, 2019 16 / 53