Existing Methods Inverse-free Preconditioned Krylov Subspace Method Block Generalization Blackbox Implementation Numerical Inverse Free Preconditioned Krylov Subspace Method for Symmetric Generalized Eigenvalue Problems Qiang Ye University of Kentucky Based on joint works with G. Golub and P . Quillen RANMEP 2008 - NCTS
Existing Methods Inverse-free Preconditioned Krylov Subspace Method Block Generalization Blackbox Implementation Numerical Outline Existing Methods 1 Inverse-free Preconditioned Krylov Subspace Method 2 Block Generalization 3 Blackbox Implementation 4 Numerical Examples 5 Interior Eigenvalues 6
Existing Methods Inverse-free Preconditioned Krylov Subspace Method Block Generalization Blackbox Implementation Numerical Consider computing p smallest eigenvalues for Ax = λ Bx , A , B symmetric , B > 0 . (Eigenvalues: λ 1 < λ 2 ≤ · · · ≤ λ n ). A and B are large the spectrum at the left is clustered
Existing Methods Inverse-free Preconditioned Krylov Subspace Method Block Generalization Blackbox Implementation Numerical Consider computing p smallest eigenvalues for Ax = λ Bx , A , B symmetric , B > 0 . (Eigenvalues: λ 1 < λ 2 ≤ · · · ≤ λ n ). A and B are large the spectrum at the left is clustered
Existing Methods Inverse-free Preconditioned Krylov Subspace Method Block Generalization Blackbox Implementation Numerical Consider computing p smallest eigenvalues for Ax = λ Bx , A , B symmetric , B > 0 . (Eigenvalues: λ 1 < λ 2 ≤ · · · ≤ λ n ). A and B are large the spectrum at the left is clustered
Existing Methods Inverse-free Preconditioned Krylov Subspace Method Block Generalization Blackbox Implementation Numerical Standard Lanczos for Ax = λ Bx : B -orthogonal basis Z m = [ z 0 , z 1 , · · · , z m ] for span { x 0 , B − 1 Ax 0 , ( B − 1 A ) 2 x 0 , · · · , ( B − 1 A ) m x 0 } Projection T m u = θ u ; ( T m = Z T m AZ m ) need B − 1 ; Convergence depends on λ 2 − λ 1 λ n − λ 1 . Several implementations, including implicitly restarted Lanczos - ARPACK
Existing Methods Inverse-free Preconditioned Krylov Subspace Method Block Generalization Blackbox Implementation Numerical Standard Lanczos for Ax = λ Bx : B -orthogonal basis Z m = [ z 0 , z 1 , · · · , z m ] for span { x 0 , B − 1 Ax 0 , ( B − 1 A ) 2 x 0 , · · · , ( B − 1 A ) m x 0 } Projection T m u = θ u ; ( T m = Z T m AZ m ) need B − 1 ; Convergence depends on λ 2 − λ 1 λ n − λ 1 . Several implementations, including implicitly restarted Lanczos - ARPACK
Existing Methods Inverse-free Preconditioned Krylov Subspace Method Block Generalization Blackbox Implementation Numerical Standard Lanczos for Ax = λ Bx : B -orthogonal basis Z m = [ z 0 , z 1 , · · · , z m ] for span { x 0 , B − 1 Ax 0 , ( B − 1 A ) 2 x 0 , · · · , ( B − 1 A ) m x 0 } Projection T m u = θ u ; ( T m = Z T m AZ m ) need B − 1 ; Convergence depends on λ 2 − λ 1 λ n − λ 1 . Several implementations, including implicitly restarted Lanczos - ARPACK
Existing Methods Inverse-free Preconditioned Krylov Subspace Method Block Generalization Blackbox Implementation Numerical Standard Lanczos for Ax = λ Bx : B -orthogonal basis Z m = [ z 0 , z 1 , · · · , z m ] for span { x 0 , B − 1 Ax 0 , ( B − 1 A ) 2 x 0 , · · · , ( B − 1 A ) m x 0 } Projection T m u = θ u ; ( T m = Z T m AZ m ) need B − 1 ; Convergence depends on λ 2 − λ 1 λ n − λ 1 . Several implementations, including implicitly restarted Lanczos - ARPACK
Existing Methods Inverse-free Preconditioned Krylov Subspace Method Block Generalization Blackbox Implementation Numerical Standard Lanczos for Ax = λ Bx : B -orthogonal basis Z m = [ z 0 , z 1 , · · · , z m ] for span { x 0 , B − 1 Ax 0 , ( B − 1 A ) 2 x 0 , · · · , ( B − 1 A ) m x 0 } Projection T m u = θ u ; ( T m = Z T m AZ m ) need B − 1 ; Convergence depends on λ 2 − λ 1 λ n − λ 1 . Several implementations, including implicitly restarted Lanczos - ARPACK
Existing Methods Inverse-free Preconditioned Krylov Subspace Method Block Generalization Blackbox Implementation Numerical Standard Lanczos for Ax = λ Bx : B -orthogonal basis Z m = [ z 0 , z 1 , · · · , z m ] for span { x 0 , B − 1 Ax 0 , ( B − 1 A ) 2 x 0 , · · · , ( B − 1 A ) m x 0 } Projection T m u = θ u ; ( T m = Z T m AZ m ) need B − 1 ; Convergence depends on λ 2 − λ 1 λ n − λ 1 . Several implementations, including implicitly restarted Lanczos - ARPACK
Existing Methods Inverse-free Preconditioned Krylov Subspace Method Block Generalization Blackbox Implementation Numerical Shift-and-invert Lanczos: Apply Lanczos to Bx = µ ( A − σ B ) x , µ = ( λ − σ ) − 1 Convergence depends on µ 2 − µ 1 µ n − µ 1 . need ( A − σ B ) − 1 ;
Existing Methods Inverse-free Preconditioned Krylov Subspace Method Block Generalization Blackbox Implementation Numerical Jacobi-Davidson / JDCG: Projection on a subspace generated from solving correction eq. Need inner iteration to solve correction eq. Sensitive to inner iterations Good local convergence but global convergence not established
Existing Methods Inverse-free Preconditioned Krylov Subspace Method Block Generalization Blackbox Implementation Numerical Jacobi-Davidson / JDCG: Projection on a subspace generated from solving correction eq. Need inner iteration to solve correction eq. Sensitive to inner iterations Good local convergence but global convergence not established
Existing Methods Inverse-free Preconditioned Krylov Subspace Method Block Generalization Blackbox Implementation Numerical Jacobi-Davidson / JDCG: Projection on a subspace generated from solving correction eq. Need inner iteration to solve correction eq. Sensitive to inner iterations Good local convergence but global convergence not established
Existing Methods Inverse-free Preconditioned Krylov Subspace Method Block Generalization Blackbox Implementation Numerical Jacobi-Davidson / JDCG: Projection on a subspace generated from solving correction eq. Need inner iteration to solve correction eq. Sensitive to inner iterations Good local convergence but global convergence not established
Existing Methods Inverse-free Preconditioned Krylov Subspace Method Block Generalization Blackbox Implementation Numerical LOBPCG - Locally optimal block preconditioned CG: Projection on span { x 0 , x 1 , M − 1 ( A − ρ 1 B ) x 1 } Simple and easy to implement Good global convergence but lack of theory
Existing Methods Inverse-free Preconditioned Krylov Subspace Method Block Generalization Blackbox Implementation Numerical LOBPCG - Locally optimal block preconditioned CG: Projection on span { x 0 , x 1 , M − 1 ( A − ρ 1 B ) x 1 } Simple and easy to implement Good global convergence but lack of theory
Existing Methods Inverse-free Preconditioned Krylov Subspace Method Block Generalization Blackbox Implementation Numerical LOBPCG - Locally optimal block preconditioned CG: Projection on span { x 0 , x 1 , M − 1 ( A − ρ 1 B ) x 1 } Simple and easy to implement Good global convergence but lack of theory
Existing Methods Inverse-free Preconditioned Krylov Subspace Method Block Generalization Blackbox Implementation Numerical Basic Inverse free method: choose x 1 ∈ span { x 0 , ( A − ρ k B ) x 0 , · · · , ( A − ρ k B ) m x 0 } ˆ to solve Ax = λ Bx . Algorithm: Input m ≥ 1 and initial x 0 with � x 0 � = 1; ρ 0 = ρ ( x 0 ) ; For k = 0 , 1 , 2 , · · · until convergence, Construct a basis Z m = [ z 0 , z 1 , · · · , z m ] for K m = span { x k , ( A − ρ k B ) x k , · · · , ( A − ρ k B ) m x k } Form A m = Z T m ( A − ρ k B ) Z m and B m = Z T m BZ m ; Find A m v 1 = µ 1 B m v 1 (smallest); ρ k + 1 = ρ k + µ 1 and x k + 1 = Z m v 1 . End
Existing Methods Inverse-free Preconditioned Krylov Subspace Method Block Generalization Blackbox Implementation Numerical Basic Inverse free method: choose x 1 ∈ span { x 0 , ( A − ρ k B ) x 0 , · · · , ( A − ρ k B ) m x 0 } ˆ to solve Ax = λ Bx . Algorithm: Input m ≥ 1 and initial x 0 with � x 0 � = 1; ρ 0 = ρ ( x 0 ) ; For k = 0 , 1 , 2 , · · · until convergence, Construct a basis Z m = [ z 0 , z 1 , · · · , z m ] for K m = span { x k , ( A − ρ k B ) x k , · · · , ( A − ρ k B ) m x k } Form A m = Z T m ( A − ρ k B ) Z m and B m = Z T m BZ m ; Find A m v 1 = µ 1 B m v 1 (smallest); ρ k + 1 = ρ k + µ 1 and x k + 1 = Z m v 1 . End
Existing Methods Inverse-free Preconditioned Krylov Subspace Method Block Generalization Blackbox Implementation Numerical Construction of a basis for K m : orthonormal: Lanczos three-term but need to form B m = Z T m BZ m B -orthonormal: Arnoldi long recurrence but B m = I Choice of m : # of iterations decreases quadratically as m ↑ optimal for small m
Existing Methods Inverse-free Preconditioned Krylov Subspace Method Block Generalization Blackbox Implementation Numerical Construction of a basis for K m : orthonormal: Lanczos three-term but need to form B m = Z T m BZ m B -orthonormal: Arnoldi long recurrence but B m = I Choice of m : # of iterations decreases quadratically as m ↑ optimal for small m
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