Hessian-based sampling in high dimensions for goal-oriented model - PowerPoint PPT Presentation

Hessian-based sampling in high dimensions for goal-oriented model order reduction Peng Chen Omar Ghattas Center for Computational Geosciences and Optimization The Institute for Computational Engineering and Sciences The University of Texas at Austin QUIET 2017 - Quantification of Uncertainty: Improving Efficiency and Technology SISSA, International School for Advanced Studies, Trieste, 18-21 July 2017 P . Chen (ICES – UT Austin) Hessian-based sampling for model order reduction 18 July, 2017 1 / 32

Parametrization Dimension Information # pixels K = 1 2 # modes K = 1 P . Chen (ICES – UT Austin) Hessian-based sampling for model order reduction 18 July, 2017 2 / 32

Parametrization Dimension Information # pixels K = 2 2 # modes K = 2 P . Chen (ICES – UT Austin) Hessian-based sampling for model order reduction 18 July, 2017 2 / 32

Parametrization Dimension Information # pixels K = 4 2 # modes K = 4 P . Chen (ICES – UT Austin) Hessian-based sampling for model order reduction 18 July, 2017 2 / 32

Parametrization Dimension Information # pixels K = 8 2 # modes K = 8 P . Chen (ICES – UT Austin) Hessian-based sampling for model order reduction 18 July, 2017 2 / 32

Parametrization Dimension Information # pixels K = 16 2 # modes K = 16 P . Chen (ICES – UT Austin) Hessian-based sampling for model order reduction 18 July, 2017 2 / 32

Parametrization Dimension Information # pixels K = 32 2 # modes K = 32 P . Chen (ICES – UT Austin) Hessian-based sampling for model order reduction 18 July, 2017 2 / 32

Parametrization Dimension Information # pixels K = 64 2 # modes K = 64 P . Chen (ICES – UT Austin) Hessian-based sampling for model order reduction 18 July, 2017 2 / 32

Parametrization Dimension Information # pixels K = 128 2 # modes K = 128 P . Chen (ICES – UT Austin) Hessian-based sampling for model order reduction 18 July, 2017 2 / 32

Parametrization Dimension Information # pixels K = 256 2 # modes K = 256 P . Chen (ICES – UT Austin) Hessian-based sampling for model order reduction 18 July, 2017 2 / 32

Parametrization Dimension Information # pixels K = 512 2 # modes K = 512 P . Chen (ICES – UT Austin) Hessian-based sampling for model order reduction 18 July, 2017 2 / 32

Parametrization Dimension Information # pixels K = 1024 2 # modes K = 1024 P . Chen (ICES – UT Austin) Hessian-based sampling for model order reduction 18 July, 2017 2 / 32

Outline Model order reduction for parametric PDEs 1 Hessian-based sampling 2 Numerical experiments 3 P . Chen (ICES – UT Austin) Hessian-based sampling for model order reduction 18 July, 2017 3 / 32

Parameters Let P ⊂ R K denote a K -dimensional parameter space, where K ∈ N ∪ ∞ . p = ( p 1 , . . . , p K ) ∈ P . √ √ 3 ] K , with uniform distribution The parameter p lives in a box, w.l.o.g., P = [ − 3 , √ √ 3 ] K ) , p ∼ µ = U ([ − 3 , with mean ¯ p = 0 , and covariance C = I . The parameter p lives in the whole space, i.e., P = R K , with Gaussian distribution p ∼ µ = N (¯ p , C ) , with mean ¯ p , and covariance C , s.p.d. Eg., C is discretized from a covariance operator C , given by C = ( − δ △ + γ I ) − α , which is self adjoint, positive, and of trace class. P . Chen (ICES – UT Austin) Hessian-based sampling for model order reduction 18 July, 2017 4 / 32

Parameters Let P ⊂ R K denote a K -dimensional parameter space, where K ∈ N ∪ ∞ . p = ( p 1 , . . . , p K ) ∈ P . √ √ 3 ] K , with uniform distribution The parameter p lives in a box, w.l.o.g., P = [ − 3 , √ √ 3 ] K ) , p ∼ µ = U ([ − 3 , with mean ¯ p = 0 , and covariance C = I . The parameter p lives in the whole space, i.e., P = R K , with Gaussian distribution p ∼ µ = N (¯ p , C ) , with mean ¯ p , and covariance C , s.p.d. Eg., C is discretized from a covariance operator C , given by C = ( − δ △ + γ I ) − α , which is self adjoint, positive, and of trace class. P . Chen (ICES – UT Austin) Hessian-based sampling for model order reduction 18 July, 2017 4 / 32

Parameters Let P ⊂ R K denote a K -dimensional parameter space, where K ∈ N ∪ ∞ . p = ( p 1 , . . . , p K ) ∈ P . √ √ 3 ] K , with uniform distribution The parameter p lives in a box, w.l.o.g., P = [ − 3 , √ √ 3 ] K ) , p ∼ µ = U ([ − 3 , with mean ¯ p = 0 , and covariance C = I . The parameter p lives in the whole space, i.e., P = R K , with Gaussian distribution p ∼ µ = N (¯ p , C ) , with mean ¯ p , and covariance C , s.p.d. Eg., C is discretized from a covariance operator C , given by C = ( − δ △ + γ I ) − α , which is self adjoint, positive, and of trace class. P . Chen (ICES – UT Austin) Hessian-based sampling for model order reduction 18 July, 2017 4 / 32

Parameters Let P ⊂ R K denote a K -dimensional parameter space, where K ∈ N ∪ ∞ . p = ( p 1 , . . . , p K ) ∈ P . √ √ 3 ] K , with uniform distribution The parameter p lives in a box, w.l.o.g., P = [ − 3 , √ √ 3 ] K ) , p ∼ µ = U ([ − 3 , with mean ¯ p = 0 , and covariance C = I . The parameter p lives in the whole space, i.e., P = R K , with Gaussian distribution p ∼ µ = N (¯ p , C ) , with mean ¯ p , and covariance C , s.p.d. Eg., C is discretized from a covariance operator C , given by C = ( − δ △ + γ I ) − α , which is self adjoint, positive, and of trace class. P . Chen (ICES – UT Austin) Hessian-based sampling for model order reduction 18 July, 2017 4 / 32

Parametric PDEs Let V denote a Hilbert space with dual V ′ . Given p ∈ P , µ -a.e., find u ∈ V such that a ( u , v ; p ) = f ( v ) ∀ v ∈ V . a ( · , · ; p ) : V × V → R is a bilinear form, e.g., � a ( u , v ; p ) = κ ( p ) ∇ u · ∇ vdx . D f ( · ) ∈ V ′ is a linear functional. s ( p ) = s ( u ( p )) ∈ R is a QoI. Ex 1. heat conduction in thermal blocks K � k − β χ D k ( x ) p k κ ( p ) = k = 1 √ √ 3 ] K ) p ∼ U ([ − 3 , K = 16 2 = 256 P . Chen (ICES – UT Austin) Hessian-based sampling for model order reduction 18 July, 2017 5 / 32

Parametric PDEs Let V denote a Hilbert space with dual V ′ . Given p ∈ P , µ -a.e., find u ∈ V such that a ( u , v ; p ) = f ( v ) ∀ v ∈ V . a ( · , · ; p ) : V × V → R is a bilinear form, e.g., � a ( u , v ; p ) = κ ( p ) ∇ u · ∇ vdx . D f ( · ) ∈ V ′ is a linear functional. s ( p ) = s ( u ( p )) ∈ R is a QoI. Ex 2. subsurface flow in a porous medium κ ( p ) = e p log-normal diffusion with p ∈ N (¯ p , C ) K = 129 2 = 16 , 641 P . Chen (ICES – UT Austin) Hessian-based sampling for model order reduction 18 July, 2017 5 / 32

Model order reduction – formulation (Maday, Patera, Rozza, et. al.) Finite element approximation Reduced basis approximation V = [ ψ 1 , . . . , ψ N ] Finite element space V h , Reduced basis space V N ⊂ V h , dim ( V h ) = N h dim ( V N ) = N V T u h = u N Given p ∈ P , find u h ∈ V h s.t. Given p ∈ P , find u N ∈ V N s.t. a ( u h , v h ; p ) = f ( v h ) ∀ v h ∈ V h a ( u N , v N ; p ) = f ( v N ) ∀ v N ∈ V N V T A h ( p ) V = A N ( p ) The algebraic system is The algebraic system is V T f h = f N A h ( p ) u h = f h A N ( p ) u N = f N P . Chen (ICES – UT Austin) Hessian-based sampling for model order reduction 18 July, 2017 6 / 32

Model order reduction – algorithms (Maday, Patera, Rozza, et. al.) POD/SVD Greedy algorithm Offline-Online Samples Affine assumption/approx. Samples a = � Q Ξ t = { p n , n = 1 , . . . , N t } Ξ t = { p n , n = 1 , . . . , N t } q = 1 θ q ( p ) a q Compute snapshots Initialize V N for N = 1 as Offline computation once U = [ u h ( p 1 ) , . . . , u h ( p N t )] A q N = V T A q V N = span { u h ( p 1 ) } h V , f N = V T f h Perform SVD Online assemble Pick next sample such that p N + 1 = argmax p ∈ Ξ t ∆ N ( p ) U = V Σ W T A N ( p ) = � Q q = 1 θ q ( p ) A q N Extract bases V [ 1 : N , :] Update bases V N + 1 as Online solve and evaluate V N ⊕ span { u h ( p N + 1 ) } A N ( p ) u N = f N , s ( p ) = s T N = argmin n E n (Σ) ≥ 1 − ε N u N Goal-oriented a-posteriori error estimate ∆ N ( p ) – dual weighted residual ∆ N ( p ) = f ( ϕ N ) − a ( u N , ϕ N ; p ) , where dual Prob.: a ( w N , ϕ N ; p ) = s ( w N ) ∀ w N ∈ W N . Q ∆ N ( p ) = ¯ f T � θ q ( p ) ϕ T N ¯ A q N u N , where ¯ f N = W T f h , and ¯ A q N = W T A q N ϕ N − h V q = 1 P . Chen (ICES – UT Austin) Hessian-based sampling for model order reduction 18 July, 2017 7 / 32

Model order reduction – samples random quasi-random centroidal Voronoi tessellation (Du, Gunzburger, et. al. ) P . Chen (ICES – UT Austin) Hessian-based sampling for model order reduction 18 July, 2017 8 / 32

Model order reduction – samples tensor grid sparse grid anisotropic sparse grid (Liao, Elman, et. al. ) (C., Schwab, et. al. ) P . Chen (ICES – UT Austin) Hessian-based sampling for model order reduction 18 July, 2017 8 / 32

Model order reduction – samples hp-adaptive-rb adaptive-add-remove hybrid goal-oriented adaptive (Eftang, Patera, et. al. ) (Hesthaven, Stamm, et. al.) (C., Quarteroni, et. al. ) P . Chen (ICES – UT Austin) Hessian-based sampling for model order reduction 18 July, 2017 8 / 32

Outline Model order reduction for parametric PDEs 1 Hessian-based sampling 2 Numerical experiments 3 P . Chen (ICES – UT Austin) Hessian-based sampling for model order reduction 18 July, 2017 9 / 32