\ An Efficient Reduced Basis Solver for SGFEM Matrix Equations. - PowerPoint PPT Presentation

\ An Efficient Reduced Basis Solver for SGFEM Matrix Equations. Catherine E. Powell School of Mathematics University of Manchester Joint work with: Valeria Simoncini, David Silvester. An Efficient Reduced Basis Solver for SGFEM Matrix Equations. – p. 1/2

\ PDEs + Random Inputs To perform forward UQ, we can may apply Stochastic FEMs: ⊲ Monte Carlo FEMs (inc QMC, MLMC, ...) ⊲ Stochastic Galerkin FEMs (SGFEMs) (this talk) ⊲ Stochastic collocation FEMs ⊲ Reduced basis FEMs ⊲ . . . SGFEMs have limitations for interesting/complex problems. n X n P � � u ( x , ξ ( ω )) ≈ u ij φ i ( x ) ψ j ( ξ ( ω )) i =1 j =1 An Efficient Reduced Basis Solver for SGFEM Matrix Equations. – p. 2/2

\ Outline Intro : standard SGFEM approximation for −∇ · a ( x , ξ ( ω )) ∇ u ( x , ξ ( ω )) = f ( x ) Matrix equation formulation of SGFEM systems Reduced basis iterative solver (MultiRB) : ⊲ Exploits low rank of solution object ⊲ Memory-efficient An Efficient Reduced Basis Solver for SGFEM Matrix Equations. – p. 3/2

\ 1. Standard SGFEM Find u ( x , y ) : D × Γ → R such that −∇ · a ( x , y ) ∇ u ( x , y ) = f ( x ) ( x , y ) ∈ D × Γ , (+ boundary conditions) where ∞ � a ( x , y ) = a 0 ( x ) + a m ( x ) y m , m =1 and y is the image of a vector of countably many random variables ξ = ( ξ 1 , ξ 2 , . . . , ) taking values in some set Γ (the parameter domain ). An Efficient Reduced Basis Solver for SGFEM Matrix Equations. – p. 4/2

\ Weak Formulation Find u ∈ V g := L 2 (Γ , H 1 g ( D )) satisfying: � � ( a ∇ u, ∇ v ) L 2 ( D ) dπ ( y ) = ( f, v ) L 2 ( D ) dπ ( y ) ∀ v ∈ V 0 . Γ Γ An Efficient Reduced Basis Solver for SGFEM Matrix Equations. – p. 5/2

\ Weak Formulation Find u ∈ V g := L 2 (Γ , H 1 g ( D )) satisfying: � � ( a ∇ u, ∇ v ) L 2 ( D ) dπ ( y ) = ( f, v ) L 2 ( D ) dπ ( y ) ∀ v ∈ V 0 . Γ Γ To construct a Galerkin approximation: Let X ⊂ H 1 g ( D ) be a finite element space on D Let P ⊂ L 2 (Γ) be a set of M -variate polynomials on Γ ⊲ total degree ≤ k ⇒ dim ( P ) = ( M + k )! M ! k ! ⊲ tensor product ⇒ dim ( P ) = Π M m =1 ( k m + 1) An Efficient Reduced Basis Solver for SGFEM Matrix Equations. – p. 5/2

\ SGFEM Linear Systems Construct u X P ∈ X ⊗ P by solving one linear system, A u = f of size n X n P = dim ( X ) × dim ( P ) where M � A = G 0 ⊗ K 0 + G ℓ ⊗ K ℓ . m =1 Matrix structure (total degree case) M = 4 and k = 1 , 2 , 3 . An Efficient Reduced Basis Solver for SGFEM Matrix Equations. – p. 6/2

\ SGFEM Linear Systems M � A = G 0 ⊗ K 0 + G m ⊗ K m m =1 G 0 , G m are associated with P (the polynomial space) and K 0 , K m are associated with X (the FEM space). All are sparse . Can solve A u = f using standard Krylov methods . Need: ⊲ multiplications with A ⊲ application of P − 1 (preconditioner) to vectors ⊲ memory to store 4 vectors of length n X n P ! An Efficient Reduced Basis Solver for SGFEM Matrix Equations. – p. 7/2

\ Example Choose D = [ − 1 , 1] × [1 , 1] and X = Q 1 with n X = 65 , 025 and 20 √ √ � � � � a ( x , y ) = 1 + σ λ m ϕ m ( x ) y m , y m ∈ − 3 , 3 , m =1 where σ = 0 . 1 and ( λ m , ϕ m ) are eigenpairs associated with � � − 1 C a ( x , x ′ ) = σ 2 exp 2 � x − x ′ � 1 M k n P Preconditioned CG 2 231 7.4e1 (6) 20 3 1,771 5.6e2 (6) 4 10 , 626 Out of Memory An Efficient Reduced Basis Solver for SGFEM Matrix Equations. – p. 8/2

\ Adaptive SGFEM (1) ⊲ start with low-dimensional spaces X (0) , P (0) and compute u (0) X P ⊲ estimate the (e.g, energy) error using a posterior estimators � � � � 1 / 2 u − u (0) � a 1 / 2 ∇ � 2 η ≈ E L 2 ( D ) X P ⊲ learn if enrichment is needed for X (0) or P (0) (or both) X P ∈ X ( ℓ ) ⊗ P ( ℓ ) , ℓ = 1 , 2 , . . . ⊲ compute u ( ℓ ) See work by: Bespalov, Powell, Silvester, Crowder (S-IFISS MATLAB Software) and Schwab, Eigel, Gittelson, Zander etc. An Efficient Reduced Basis Solver for SGFEM Matrix Equations. – p. 9/2

\ Adaptive SGFEM (2) ⊲ start with standard (probably too large) spaces X, P ⊲ convert linear system A u = f into a matrix equation, with solution U ⊲ apply an iterative method to generate U k ≈ U, k = 0 , 1 , 2 , . . . where V k ∈ R n X × n R , Y k ∈ R n R × n P U k = V k Y k , with n R << n X Note: the product V k Y k is never formed! An Efficient Reduced Basis Solver for SGFEM Matrix Equations. – p. 10/2

\ 2. Matrix Equation Formulation Define the n X × n P solution matrix U = [ u 1 , u 2 , . . . , u n P ] , u = vec ( U ) . Rewrite A u = f as a multi-term matrix equation M � K 0 UG 0 + K m UG m = F. m =1 Key fact: U is often a low-rank matrix . Standard Krylov iterative methods like CG do not take advantage of this. An Efficient Reduced Basis Solver for SGFEM Matrix Equations. – p. 11/2

\ Low Rank Example Let D = [0 , 1] × [0 , 1] with a 0 = 1 , y m ∈ [ − 1 , 1] and γ m = O ( m − 4 ) a m ( x ) = γ m cos (2 πβ 1 x 1 ) cos (2 πβ 2 x 2 ) , ( fast decay coefficients). Tol= 10 − 6 Tol= 10 − 7 Tol= 10 − 8 M k n P rank rank rank 5 3 56 19 24 30 4 126 23 29 37 9 3 220 21 29 34 4 715 23 32 41 Approximate ranks of the SGFEM solution matrix U ( n X = 4 , 096 ). An Efficient Reduced Basis Solver for SGFEM Matrix Equations. – p. 12/2

\ Singular Values ( n X = 4 , 096 , n P = 220 ) Singular values of U (blue), and a reduced solution matrix (red) of size n R × n P for n R = 20 (left) and n R = 30 (right). 0 0 10 10 −5 −5 10 10 −10 −10 10 10 −15 −15 10 10 −20 −20 10 10 −25 −25 10 10 0 50 100 150 200 250 0 50 100 150 200 250 ⊲ An Efficient Reduced Basis Solver for SGFEM Matrix Equations. – p. 13/2

\ Reformulated Matrix Equation M � K m UG m = f 0 g ⊤ K 0 UG 0 + 0 m =1 An Efficient Reduced Basis Solver for SGFEM Matrix Equations. – p. 14/2

\ Reformulated Matrix Equation M � K m UG m = f 0 g ⊤ K 0 UG 0 + 0 m =1 ⊲ Using G 0 = I and Cholesky factorisation K 0 = LL ⊤ : M � K m XG m = ˆ ˆ f 0 g ⊤ X + 0 m =1 K m = L − 1 K m L −⊤ (preconditioning) . where X := L ⊤ U , ˆ An Efficient Reduced Basis Solver for SGFEM Matrix Equations. – p. 14/2

\ Reformulated Matrix Equation M � K m UG m = f 0 g ⊤ K 0 UG 0 + 0 m =1 ⊲ Using G 0 = I and Cholesky factorisation K 0 = LL ⊤ : M � K m XG m = ˆ ˆ f 0 g ⊤ X + 0 m =1 K m = L − 1 K m L −⊤ (preconditioning) . where X := L ⊤ U , ˆ ⊲ Introduce shifts so FEM matrices are positive definite : � � M M � � � � ˆ XG m = ˆ f 0 g ⊤ X I − α m G m + K m + α m I 0 m =1 m =1 An Efficient Reduced Basis Solver for SGFEM Matrix Equations. – p. 14/2

\ Reformulated Matrix Equation M � K m UG m = f 0 g ⊤ K 0 UG 0 + 0 m =1 ⊲ Using G 0 = I and Cholesky factorisation K 0 = LL ⊤ : M � K m XG m = ˆ ˆ f 0 g ⊤ X + 0 m =1 K m = L − 1 K m L −⊤ (preconditioning) . where X := L ⊤ U , ˆ ⊲ Introduce shifts so FEM matrices are positive definite : M � A m XB m = ˆ f 0 g ⊤ XB 0 + 0 m =1 An Efficient Reduced Basis Solver for SGFEM Matrix Equations. – p. 15/2

\ 3. Reduced Basis Approximation Given K R ⊂ R n R with n R ≪ n X and an orthonormal basis V R = [ v 1 , . . . , v n R ] , X ≈ X R := V R Y R , where the n R × n P reduced solution Y R satisfies V ⊤ R R R = 0 where R R is the residual. An Efficient Reduced Basis Solver for SGFEM Matrix Equations. – p. 16/2

\ 3. Reduced Basis Approximation Given K R ⊂ R n R with n R ≪ n X and an orthonormal basis V R = [ v 1 , . . . , v n R ] , X ≈ X R := V R Y R , where the n R × n P reduced solution Y R satisfies V ⊤ R R R = 0 where R R is the residual. Equivalently, M � � � � � � � R ˆ V ⊤ V ⊤ V ⊤ f ⊤ R V R Y R B 0 + R A m V R Y R B m = g 0 . 0 � �� � � �� � m =1 � �� � n R × n R n R × n R n R × 1 An Efficient Reduced Basis Solver for SGFEM Matrix Equations. – p. 16/2

\ MultiRB Iterative Method ⊲ Start with V 0 = span { v 0 } ⊲ For j = 1 , 2 , . . . (until convergence) - Augment V j − 1 with at most M new vectors ( A m + s j I ) − 1 v j − 1 ∈ R n X , m = 1 , . . . , M - Truncate SVD & orthonormalise to obtain V j - Solve reduced problem to find Y j Requires O (( n X + n P ) · M ) memory rather than O ( n X · n P ) (Motivated by rational Krylov methods for Sylvester equations ( M = 1) ). An Efficient Reduced Basis Solver for SGFEM Matrix Equations. – p. 17/2

\ Krylov subspaces Recall, the standard Krylov space of dimension k associated with a vector v 0 and matrix A is � � v 0 , A v 0 , A 2 v 0 , . . . , A k − 1 v 0 K k ( A, v 0 ) = span . An Efficient Reduced Basis Solver for SGFEM Matrix Equations. – p. 18/2

\ Krylov subspaces Recall, the standard Krylov space of dimension k associated with a vector v 0 and matrix A is � � v 0 , A v 0 , A 2 v 0 , . . . , A k − 1 v 0 K k ( A, v 0 ) = span . For Sylvester equations (M=1), we use rational Krylov spaces � v 0 , ( A + s 1 I ) − 1 v 0 , ( A + s 2 I ) − 1 ( A + s 1 I ) − 1 v 0 , K k ( A, v 0 , s ) = span . . . . . . , Π k j =1 ( A + s j I ) − 1 v 0 } . where s = ( s 1 , s 2 , . . . ) are parameters . An Efficient Reduced Basis Solver for SGFEM Matrix Equations. – p. 18/2