Reduced Basis Collocation Methods for Partial Differential Equations - PowerPoint PPT Presentation

Preliminary: Spectral Methods for PDEs with Uncertain Coefficients Reduced Basis Methods Reduced Basis + Sparse Grid Collocation Iterative Solution of Reduced Problem Concluding Remarks Reduced Basis Collocation Methods for Partial Differential Equations with Random Coefficients Howard C. Elman Department of Computer Science University of Maryland at College Park Collaborators: Qifeng Liao , Shanghai Tech University Virginia Forstall , University of Maryland 0 / 42 H. C. Elman Reduced Basis Collocation for PDEs

Preliminary: Spectral Methods for PDEs with Uncertain Coefficients Reduced Basis Methods Reduced Basis + Sparse Grid Collocation Iterative Solution of Reduced Problem Concluding Remarks Preliminary: Spectral Methods for PDEs with Uncertain Coefficients 1 Problem Definition Solution Methods Reduced Basis Methods 2 Offline Computations Reduced Problem Reduced Problem: Costs Reduced Problem: Capturing Features of Model 3 Reduced Basis + Sparse Grid Collocation Introduction Performance for Diffusion Equation Application to the Navier-Stokes Equations 4 Iterative Solution of Reduced Problem Introduction Implementation Performance Concluding Remarks 5 1 / 42 H. C. Elman Reduced Basis Collocation for PDEs

Preliminary: Spectral Methods for PDEs with Uncertain Coefficients Reduced Basis Methods Problem Definition Reduced Basis + Sparse Grid Collocation Solution Methods Iterative Solution of Reduced Problem Concluding Remarks Partial Differential Equations with Uncertain Coefficients Examples: Diffusion equation: −∇ · ( a ( x , ξ ) ∇ u ) = f u + ∇ p = � Navier-Stokes equations: −∇ · ( a ( x , ξ ) ∇ � u ) + ( � u · ∇ ) � f ∇ · � u = 0 Posed on D ⊂ R d with suitable boundary conditions Sources: models of diffusion in media with uncertain permeabilities multiphase flows Uncertainty / randomness: a = a ( x , ξ ) is a random field : for each fixed x ∈ D , a ( x , ξ ) is a random variable depending on m random parameters ξ 1 , . . . , ξ m In this study: a ( x , ξ ) = a 0 ( x ) + � m r =1 a r ( x ) ξ r Possible sources: Karhunen-Lo` eve or Piecewise constant expansion coefficients on D 2 / 42 H. C. Elman Reduced Basis Collocation for PDEs

Preliminary: Spectral Methods for PDEs with Uncertain Coefficients Reduced Basis Methods Problem Definition Reduced Basis + Sparse Grid Collocation Solution Methods Iterative Solution of Reduced Problem Concluding Remarks The Stochastic Galerkin Method Standard weak diffusion problem: find u ∈ H 1 E ( D ) s.t. � � ∀ v ∈ H 1 a ( u , v ) = a ∇ u · ∇ vdx = 0 ( D ) f v dx D D Extended ( stochastic ) weak formulation: find u ∈ H 1 E ( D ) ⊗ L 2 (Ω) s.t. � � � � ∀ v ∈ H 1 a ∇ u ·∇ v dx dP (Ω) = f v dx dP (Ω) 0 ( D ) ⊗ L 2 (Ω) Ω D Ω D � �� � � �� � � � � � a ( x , ξ ) ∇ u ·∇ v d x ρ ( ξ ) d ξ f v dx ρ ( ξ ) d ξ (Γ = ξ (Ω)) Γ D Γ D Discretization in physical space: S ( h ) ⊂ H 1 E ( D ), basis { φ j } N j =1 E Example: piecewise linear “hat functions” Discretization in space of random variables: T ( p ) ⊂ L 2 (Γ), basis { ψ ℓ } M ℓ =1 Example: m -variate polynomials in ξ of total degree p 3 / 42 H. C. Elman Reduced Basis Collocation for PDEs

Preliminary: Spectral Methods for PDEs with Uncertain Coefficients Reduced Basis Methods Problem Definition Reduced Basis + Sparse Grid Collocation Solution Methods Iterative Solution of Reduced Problem Concluding Remarks Discrete solution: u hp ( x , ξ ) = � N � M ℓ =1 u j ℓ φ j ( x ) ψ ℓ ( ξ ) j =1 Requires solution of large coupled system Matrix (right): G 0 ⊗ A 0 + � m r =1 G r ⊗ A r � m + p � “Stochastic dimension”: M = p (Ghanem, Spanos, Babuˇ ska, Deb, Oden, Matthies, Keese, Karniadakis, Xue, Schwab, Todor) 4 / 42 H. C. Elman Reduced Basis Collocation for PDEs

Preliminary: Spectral Methods for PDEs with Uncertain Coefficients Reduced Basis Methods Problem Definition Reduced Basis + Sparse Grid Collocation Solution Methods Iterative Solution of Reduced Problem Concluding Remarks The Stochastic Collocation Method Monte-Carlo (sampling) method: find u ∈ H 1 E ( D ) s.t. � a ( x , ξ ( k ) ) ∇ u ·∇ vdx for all v ∈ H 1 E 0 ( D ) D for a collection of samples { ξ ( k ) } ∈ L 2 (Γ) Collocation (Xiu, Hesthaven, Babuˇ ska, Nobile, Tempone, Webster) Choose { ξ ( k ) } in a special way (sparse grids), then construct construct discrete solution u hp ( x , ξ ) to interpolate { u h ( x , ξ ( k ) ) } Structure of collocation solution: u hp ( x , ξ ) := � ξ ( k ) ∈ Θ p u c ( x , ξ ( k ) ) L ξ ( k ) ( ξ ) Features: Decouples algebraic system (like MC) Applies in a straightforward way to nonlinear random terms Coefficients { u c ( x , ξ ( k ) ) } obtained from large-scale PDE solve Expensive when number of points | Θ p | is large 5 / 42 H. C. Elman Reduced Basis Collocation for PDEs

Preliminary: Spectral Methods for PDEs with Uncertain Coefficients Reduced Basis Methods Problem Definition Reduced Basis + Sparse Grid Collocation Solution Methods Iterative Solution of Reduced Problem Concluding Remarks Properties of These Methods For both Galerkin and collocation Each computes a discrete function u hp Moments of u estimated using moments of u hp (cheap) Convergence: � E ( u ) − E ( u hp ) � H 1 ( D ) ≤ c 1 h + c 2 r p , r < 1 Exponential in polynomial degree Contrast with Monte Carlo: Perform N MC (discrete) PDE solves to obtain samples { u ( s ) h } N MC s =1 � N MC s =1 u ( s ) Moments from averaging, e.g., ˆ 1 E ( u h ) = Error ∼ 1 / √ N MC N MC h One other thing: “ p ” has different meaning for Galerkin and collocation Disadvantage of collocation: For comparable accuracy # stochastic dof (collocation) ≈ 2 p (# stochastic dof (Galerkin)) 6 / 42 H. C. Elman Reduced Basis Collocation for PDEs

Preliminary: Spectral Methods for PDEs with Uncertain Coefficients Reduced Basis Methods Problem Definition Reduced Basis + Sparse Grid Collocation Solution Methods Iterative Solution of Reduced Problem Concluding Remarks Representative Comparison for Diffusion Equation Representative comparative performance (E., Miller, Phipps, Tuminaro) p = 6 p = 5 Using mean-based preconditioner p = 4 for Galerkin system p = 3 Kruger, Pellisetti, Ghanem m = 5 p = 2 uniform Le Maˆ ıtre, et al., E. & Powell density p = 1 Error Question: Can costs of collocation be reduced? 7 / 42 H. C. Elman Reduced Basis Collocation for PDEs

Preliminary: Spectral Methods for PDEs with Uncertain Coefficients Offline Computations Reduced Basis Methods Reduced Problem Reduced Basis + Sparse Grid Collocation Reduced Problem: Costs Iterative Solution of Reduced Problem Reduced Problem: Capturing Features of Model Concluding Remarks Preliminary: Spectral Methods for PDEs with Uncertain Coefficients 1 Reduced Basis Methods 2 Offline Computations Reduced Problem Reduced Problem: Costs Reduced Problem: Capturing Features of Model 3 Reduced Basis + Sparse Grid Collocation Iterative Solution of Reduced Problem 4 Concluding Remarks 5 8 / 42 H. C. Elman Reduced Basis Collocation for PDEs

Preliminary: Spectral Methods for PDEs with Uncertain Coefficients Offline Computations Reduced Basis Methods Reduced Problem Reduced Basis + Sparse Grid Collocation Reduced Problem: Costs Iterative Solution of Reduced Problem Reduced Problem: Capturing Features of Model Concluding Remarks Reduced Basis Methods Starting point: Parameter-dependent PDE L ξ u = f √ λ r a r ( x ) ξ r ) ∇ L ξ = −∇ · ( a 0 + σ � m In examples given: r =1 Discretize: Discrete system L h , ξ ( u h ) = f Algebraic system F ξ ( u h ) = 0 ( A ξ u h = f ) of order N Complication: Expensive if many realizations (samples of ξ ) are required Idea (Patera, Boyaval, Bris, Leli` evre, Maday, Nguyen, . . . ): Solve the problem on a reduced space That is: by some means, choose ξ (1) , ξ (2) , . . . , ξ ( n ) , n ≪ N Solve F ξ ( i ) ( u ( i ) h ) = 0, u ( i ) = u h ( · , ξ ( i ) ), i = 1 , . . . , n h u h ( · , ξ ) ∈ span { u (1) h , . . . , u ( n ) For other ξ , approximate u h ( · , ξ ) by ˜ h } Terminology: { u (1) h , . . . , u ( n ) h } called snapshots 9 / 42 H. C. Elman Reduced Basis Collocation for PDEs

Preliminary: Spectral Methods for PDEs with Uncertain Coefficients Offline Computations Reduced Basis Methods Reduced Problem Reduced Basis + Sparse Grid Collocation Reduced Problem: Costs Iterative Solution of Reduced Problem Reduced Problem: Capturing Features of Model Concluding Remarks Offline Computations Strategy for generating a basis / choosing snapshots (Patera, et al.): For ˜ u h ( · , ξ ) ≈ u h ( · , ξ ) (equivalently, ˜ u ξ ≈ u ξ ), use an error indicator η (˜ u h ) ≈ � e h � , e h = u h − ˜ u h Given: a set of candidate parameters X = { ξ } , an initial choice ξ (1) ∈ X , and u (1) = u ( · , ξ (1) ) Q = u (1) Set while max ξ ∈X ( η (˜ u h ( · , ξ ))) > τ compute ˜ u h ( · , ξ ), η (˜ u h ( · , ξ )), ∀ ξ ∈ X % use current reduced let ξ ∗ = argmax ξ ∈X ( η (˜ u h ( · , ξ )) % basis u h ( · , ξ ∗ )) > τ then if η (˜ augment basis with u h ( · , ξ ∗ ), update Q with u ξ ∗ endif end Potentially expensive, but viewed as “offline” preprocessing “ Online ” simulation done using reduced basis 10 / 42 H. C. Elman Reduced Basis Collocation for PDEs