The problem RQI and JD The tuned preconditioner Prec. RQI and JD Extensions Conclusions Rayleigh Quotient iteration and simplified Jacobi-Davidson method with preconditioned iterative solves Melina Freitag Department of Mathematical Sciences University of Bath 10th Copper Mountain Conference on Iterative Methods, 2008 Copper Mountain, Colorado 10th April 2008 joint work with Alastair Spence (Bath) Melina Freitag University of Bath RQI and JD method with tuned preconditioners for eigensolvers The problem RQI and JD The tuned preconditioner Prec. RQI and JD Extensions Conclusions Problem and iterative methods Find a small number of eigenvalues close to a shift σ and corresponding eigenvectors of: Ax = λx, λ ∈ C,x ∈ Cn A is large, sparse, nonsymmetric Melina Freitag University of Bath RQI and JD method with tuned preconditioners for eigensolvers The problem RQI and JD The tuned preconditioner Prec. RQI and JD Extensions Conclusions Problem and iterative methods Find a small number of eigenvalues close to a shift σ and corresponding eigenvectors of: Ax = λx, λ ∈ C,x ∈ Cn A is large, sparse, nonsymmetric Iterative solves (e.g. Power method) Melina Freitag University of Bath RQI and JD method with tuned preconditioners for eigensolvers The problem RQI and JD The tuned preconditioner Prec. RQI and JD Extensions Conclusions Problem and iterative methods Find a small number of eigenvalues close to a shift σ and corresponding eigenvectors of: Ax = λx, λ ∈ C,x ∈ Cn A is large, sparse, nonsymmetric Iterative solves (e.g. Power method) Problem becomes 1 (A − σI)−1x = x λ − σ each step of the iterative method involves repeated application of (A − σI)−1 to a vector (inverse iteration) Melina Freitag University of Bath RQI and JD method with tuned preconditioners for eigensolvers The problem RQI and JD The tuned preconditioner Prec. RQI and JD Extensions Conclusions Problem and iterative methods Find a small number of eigenvalues close to a shift σ and corresponding eigenvectors of: Ax = λx, λ ∈ C,x ∈ Cn A is large, sparse, nonsymmetric Iterative solves (e.g. Power method) Problem becomes 1 (A − σI)−1x = x λ − σ each step of the iterative method involves repeated application of (A − σI)−1 to a vector (inverse iteration) Inner iterative solve: (A − σI)y = x using Krylov or Galerkin-Krylov method for linear systems. leading to inner-outer iterative method. Melina Freitag University of Bath RQI and JD method with tuned preconditioners for eigensolvers The problem RQI and JD The tuned preconditioner Prec. RQI and JD Extensions Conclusions Shift-invert strategy This talk: Inner Iteration and preconditioning Comparison of two methods: Inexact Rayleigh quotient iteration and Simplified Jacobi-Davidson method Melina Freitag University of Bath RQI and JD method with tuned preconditioners for eigensolvers The problem RQI and JD The tuned preconditioner Prec. RQI and JD Extensions Conclusions The algorithms Inexact Rayleigh quotient iteration solve (A − ρ(x(i))I)y(i) = x(i), inexactly. (i) y H Rescale x(i+1) = and update ρ(x(i+1))= x(i+1) Ax(i+1), ky(i)k Test: eigenvalue residual r(i+1) =(A − ρ(x(i+1))I)x(i+1). Melina Freitag University of Bath RQI and JD method with tuned preconditioners for eigensolvers The problem RQI and JD The tuned preconditioner Prec. RQI and JD Extensions Conclusions The algorithms Inexact Rayleigh quotient iteration solve (A − ρ(x(i))I)y(i) = x(i), inexactly. (i) y H Rescale x(i+1) = and update ρ(x(i+1))= x(i+1) Ax(i+1), ky(i)k Test: eigenvalue residual r(i+1) =(A − ρ(x(i+1))I)x(i+1). Inexact simplified Jacobi-Davidson method (without subspace expansion) solve H H (I − x(i)x(i) )(A − ρ(x(i))I)(I − x(i)x(i) )s(i) = −r(i), (i) (i) x + s H Rescale x(i+1) = and update ρ(x(i+1))= x(i+1) Ax(i+1), kx(i) + s(i)k Test: eigenvalue residual r(i+1) =(A − ρ(x(i+1))I)x(i+1). Melina Freitag University of Bath RQI and JD method with tuned preconditioners for eigensolvers The problem RQI and JD The tuned preconditioner Prec. RQI and JD Extensions Conclusions An experiment Matrix sherman5.mtx Rayleigh quotient shift; FOM as inner solver −1 10 Rayleigh Quotient iteration without preconditioner −2 10 −3 10 −4 10 eigenvalue residual −5 10 −6 10 0 2 4 6 8 10 12 14 16 18 20 outer iteration Figure: Rayleigh quotient iteration and Jacobi-Davidson method - without preconditioner Melina Freitag University of Bath RQI and JD method with tuned preconditioners for eigensolvers The problem RQI and JD The tuned preconditioner Prec. RQI and JD Extensions Conclusions An experiment Matrix sherman5.mtx Rayleigh quotient shift; FOM as inner solver −1 −1 10 10 Rayleigh Quotient iteration without preconditioner simplified Jacobi−Davidson without preconditoner −2 −2 10 10 −3 −3 10 10 −4 −4 10 10 eigenvalue residual eigenvalue residual −5 −5 10 10 −6 −6 10 10 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 outer iteration outer iteration Figure: Rayleigh quotient iteration and Jacobi-Davidson method - without preconditioner Melina Freitag University of Bath RQI and JD method with tuned preconditioners for eigensolvers The problem RQI and JD The tuned preconditioner Prec. RQI and JD Extensions Conclusions An experiment Matrix sherman5.mtx Rayleigh quotient shift; FOM as inner solver −1 10 simplified Jacobi−Davidson without preconditoner Rayleigh Quotient iteration without preconditioner −2 10 −3 10 −4 10 eigenvalue residual −5 10 −6 10 0 2 4 6 8 10 12 14 16 18 20 outer iteration Figure: Rayleigh quotient iteration and Jacobi-Davidson method - without preconditioner Melina Freitag University of Bath RQI and JD method with tuned preconditioners for eigensolvers The problem RQI and JD The tuned preconditioner Prec. RQI and JD Extensions Conclusions An experiment Matrix sherman5.mtx Rayleigh quotient shift; preconditioned FOM as inner solver Rayleigh Quotient iteration with standard preconditioner −1.1 10 −1.2 10 −1.3 10 eigenvalue residual −1.4 10 0 2 4 6 8 10 12 14 16 18 20 outer iteration Figure: Rayleigh quotient iteration and Jacobi-Davidson method - with preconditioner P1 Melina Freitag University of Bath RQI and JD method with tuned preconditioners for eigensolvers The problem RQI and JD The tuned preconditioner Prec. RQI and JD Extensions Conclusions An experiment Matrix sherman5.mtx Rayleigh quotient shift; preconditioned FOM as inner solver −1 10 Rayleigh Quotient iteration with standard preconditioner simplified Jacobi−Davidson without preconditoner −2 10 −1.1 10 −3 10 −4 10 −1.2 −5 10 10 −6 10 −1.3 −7 10 10 eigenvalue residual eigenvalue residual −8 10 −9 −1.4 10 10 −10 10 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 outer iteration outer iteration Figure: Rayleigh quotient iteration and Jacobi-Davidson method - with preconditioner P1 Melina Freitag University of Bath RQI and JD method with tuned preconditioners for eigensolvers The problem RQI and JD The tuned preconditioner Prec. RQI and JD Extensions Conclusions An experiment Matrix sherman5.mtx Rayleigh quotient shift; preconditioned FOM as inner solver −1 10 −2 10 −3 10 simplified Jacobi−Davidson with standard preconditoner −4 Rayleigh Quotient iteration with standard preconditioner 10 −5 10 −6 10 −7 10 eigenvalue residual −8 10 −9 10 −10 10 0 2 4 6 8 10 12 14 16 18 20 outer iteration Figure: Rayleigh quotient iteration and Jacobi-Davidson method - with preconditioner P1 Melina Freitag University of Bath RQI and JD method with tuned preconditioners for eigensolvers The problem RQI and JD The tuned preconditioner Prec. RQI and JD Extensions Conclusions An experiment Matrix sherman5.mtx Rayleigh quotient shift; preconditioned FOM as inner solver −1 10 Rayleigh Quotient iteration with tuned preconditioner −2 10 −3 10 −4 10 −5 10 −6 10 −7 10 eigenvalue residual −8 10 −9 10 −10 10 0 2 4 6 8 10 12 14 16 18 20 outer iteration Figure: Rayleigh quotient iteration and Jacobi-Davidson method - with preconditioner P2 Melina Freitag University of Bath RQI and JD method with tuned preconditioners for eigensolvers The problem RQI and JD The tuned preconditioner Prec. RQI and JD Extensions Conclusions An experiment Matrix sherman5.mtx Rayleigh quotient shift; preconditioned FOM as inner solver −1 −1 10 10 Rayleigh Quotient iteration with tuned preconditioner simplified Jacobi−Davidson without preconditoner −2 −2 10 10 −3 −3 10 10 −4 −4 10 10 −5 −5 10 10 −6 −6 10 10 −7 −7 10 10 eigenvalue residual eigenvalue residual −8 −8 10 10 −9 −9 10 10 −10 −10 10 10 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 outer iteration outer iteration Figure: Rayleigh quotient iteration and Jacobi-Davidson method - with preconditioner P2 Melina Freitag University of Bath RQI and JD method with tuned preconditioners for eigensolvers The problem RQI and JD The tuned preconditioner Prec. RQI and JD Extensions Conclusions An experiment Matrix sherman5.mtx Rayleigh quotient shift; preconditioned FOM as inner solver −1 10 simplified Jacobi−Davidson with tuned preconditoner −2 Rayleigh Quotient iteration with tuned preconditioner 10 −3 10 −4 10 −5 10 −6 10 −7 10 eigenvalue residual −8 10 −9 10 −10 10 0 2 4 6 8 10 12 14 16 18 20 outer iteration Figure: Rayleigh quotient iteration and Jacobi-Davidson method - with preconditioner P2 Melina Freitag University of Bath RQI and JD method with tuned preconditioners for eigensolvers The problem RQI and JD The tuned preconditioner Prec.