Krylov subspace methods for eigenvalue problems David S. Watkins - PowerPoint PPT Presentation

Second Application Nonlinear Optics Schrödinger eigenvalue problem − � 2 2 m ∇ 2 ψ + V ψ = λψ Solve numerically (finite elements) October 16, 2008 – p. 1

Second Application Nonlinear Optics Schrödinger eigenvalue problem − � 2 2 m ∇ 2 ψ + V ψ = λψ Solve numerically (finite elements) K = K T > 0 , M = M T > 0 Kv = λMv October 16, 2008 – p. 1

Second Application Nonlinear Optics Schrödinger eigenvalue problem − � 2 2 m ∇ 2 ψ + V ψ = λψ Solve numerically (finite elements) K = K T > 0 , M = M T > 0 Kv = λMv Matrices are large and sparse. October 16, 2008 – p. 1

Kv = λMv October 16, 2008 – p. 1

Kv = λMv Want few smallest eigenvalues and associated eigenvectors. October 16, 2008 – p. 1

Kv = λMv Want few smallest eigenvalues and associated eigenvectors. Invert the problem. October 16, 2008 – p. 1

Kv = λMv Want few smallest eigenvalues and associated eigenvectors. Invert the problem. K = R T R October 16, 2008 – p. 1

Kv = λMv Want few smallest eigenvalues and associated eigenvectors. Invert the problem. K = R T R R T Rv = λMv October 16, 2008 – p. 1

Kv = λMv Want few smallest eigenvalues and associated eigenvectors. Invert the problem. K = R T R R T Rv = λMv R − T MR − 1 ( Rv ) = λ − 1 ( Rv ) October 16, 2008 – p. 1

Kv = λMv Want few smallest eigenvalues and associated eigenvectors. Invert the problem. K = R T R R T Rv = λMv R − T MR − 1 ( Rv ) = λ − 1 ( Rv ) A T = A > 0 A = R − T MR − 1 , October 16, 2008 – p. 1

Kv = λMv Want few smallest eigenvalues and associated eigenvectors. Invert the problem. K = R T R R T Rv = λMv R − T MR − 1 ( Rv ) = λ − 1 ( Rv ) A T = A > 0 A = R − T MR − 1 , x �→ Ax October 16, 2008 – p. 1

Kv = λMv Want few smallest eigenvalues and associated eigenvectors. Invert the problem. K = R T R R T Rv = λMv R − T MR − 1 ( Rv ) = λ − 1 ( Rv ) A T = A > 0 A = R − T MR − 1 , backsolve x �→ Ax October 16, 2008 – p. 1

Kv = λMv Want few smallest eigenvalues and associated eigenvectors. Invert the problem. K = R T R R T Rv = λMv R − T MR − 1 ( Rv ) = λ − 1 ( Rv ) A T = A > 0 A = R − T MR − 1 , backsolve x �→ Ax Do not form A explicitly. October 16, 2008 – p. 1

What our applications have in common October 16, 2008 – p. 1

What our applications have in common large, sparse matrices October 16, 2008 – p. 1

What our applications have in common large, sparse matrices use of matrix factorization (Cholesky decomposition) October 16, 2008 – p. 1

What our applications have in common large, sparse matrices use of matrix factorization (Cholesky decomposition) some kind of structure October 16, 2008 – p. 1

What our applications have in common large, sparse matrices use of matrix factorization (Cholesky decomposition) some kind of structure . . . not enough time to discuss this October 16, 2008 – p. 1

Classification of Eigenvalue Problems October 16, 2008 – p. 1

Classification of Eigenvalue Problems small October 16, 2008 – p. 1

Classification of Eigenvalue Problems small medium October 16, 2008 – p. 1

Classification of Eigenvalue Problems small medium large October 16, 2008 – p. 1

Small Matrices October 16, 2008 – p. 1

Small Matrices store conventionally October 16, 2008 – p. 1

Small Matrices store conventionally similarity transformations October 16, 2008 – p. 1

Small Matrices store conventionally similarity transformations QR algorithm October 16, 2008 – p. 1

Small Matrices store conventionally similarity transformations QR algorithm get all eigenvalues/vectors October 16, 2008 – p. 1

Small Matrices store conventionally similarity transformations QR algorithm get all eigenvalues/vectors n ≈ 10 3 October 16, 2008 – p. 1

Medium Matrices October 16, 2008 – p. 1

Medium Matrices store as sparse matrix October 16, 2008 – p. 1

Medium Matrices store as sparse matrix no similarity transformations October 16, 2008 – p. 1

Medium Matrices store as sparse matrix no similarity transformations matrix factorization okay October 16, 2008 – p. 1

Medium Matrices store as sparse matrix no similarity transformations matrix factorization okay shift and invert October 16, 2008 – p. 1

0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 nz = 670 October 16, 2008 – p. 1

0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 nz = 1815 October 16, 2008 – p. 2

Medium Matrices, Continued October 16, 2008 – p. 2

Medium Matrices, Continued store matrix factor as sparse matrix October 16, 2008 – p. 2

Medium Matrices, Continued store matrix factor as sparse matrix n ≈ 10 5 October 16, 2008 – p. 2

Medium Matrices, Continued store matrix factor as sparse matrix n ≈ 10 5 get selected eigenvalues/vectors October 16, 2008 – p. 2

Medium Matrices, Continued store matrix factor as sparse matrix n ≈ 10 5 get selected eigenvalues/vectors Krylov subspace methods October 16, 2008 – p. 2

Medium Matrices, Continued store matrix factor as sparse matrix n ≈ 10 5 get selected eigenvalues/vectors Krylov subspace methods Jacobi-Davidson methods October 16, 2008 – p. 2