Solvers for large linear systems arising in the Stochastic Finite - PowerPoint PPT Presentation

Institut f¨ ur Numerische Mathematik und Optimierung Solvers for large linear systems arising in the Stochastic Finite Element Method Elisabeth Ullmann

1 Collaborators • Catherine Powell, David Silvester University of Manchester, School of Mathematics, Manchester, UK • Oliver Ernst TU Bergakademie Freiberg, Freiberg, Germany Support: DAAD/British Council Grant Uncertainty quantification in computer simulations of groundwater flow problems with emphasis on contaminant transport Solvers for large linear systems arising in SFEM Computational Methods with Applications, Harrachov 2007

2 Outline • Stochastic Finite Element Method (SFEM) ◦ Variational formulation of elliptic SPDEs ◦ Structure of Galerkin matrix • Aspects of the stochastic discretization ◦ Multivariate basis functions: complete/tensor polynomials ◦ Structure and spectral properties of stochastic Galerkin matrices • Solvers for the global Galerkin system ◦ block-diagonal Galerkin system: Krylov subspace recycling method ◦ fully-coupled Galerkin system: Mean-based preconditioner • Numerical examples Solvers for large linear systems arising in SFEM Computational Methods with Applications, Harrachov 2007

Review of SFEM 3 1 Review of SFEM [Ghanem & Spanos, 1991] Stochastic elliptic boundary value problem Given: bounded spatial domain D ⊂ R 2 with boundary Γ = Γ D ∪ Γ N and a complete probability space (Ω , A , P ) . Task: solve the second order elliptic stochastic boundary value problem −∇ · ( T ( x , ω ) ∇ p ( x , ω )) = F ( x ) , x ∈ D, P. − a.s. (1a) p ( x ) = p D ( x ) , x ∈ Γ D , (1b) n · ( T ∇ p )( x ) = p N ( x ) , x ∈ Γ N . (1c) Note: T and therefore p are random fields . Solvers for large linear systems arising in SFEM Computational Methods with Applications, Harrachov 2007

Review of SFEM 4 Mixed formulation of stochastic elliptic bvp u ( x , ω ) = − T ( x , ω ) ∇ p ( x , ω ) , x ∈ D, P. − a.s (2a) ∇ · u ( x , ω ) = F ( x ) , x ∈ D, P. − a.s. (2b) p ( x ) = p D ( x ) , x ∈ Γ D , (2c) n · u ( x ) = − p N ( x ) , x ∈ Γ N . (2d) Solvers for large linear systems arising in SFEM Computational Methods with Applications, Harrachov 2007

Review of SFEM 5 Mixed stochastic variational formulation Find functions u ∈ H Γ N ( div , D ) ⊗ L 2 P (Ω) and p ∈ L 2 ( D ) ⊗ L 2 P (Ω) such that for all test functions v ∈ H Γ N ( div , D ) ⊗ L 2 P (Ω) and q ∈ L 2 ( D ) ⊗ L 2 P (Ω) there holds �� � �� � �� � T − 1 u · v d x − p ∇ · v d x = − p D n · v (3a) D D Γ D � � �� � � − ∇ · u q d x = − Fq d x . (3b) D D Notation: �·� denotes the expectation operator w.r.t. the measure P � � ξ � := ξ ( ω ) dP ( ω ) , Ω L 2 Ω ξ 2 ( ω ) dP ( ω ) < ∞ ξ : � ξ 2 � < ∞ � � � � � P (Ω) := ξ ( ω ) : = Solvers for large linear systems arising in SFEM Computational Methods with Applications, Harrachov 2007

Review of SFEM 6 Discretization steps of stochastic variational formulation ◦ Input random fields depend on M mutually independent random variables { ξ m } M m =1 with given probability density functions ρ m : Γ m → [0 , ∞ ) . ρ ( ξ ) := ρ 1 ( ξ 1 ) · · · ρ M ( ξ M ) , ξ ∈ Γ := Γ 1 × · · · × Γ M . ◦ Reformulation of (3): Identify L 2 P (Ω) with L 2 � ρ (Γ) and �·� with Γ ρ ( ξ ) · d ξ . ◦ Deterministic discretization: choose finite dimensional spaces X h := span { φ 1 , φ 2 , . . . , φ N u x } ⊂ H Γ N ( div , D ) Y h := span { π 1 , π 2 , . . . , π N p x } ⊂ L 2 ( D ) ◦ Stochastic discretization: W h := span { ψ 1 ( ξ ) , ψ 2 ( ξ ) , . . . , ψ N ξ ( ξ ) } ⊂ L 2 ρ (Γ) . ◦ Variational space: ( X h × Y h ) ⊗ W h ⊂ ( H Γ N ( div , D ) × L 2 ( D )) ⊗ L 2 ρ (Γ) Solvers for large linear systems arising in SFEM Computational Methods with Applications, Harrachov 2007

Review of SFEM 7 Representation of input random field T − 1 ( x , ξ ) = � N ξ n =1 T n ( x ) ψ n ( ξ ) Karhunen-Lo` eve expansion Wiener’s polynomial chaos expansion T − 1 = � T − 1 � + � M T − 1 = � m =1 T m ( x ) ξ m α ∈ I T α ( x ) H α ( ξ ) I := { α ∈ N M For details see O. Ernst’s talk. 0 , | α | ≤ d } linear in ξ nonlinear in ξ for d > 1 � M + d � M + 1 terms terms M Example: Gaussian random fields Example: lognormal random fields Solvers for large linear systems arising in SFEM Computational Methods with Applications, Harrachov 2007

Review of SFEM 8 Structure of Galerkin matrix ( ψ 1 ≡ 1) N ξ � A = G n ⊗ K n n =1 Stochastic part: [ G n ] ℓ,j = � ψ n ψ ℓ ψ j � , n, j, ℓ = 1 , . . . , N ξ . Deterministic part:     B T  A 1  A n O K 1 = K n = n = 2 , 3 , . . . , N ξ ,   B O O O � x , k = 1 , 2 , . . . , N p [ B ] i,k = − ∇ · φ i ( x ) π k ( x ) d x , i = 1 , 2 , . . . , N u x , D � [ A n ] i,k = T n ( x ) φ i ( x ) · φ k ( x ) d x , i, k = 1 , 2 , . . . , N u x , n = 1 , . . . , N ξ . D Solvers for large linear systems arising in SFEM Computational Methods with Applications, Harrachov 2007

Stochastic discretization 9 2 The stochastic discretization m =1 p ( m ) ρ (Γ) : ψ α ( ξ ) = � M Basis functions for W h ⊂ L 2 α m ( ξ m ) � � p ( m ) p ( m ) = δ ij and α = ( α 1 , α 2 , . . . , α M ) ∈ N M where is a multi-index. 0 i j tensor polynomials (TP) complete polynomials (CP) � � � � p ( m ) p ( m ) � M deg ≤ d , m = 1 , . . . , M m =1 deg ≤ d α m α m W h = Q d W h = P d � M + d dim( Q d ) = ( d + 1) M � dim( P d ) = M Numbering convention of basis polynomials: n ↔ α , G n ↔ G α (TP) α 1 = 0 , α 1 = 1 , . . . , α 1 = d, α 2 = 0 , α 2 = 1 , . . . , α 2 = d, . . . , α M = d (CP) Same as for (TP), but drop multi-indices with | α | > d . Solvers for large linear systems arising in SFEM Computational Methods with Applications, Harrachov 2007

Stochastic discretization 10 α ψ α In P d ? (0,0) p 0 ( ξ 1 ) p 0 ( ξ 2 ) � (1,0) p 1 ( ξ 1 ) p 0 ( ξ 2 ) � (2,0) p 2 ( ξ 1 ) p 0 ( ξ 2 ) � (0,1) p 0 ( ξ 1 ) p 1 ( ξ 2 ) � (1,1) p 1 ( ξ 1 ) p 1 ( ξ 2 ) � (2,1) p 2 ( ξ 1 ) p 1 ( ξ 2 ) x (0,2) p 0 ( ξ 1 ) p 2 ( ξ 2 ) � (1,2) p 1 ( ξ 1 ) p 2 ( ξ 2 ) x (2,2) p 2 ( ξ 1 ) p 2 ( ξ 2 ) x Table 1: Dropping of basis polynomials for M = 2 , d = 2 . Solvers for large linear systems arising in SFEM Computational Methods with Applications, Harrachov 2007

Stochastic discretization 11 Matrix structure and eigenvalue bounds G α λ max ( G α ) M = 1 � p n p i p j � = [ U n ] ij n = 0 I d +1 = 1 n = 1 � ξp i p j � largest root of p d +1 [Golub, Welsch] n = 2 � p 2 p i p j � O ( d ) [U.] for Gaussian ξ �� M � U ( M ) α M ⊗ · · · ⊗ U (1) m =1 λ m , λ m ∈ Λ( U ( m ) = max α m ) Q d α 1 � � U ( m ) | α | = 1 - = λ max , α m = 1 1 �� M � m =1 λ m , λ m ∈ Λ( U ( m ) - ≤ max α m ) P d | α | = 1 - = λ max ( U 1 ) [U.], [Elman, Powell] for { ξ i } M i =1 iid Solvers for large linear systems arising in SFEM Computational Methods with Applications, Harrachov 2007

Solution strategies 12 3 Solution strategies The input random field T is • linear ⇒ G α , | α | ≤ 1 . The stochastic basis functions are ◮ complete polynomials: Solve system in N x · N ξ unknowns. [Ghanem & Pellissetti], [Ghanem & Kruger], [Le Maˆ ıtre et al.], [Matthies & Keese], [Seynaeve et al.], [Elman & Furnival], [Elman & Powell], [Rosseel et al.] ◮ tensor polynomials: Construct biorthogonal stochastic basis functions. Solve N ξ systems in N x unknowns in parallel or use Krylov subspace recycling techniques. [Eiermann, Ernst & U.], [Cai et al.], [Ernst, U. et al.] • nonlinear ⇒ G α , | α | > 1 . The stochastic basis functions are ◮ complete polynomials: Solve system in N x · N ξ unknowns. [Matthies & Keese], [Rosseel et al.], [Ernst, U. et al.] ◮ tensor polynomials: Solve system in N x · N ξ unknowns. [?] Solvers for large linear systems arising in SFEM Computational Methods with Applications, Harrachov 2007

Solution strategies 12 3 Solution strategies The input random field T is • linear ⇒ G α , | α | ≤ 1 . The stochastic basis functions are ◮ complete polynomials: Solve system in N x · N ξ unknowns. [Ghanem & Pellissetti], [Ghanem & Kruger], [Le Maˆ ıtre et al.], [Matthies & Keese], [Seynaeve et al.], [Elman & Furnival], [Elman & Powell], [Rosseel et al.] ◮ tensor polynomials: Construct biorthogonal stochastic basis functions. Solve N ξ systems in N x unknowns in parallel or use Krylov subspace recycling techniques. [Eiermann, Ernst & U.], [Cai et al.], [Ernst, U. et al.] • nonlinear ⇒ G α , | α | > 1 . The stochastic basis functions are ◮ complete polynomials: Solve system in N x · N ξ unknowns. [Matthies & Keese], [Rosseel et al.], [Ernst, U. et al.] ◮ tensor polynomials: Solve system in N x · N ξ unknowns. [?] Solvers for large linear systems arising in SFEM Computational Methods with Applications, Harrachov 2007