Adaptive Surrogate Modeling for Response Surface Approximations with - PowerPoint PPT Presentation

Adaptive Surrogate Modeling for Response Surface Approximations with Application to Bayesian Inference Serge Prudhomme a and C. Bryant b a Department of Mathematics, Ecole Polytechnique de Montr´ eal SRI Center for Uncertainty Quantification, KAUST b ICES, The University of Texas at Austin, USA Advances in UQ Methods, Algorithms, and Applications KAUST, Thuwal, Saudi Arabia January 6-9, 2015 S. Prudhomme and C. Bryant Adaptive Surrogate Modeling January 6-9, 2015 1 / 35

Outline Outline • Introduction. • Error estimation for PDEs with uncertain coefficients. • Adaptive scheme. • Numerical examples. • Application to Bayesian Inference. “. . . 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 S. Prudhomme and C. Bryant Adaptive Surrogate Modeling January 6-9, 2015 2 / 35

Introduction Introduction 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 N Surrogate model M h,N M h ≈ M h,N ( λ ) = Q ( u h,N ) M ≈ M N ( λ ) = Q ( u N ) S. Prudhomme and C. Bryant Adaptive Surrogate Modeling January 6-9, 2015 3 / 35

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 . . . S. Prudhomme and C. Bryant Adaptive Surrogate Modeling January 6-9, 2015 4 / 35

Model Problem Model Problem and Discretization Model Problem: A ( λ ; u ) = f ( λ ) , ∀ x ∈ D Assume: λ = Λ( θ ) = � k ∈I N λ k Ψ k ( ξ ( θ )) Non-intrusive approach (“pseudo-spectral projection method”): � � u m k ( x )Ψ k ( ξ ) := u N ( x , ξ ) u ( x , ξ ) ≈ u k ( x )Ψ k ( ξ ) ≈ k ∈I N k ∈I N with � u k ( x ) = � u ( x , · ) , Ψ k � := u ( x , ξ )Ψ k ( ξ ) ρ ( ξ ) d ξ Ω m ( N ) � u ( x , ξ j ) Ψ k ( ξ j ) w j := u m ≈ k ( x ) j =1 Pros: fast convergence, sampling-like u ( x , ξ j ) , choice of ξ j S. Prudhomme and C. Bryant Adaptive Surrogate Modeling January 6-9, 2015 5 / 35

Model Problem Model Problem and Discretization Gaussian quadrature: • select quadrature rule { ξ j , w j } m ( N ) according to ρ ξ j =1 • integrand ( u N Ψ N ) is at least of order 2 N in each dimension m ( N ) ≥ ( N + 1) n Parameterized discrete solution (the surrogate model): Solve for u h ( x , ξ j ) → u h,m ( x ) = � m ( N ) u h ( x , ξ j ) Ψ k ( ξ j ) w j k j =1 � u h,N ( x , ξ ) = u h,m ( x )Ψ k ( ξ ) k k ∈I N Evaluate: � u h,N � Q ξ ( u ) − Q ξ S. Prudhomme and C. Bryant Adaptive Surrogate Modeling January 6-9, 2015 6 / 35

Goal-oriented error estimation Goal-oriented error estimation Weak formulation of A ( λ ( ξ ); u ) = f ( ξ ) Find u h ( ξ ) ∈ V h ⊂ V such that Find u ( ξ ) ∈ V such that ∀ v h ∈ V h B ξ ( u h , v h ) = F ξ ( v h ) B ξ ( u, v ) = F ξ ( v ) ∀ v ∈ V Quantity of interest (QoI): Adjoint problem: � Find p ( · , ξ ) ∈ V such that Q ξ ( u ) = q ( x ) u ( x , ξ ) d x B ξ ( v, p ) = Q ξ ( v ) ∀ v ∈ V D Error representation Q ξ ( u ) − Q ξ ( u h ) = F ξ ( p ) − B ξ ( u h , p ) := R ξ ( u h ; p ) � � Note: Q ξ ( u ) − Q ξ ( u h ) = R ξ ( u h ; p ) + ∆ B ≈ R ξ ( u h ; p ) S. Prudhomme and C. Bryant Adaptive Surrogate Modeling January 6-9, 2015 7 / 35

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: Compute p + ( ξ ) ∈ V + , V h ⊂ V + ⊂ V and η ( ξ ) = R ξ ( u h ; p + ) Other choices • Local interpolation: R ξ ( u h ; p ) ≈ R ξ ( u h ; π + p h − p 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 and C. Bryant Adaptive Surrogate Modeling January 6-9, 2015 8 / 35

Goal-oriented error estimation Case with Uncertain Parameters Since u h,N ( · , ξ ) ∈ V h ⊂ V , the adjoint equation still holds � � � u h,N � u h,N , p B ξ = Q ξ New error representation: � � � u h,N � Q ξ u − Q ξ � � u h,N ; p = R ξ � u h,N ; p + � � u h,N ; p − p + � = R ξ + R ξ � u h,N ; p + ,N � � u h,N ; p + − p + ,N � � u h,N ; p − p + � = R ξ + R ξ + R ξ Total Error Estimate: � � � u h,N � � u h,N ; p + ,N � Q ξ u − Q ξ ≈ R ξ 1 Butler et al., 2012, Almeida and Oden, 2010 S. Prudhomme and C. Bryant Adaptive Surrogate Modeling January 6-9, 2015 9 / 35

Goal-oriented error estimation Proposed error decomposition u h − u h,N � � � � u h,N � � u − u h � � � � Q ξ u − Q ξ = Q ξ + Q ξ + ∆ Q ξ � �� � � �� � error due to error due to approx physical discretization in parameter space Total error: � 2 × poly. eval � � � u h,N � � u h,N ; p + ,N � Q ξ u − Q ξ ≈ R ξ := E ( ξ ) 1 × inner product Physical space discretization error: � 2 × pde solve � � � u h � � u h ; p + � := E D ( ξ ) Q ξ u − Q ξ ≈ R ξ 1 × inner product Parameter space discretization error: � u h − u h,N � � u h,N ; p + ,N � � u h ; p + � := E Ω ( ξ ) Q ξ ≈ R ξ − R ξ S. Prudhomme and C. Bryant Adaptive Surrogate Modeling January 6-9, 2015 10 / 35

Goal-oriented error estimation Summary of the procedure • Solve forward and adjoint problems at quadrature points, � � m ( N ) u h ( x , ξ j ) j =1 � u h ; p + � → R ξ j { p + ( x , ξ j ) } m ( N ) j =1 • Construct fully discrete solutions and PC expansion for E D u h,N ( x , ξ ) = � k ∈I N u h,m ( x )Ψ k ( ξ ) � k E D ( ξ ) = e D k Ψ k ( ξ ) p + ,N ( x , ξ ) = � k ∈I N p + ,m ( x )Ψ k ( ξ ) k ∈I N k • Construct E and E Ω � E Ω ( ξ ) = E ( ξ ) − E D ( ξ ) E ( ξ ) = e k Ψ k ( ξ ) k ∈I M S. Prudhomme and C. Bryant Adaptive Surrogate Modeling January 6-9, 2015 11 / 35

Adaptive Strategy Adaptivity Strategy � � E D � � � E Ω � � > if � 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 and C. Bryant Adaptive Surrogate Modeling January 6-9, 2015 12 / 35

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 and C. Bryant Adaptive Surrogate Modeling January 6-9, 2015 13 / 35

Numerical results Example 1: Effectivity indices �E� L 2 �E Ω � L 2 �E D � L 2 �E� L 2 Ω � Q ( u ) − Q ( u h,p,N ) � L 2 Ω Ω Ω Ω 5.12427e-01 3.28574e-03 4.34727e-01 .851 1.79962e-01 3.48349e-03 1.95149e-01 .782 5.23817e-02 6.59002e-03 4.25596e-02 .921 2.30547e-02 3.77558e-03 2.85842e-02 .949 6.17006e-03 5.77325e-03 8.41438e-03 .998 2.21929e-03 4.48790e-03 7.25161e-03 .987 2.20458e-03 3.98680e-04 2.80610e-03 .984 7.00606e-04 4.31703e-04 9.24221e-04 .990 3.58282e-04 4.13817e-04 8.06397e-04 1.01 3.58118e-04 1.47612e-04 5.17592e-04 1.03 1.38497e-04 1.49756e-04 2.71081e-04 1.11 8.78811e-05 2.61145e-05 1.06502e-04 1.02 5.10997e-05 2.59334e-05 7.73100e-05 1.00 1.34534e-05 2.59640e-05 3.87553e-05 .985 1.33674e-05 1.22096e-05 2.57607e-05 .981 S. Prudhomme and C. Bryant Adaptive Surrogate Modeling January 6-9, 2015 14 / 35

Numerical results Example 2: Response surface with discontinuity Convection-diffusion model in 2D: � sin( 3 π � 2 ξ 1 ) in D = (0 , 1) 2 − 2∆ u + · ∇ u = f ( ξ ) 4 ⌊ ξ 2 − ξ 1 ⌋ u = 0 on ∂D Loading f is chosen so that � 3 π �� � · ( x − x 2 )( y − y 2 ) u ( x, y, ξ ) = 10 sin 2 ξ 1 4 ⌊ ξ 2 − ξ 1 ⌋ where � 0 ξ 1 ≤ ξ 2 ⌊ ξ 2 − ξ 1 ⌋ = − 1 ξ 1 > ξ 2 with ξ 1 , ξ 2 ∼ U (0 , 1) . S. Prudhomme and C. Bryant Adaptive Surrogate Modeling January 6-9, 2015 15 / 35