Rational Krylov Methods for Solving Nonlinear Eigenvalue Problems - PowerPoint PPT Presentation

Rational Krylov Methods for Solving Nonlinear Eigenvalue Problems Roel Van Beeumen rvanbeeumen@lbl.gov Computational Research Division Lawrence Berkeley National Laboratory BASCD 2016 Stanford – December 3, 2016

Quadratic eigenvalue problem Vibration analysis in structural analysis gives rise to ( λ 2 M + λ C + K ) x = 0 where λ is an eigenvalue x is an eigenvector M is the mass matrix C is the damping matrix K is the stiffness matrix R. Van Beeumen (Berkeley Lab) Rational Krylov methods for NLEPs Stanford – December 3, 2016 1

Motivation: Nonlinear damping Clamped beam: Clamped sandwich beam: � � � � λ 2 M + λ C ( λ ) + K λ 2 M + λ C + K x = 0 x = 0 | C | | C ( λ ) | | λ | | λ | for λ on the imaginary axis R. Van Beeumen (Berkeley Lab) Rational Krylov methods for NLEPs Stanford – December 3, 2016 2

Motivation: Active damping Active damping in cars: input output System Controller Delay eigenvalue problem � � λ 2 M + λ C + K + e − λτ E x = 0 R. Van Beeumen (Berkeley Lab) Rational Krylov methods for NLEPs Stanford – December 3, 2016 3

Motivation: Nonlinear boundary conditions Cavity design of a linear accelerator � 1 � ∇ × µ ∇ × E − λε E = 0 Maxwell’s equations + nonlinear waveguide boundary conditions:   k � � λ − κ 2  x = 0  K − λ M + i c , j W j j =1 where λ = ω 2 / c 2 κ c , j are the cutoff values R. Van Beeumen (Berkeley Lab) Rational Krylov methods for NLEPs Stanford – December 3, 2016 4

Outline 1 Motivation 2 Solving Nonlinear Eigenvalue Problems Approximation Linearization pencils Solving linear eigenvalue problem 3 Numerical Experiment R. Van Beeumen (Berkeley Lab) Rational Krylov methods for NLEPs Stanford – December 3, 2016 5

Nonlinear eigenvalue problem (NLEP) NLEP The nonlinear eigenvalue problem: A ( λ ) x = 0 where λ ∈ Ω ⊆ C : eigenvalue x ∈ C n \{ 0 } : eigenvector A : Ω → C n × n : matrix-valued function R. Van Beeumen (Berkeley Lab) Rational Krylov methods for NLEPs Stanford – December 3, 2016 6

Nonlinear eigenvalue problem (NLEP) NLEP The nonlinear eigenvalue problem: A ( λ ) x = 0 where λ ∈ Ω ⊆ C : eigenvalue x ∈ C n \{ 0 } : eigenvector A : Ω → C n × n : matrix-valued function Note that the NLEP is � nonlinear in eigenvalue λ , � linear in eigenvector x . R. Van Beeumen (Berkeley Lab) Rational Krylov methods for NLEPs Stanford – December 3, 2016 6

Solving NLEPs NLEP A ( λ ) x = 0 ⇓ 1 approximation via interpolation PEP P d ( λ ) x = 0 ⇓ 2 linearization GEP L ( λ ) x = 0 ⇓ 3 solving linear eigenvalue problem Solution R. Van Beeumen (Berkeley Lab) Rational Krylov methods for NLEPs Stanford – December 3, 2016 7

Approximation NLEP Step 1: Polynomial interpolation A ( λ ) x = 0 A ( λ ) ≈ P d ( λ ) = A 0 + A 1 λ + A 2 λ 2 + · · · + A d λ d ⇓ PEP P d ( λ ) x = 0 ⇓ GEP L ( λ ) x = 0 ⇓ Solution R. Van Beeumen (Berkeley Lab) Rational Krylov methods for NLEPs Stanford – December 3, 2016 8

Approximation NLEP Step 1: Polynomial interpolation A ( λ ) x = 0 A ( λ ) ≈ P d ( λ ) = A 0 + A 1 λ + A 2 λ 2 + · · · + A d λ d ⇓ PEP 1 P d ( λ ) x = 0 0 . 5 ⇓ GEP λ 0 L ( λ ) x = 0 1 2 3 4 ⇓ − 0 . 5 Solution R. Van Beeumen (Berkeley Lab) Rational Krylov methods for NLEPs Stanford – December 3, 2016 8

Approximation NLEP Step 1: Polynomial interpolation A ( λ ) x = 0 A ( λ ) ≈ P d ( λ ) = A 0 + A 1 λ + A 2 λ 2 + · · · + A d λ d ⇓ PEP 1 P d ( λ ) x = 0 0 . 5 ⇓ GEP λ 0 L ( λ ) x = 0 1 2 3 4 ⇓ − 0 . 5 Solution R. Van Beeumen (Berkeley Lab) Rational Krylov methods for NLEPs Stanford – December 3, 2016 8

Approximation NLEP Step 1: Polynomial interpolation A ( λ ) x = 0 A ( λ ) ≈ P d ( λ ) = A 0 + A 1 λ + A 2 λ 2 + · · · + A d λ d ⇓ PEP 1 P d ( λ ) x = 0 0 . 5 ⇓ GEP λ 0 L ( λ ) x = 0 1 2 3 4 ⇓ − 0 . 5 Solution R. Van Beeumen (Berkeley Lab) Rational Krylov methods for NLEPs Stanford – December 3, 2016 8

Approximation NLEP Step 1: Polynomial interpolation A ( λ ) x = 0 A ( λ ) ≈ P d ( λ ) = A 0 + A 1 λ + A 2 λ 2 + · · · + A d λ d ⇓ PEP 1 P d ( λ ) x = 0 0 . 5 ⇓ GEP λ 0 L ( λ ) x = 0 1 2 3 4 ⇓ − 0 . 5 Solution R. Van Beeumen (Berkeley Lab) Rational Krylov methods for NLEPs Stanford – December 3, 2016 8

Approximation: Newton interpolation Dynamic polynomial interpolation (Newton) A ( λ ) ≈ A 0 n 0 ( λ ) + A 1 n 1 ( λ ) 1 0 . 5 λ 0 1 2 3 4 − 0 . 5 R. Van Beeumen (Berkeley Lab) Rational Krylov methods for NLEPs Stanford – December 3, 2016 9

Approximation: Newton interpolation Dynamic polynomial interpolation (Newton) A ( λ ) ≈ A 0 n 0 ( λ ) + A 1 n 1 ( λ ) 1 0 . 5 λ 0 1 2 3 4 − 0 . 5 R. Van Beeumen (Berkeley Lab) Rational Krylov methods for NLEPs Stanford – December 3, 2016 9

Approximation: Newton interpolation Dynamic polynomial interpolation (Newton) A ( λ ) ≈ A 0 n 0 ( λ ) + A 1 n 1 ( λ ) + A 2 n 2 ( λ ) 1 0 . 5 λ 0 1 2 3 4 − 0 . 5 R. Van Beeumen (Berkeley Lab) Rational Krylov methods for NLEPs Stanford – December 3, 2016 9

Approximation: Newton interpolation Dynamic polynomial interpolation (Newton) A ( λ ) ≈ A 0 n 0 ( λ ) + A 1 n 1 ( λ ) + · · · + A 3 n 3 ( λ ) 1 0 . 5 λ 0 1 2 3 4 − 0 . 5 R. Van Beeumen (Berkeley Lab) Rational Krylov methods for NLEPs Stanford – December 3, 2016 9

Approximation: Newton interpolation Dynamic polynomial interpolation (Newton) A ( λ ) ≈ A 0 n 0 ( λ ) + A 1 n 1 ( λ ) + · · · + A 4 n 4 ( λ ) 1 0 . 5 λ 0 1 2 3 4 − 0 . 5 R. Van Beeumen (Berkeley Lab) Rational Krylov methods for NLEPs Stanford – December 3, 2016 9

Approximation: Newton interpolation Dynamic polynomial interpolation (Newton) A ( λ ) ≈ A 0 n 0 ( λ ) + A 1 n 1 ( λ ) + · · · + A 5 n 5 ( λ ) 1 0 . 5 λ 0 1 2 3 4 − 0 . 5 R. Van Beeumen (Berkeley Lab) Rational Krylov methods for NLEPs Stanford – December 3, 2016 9

Approximation: Newton interpolation Dynamic polynomial interpolation (Newton) A ( λ ) ≈ A 0 n 0 ( λ ) + A 1 n 1 ( λ ) + · · · + A 6 n 6 ( λ ) 1 0 . 5 λ 0 1 2 3 4 − 0 . 5 R. Van Beeumen (Berkeley Lab) Rational Krylov methods for NLEPs Stanford – December 3, 2016 9

Approximation: Newton interpolation Dynamic polynomial interpolation (Newton) A ( λ ) ≈ A 0 n 0 ( λ ) + A 1 n 1 ( λ ) + · · · + A 7 n 7 ( λ ) 1 0 . 5 λ 0 1 2 3 4 − 0 . 5 R. Van Beeumen (Berkeley Lab) Rational Krylov methods for NLEPs Stanford – December 3, 2016 9

Approximation: Newton interpolation Dynamic polynomial interpolation (Newton) A ( λ ) ≈ A 0 n 0 ( λ ) + A 1 n 1 ( λ ) + · · · + A 8 n 8 ( λ ) 1 0 . 5 λ 0 1 2 3 4 − 0 . 5 R. Van Beeumen (Berkeley Lab) Rational Krylov methods for NLEPs Stanford – December 3, 2016 9

Approximation: Newton interpolation Dynamic polynomial interpolation (Newton) A ( λ ) ≈ A 0 n 0 ( λ ) + A 1 n 1 ( λ ) + · · · + A 9 n 9 ( λ ) 1 0 . 5 λ 0 1 2 3 4 − 0 . 5 R. Van Beeumen (Berkeley Lab) Rational Krylov methods for NLEPs Stanford – December 3, 2016 9

Approximation: Polynomial versus Rational Scalar nonlinear function: √ A ( λ ) = 0 . 2 λ − 0 . 6 sin(2 λ ) = 0 with target set: Σ = [0 . 01 , 4] A ( λ ) 0 . 5 λ 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 R. Van Beeumen (Berkeley Lab) Rational Krylov methods for NLEPs Stanford – December 3, 2016 10

Approximation: Polynomial versus Rational Scalar nonlinear function: √ A ( λ ) = 0 . 2 λ − 0 . 6 sin(2 λ ) = 0 with target set: Σ = [0 . 01 , 4] A ( λ ) Leja points 0 . 5 λ 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 R. Van Beeumen (Berkeley Lab) Rational Krylov methods for NLEPs Stanford – December 3, 2016 10

Approximation: Polynomial versus Rational Scalar nonlinear function: √ A ( λ ) = 0 . 2 λ − 0 . 6 sin(2 λ ) = 0 with target set: Σ = [0 . 01 , 4] A ( λ ) Leja points 0 . 5 λ 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 A ( λ ) Leja–Bagby points 0 . 5 λ 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4 R. Van Beeumen (Berkeley Lab) Rational Krylov methods for NLEPs Stanford – December 3, 2016 10

Approximation: Polynomial versus Rational Scalar nonlinear function: √ A ( λ ) = 0 . 2 λ − 0 . 6 sin(2 λ ) = 0 interpolation error 10 0 pol. Leja rat. Leja–Bagby 10 - 5 10 - 10 10 - 15 0 20 40 60 80 100 number of interpolation nodes R. Van Beeumen (Berkeley Lab) Rational Krylov methods for NLEPs Stanford – December 3, 2016 11

Approximation: Polynomial versus Rational Scalar nonlinear function: √ A ( λ ) = 0 . 2 λ − 0 . 6 sin(2 λ ) = 0 interpolation error convergence of eigenvalues 10 0 10 0 pol. Leja Newton Rational Krylov rat. Leja–Bagby Fully Rational Krylov 10 - 5 10 - 5 10 - 10 10 - 10 10 - 15 10 - 15 0 20 40 60 80 100 0 20 40 60 80 100 number of interpolation nodes iteration R. Van Beeumen (Berkeley Lab) Rational Krylov methods for NLEPs Stanford – December 3, 2016 11

Linearization pencils NLEP A ( λ ) x = 0 ⇓ PEP Step 2: Linearization P d ( λ ) x = 0 P d ( λ ) x = 0 ⇓ ⇓ GEP L ( λ ) x = ( A − λ B ) x = 0 L ( λ ) x = 0 ⇓ Solution R. Van Beeumen (Berkeley Lab) Rational Krylov methods for NLEPs Stanford – December 3, 2016 12