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MAX PLANCK INSTITUTE
FOR DYNAMICS OF COMPLEX
TECHNICAL SYSTEMS
MAGDEBURG
November 5, 2010
Modern Numerical Methods for Large–ScaleEigenvalue Problems
Patrick Kurschner
Max Planck Institute for Dynamics of Complex Dynamical SystemsComputational Methods in Systems and Control Theory
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 1/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
Overview
1 Introduction
2 Algorithms for Linear Problems
3 Methods for Nonlinear Eigenvalue Problems
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 2/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
IntroductionWhat are eigenvalue problems?
Recall the standard linear eigenvalue problem from undergraduate linearalgebra courses:
= •
A x = λ x
n× n matrix A
(right) eigenvectorx ∈ Cn\0 of A
eigenvalueλ ∈ C of A
• λ is an eigenvalue of A
⇔ λ is a root of thecharacteristic polynomialχA(λ) := det (A− λI ) = 0
⇔ A− λI is singular
• left eigenvectors:yHA = yHλ, yHx = 1
This is only the beginning!
Some facts
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 3/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
IntroductionWhat are eigenvalue problems?
Recall the standard linear eigenvalue problem from undergraduate linearalgebra courses:
= •
A x = λ x
n× n matrix A
(right) eigenvectorx ∈ Cn\0 of A
eigenvalueλ ∈ C of A
• λ is an eigenvalue of A
⇔ λ is a root of thecharacteristic polynomialχA(λ) := det (A− λI ) = 0
⇔ A− λI is singular
• left eigenvectors:yHA = yHλ, yHx = 1
This is only the beginning!
Some facts
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 3/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
IntroductionWhat are eigenvalue problems?
Recall the standard linear eigenvalue problem from undergraduate linearalgebra courses:
= •
A x = λ x
n× n matrix A
(right) eigenvectorx ∈ Cn\0 of A
eigenvalueλ ∈ C of A
• λ is an eigenvalue of A
⇔ λ is a root of thecharacteristic polynomialχA(λ) := det (A− λI ) = 0
⇔ A− λI is singular
• left eigenvectors:yHA = yHλ, yHx = 1
This is only the beginning!
Some facts
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 3/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
IntroductionWhat are eigenvalue problems?
Recall the standard linear eigenvalue problem from undergraduate linearalgebra courses:
= •
A x = λ x
n× n matrix A
(right) eigenvectorx ∈ Cn\0 of A
eigenvalueλ ∈ C of A
• λ is an eigenvalue of A
⇔ λ is a root of thecharacteristic polynomialχA(λ) := det (A− λI ) = 0
⇔ A− λI is singular
• left eigenvectors:yHA = yHλ, yHx = 1
This is only the beginning!
Some facts
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 3/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
IntroductionWhat are eigenvalue problems?
Recall the standard linear eigenvalue problem from undergraduate linearalgebra courses:
= •
A x = λ x
n× n matrix A
(right) eigenvectorx ∈ Cn\0 of A
eigenvalueλ ∈ C of A
• λ is an eigenvalue of A
⇔ λ is a root of thecharacteristic polynomialχA(λ) := det (A− λI ) = 0
⇔ A− λI is singular
• left eigenvectors:yHA = yHλ, yHx = 1
This is only the beginning!
Some facts
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 3/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
IntroductionWhat are eigenvalue problems?
Recall the standard linear eigenvalue problem from undergraduate linearalgebra courses:
= •
A x = λ x
n× n matrix A
(right) eigenvectorx ∈ Cn\0 of A
eigenvalueλ ∈ C of A
• λ is an eigenvalue of A
⇔ λ is a root of thecharacteristic polynomialχA(λ) := det (A− λI ) = 0
⇔ A− λI is singular
• left eigenvectors:yHA = yHλ, yHx = 1
This is only the beginning!
Some facts
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 3/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
IntroductionWhat are eigenvalue problems?
Recall the standard linear eigenvalue problem from undergraduate linearalgebra courses:
= •
A x = λ x
n× n matrix A
(right) eigenvectorx ∈ Cn\0 of A
eigenvalueλ ∈ C of A
• λ is an eigenvalue of A
⇔ λ is a root of thecharacteristic polynomialχA(λ) := det (A− λI ) = 0
⇔ A− λI is singular
• left eigenvectors:yHA = yHλ, yHx = 1
This is only the beginning!
Some facts
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 3/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
IntroductionWhat are eigenvalue problems?
standard problems: Ax = λx
generalized problems: (A−λE )x = 0
Linear eigenvalue problems
Different kinds ofeigenvalue problems
T (λ)x = 0, where T (λ) is an arbitrary nonlinear matrix-valuedoperator T (·) : C→ Cn×n.
Polynomial problems:
T (λ)x =p∑
i=0
λiAix = 0, e.g., (λ2M + λD + K )x = 0
Rational problems:
T (λ)x =
(A + λB +
p∑i=0
1λ−µi
Ci
)x = 0
Transcendent problems:T (λ)x =
[λ2M + λD + K − e−λ(U + λV )
]x = 0
Nonlinear eigenvalue problems
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 4/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
IntroductionWhat are eigenvalue problems?
standard problems: (A− λI )x = 0
generalized problems: (A−λE )x = 0
Linear eigenvalue problems
Different kinds ofeigenvalue problems
T (λ)x = 0, where T (λ) is an arbitrary nonlinear matrix-valuedoperator T (·) : C→ Cn×n.
Polynomial problems:
T (λ)x =p∑
i=0
λiAix = 0, e.g., (λ2M + λD + K )x = 0
Rational problems:
T (λ)x =
(A + λB +
p∑i=0
1λ−µi
Ci
)x = 0
Transcendent problems:T (λ)x =
[λ2M + λD + K − e−λ(U + λV )
]x = 0
Nonlinear eigenvalue problems
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 4/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
IntroductionWhat are eigenvalue problems?
standard problems: (A− λI )x = 0
generalized problems: (A−λE )x = 0
Linear eigenvalue problems
Different kinds ofeigenvalue problems
T (λ)x = 0, where T (λ) is an arbitrary nonlinear matrix-valuedoperator T (·) : C→ Cn×n.
Polynomial problems:
T (λ)x =p∑
i=0
λiAix = 0, e.g., (λ2M + λD + K )x = 0
Rational problems:
T (λ)x =
(A + λB +
p∑i=0
1λ−µi
Ci
)x = 0
Transcendent problems:T (λ)x =
[λ2M + λD + K − e−λ(U + λV )
]x = 0
Nonlinear eigenvalue problems
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 4/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
IntroductionWhat are eigenvalue problems?
standard problems: (A− λI )x = 0
generalized problems: (A−λE )x = 0
Linear eigenvalue problems
Different kinds ofeigenvalue problems
T (λ)x = 0, where T (λ) is an arbitrary nonlinear matrix-valuedoperator T (·) : C→ Cn×n.
Polynomial problems:
T (λ)x =p∑
i=0
λiAix = 0, e.g., (λ2M + λD + K )x = 0
Rational problems:
T (λ)x =
(A + λB +
p∑i=0
1λ−µi
Ci
)x = 0
Transcendent problems:T (λ)x =
[λ2M + λD + K − e−λ(U + λV )
]x = 0
Nonlinear eigenvalue problems
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 4/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
IntroductionWhat are eigenvalue problems?
standard problems: (A− λI )x = 0
generalized problems: (A−λE )x = 0
Linear eigenvalue problems
Different kinds ofeigenvalue problems
T (λ)x = 0, where T (λ) is an arbitrary nonlinear matrix-valuedoperator T (·) : C→ Cn×n.
Polynomial problems:
T (λ)x =p∑
i=0
λiAix = 0, e.g., (λ2M + λD + K )x = 0
Rational problems:
T (λ)x =
(A + λB +
p∑i=0
1λ−µi
Ci
)x = 0
Transcendent problems:T (λ)x =
[λ2M + λD + K − e−λ(U + λV )
]x = 0
Nonlinear eigenvalue problems
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 4/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
IntroductionWhat are eigenvalue problems?
standard problems: (A− λI )x = 0
generalized problems: (A−λE )x = 0
Linear eigenvalue problems
Different kinds ofeigenvalue problems
T (λ)x = 0, where T (λ) is an arbitrary nonlinear matrix-valuedoperator T (·) : C→ Cn×n.
Polynomial problems:
T (λ)x =p∑
i=0
λiAix = 0, e.g., (λ2M + λD + K )x = 0
Rational problems:
T (λ)x =
(A + λB +
p∑i=0
1λ−µi
Ci
)x = 0
Transcendent problems:T (λ)x =
[λ2M + λD + K − e−λ(U + λV )
]x = 0
Nonlinear eigenvalue problems
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 4/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
IntroductionWhat are eigenvalue problems?
standard problems: (A− λI )x = 0
generalized problems: (A−λE )x = 0
Linear eigenvalue problems
Different kinds ofeigenvalue problems
T (λ)x = 0, where T (λ) is an arbitrary nonlinear matrix-valuedoperator T (·) : C→ Cn×n.
Polynomial problems:
T (λ)x =p∑
i=0
λiAix = 0, e.g., (λ2M + λD + K )x = 0
Rational problems:
T (λ)x =
(A + λB +
p∑i=0
1λ−µi
Ci
)x = 0
Transcendent problems:T (λ)x =
[λ2M + λD + K − e−λ(U + λV )
]x = 0
Nonlinear eigenvalue problems
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 4/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
IntroductionWhat are eigenvalue problems?
standard problems: (A− λI )x = 0
generalized problems: (A−λE )x = 0
Linear eigenvalue problems
Different kinds ofeigenvalue problems
T (λ)x = 0, where T (λ) is an arbitrary nonlinear matrix-valuedoperator T (·) : C→ Cn×n.
Polynomial problems:
T (λ)x =p∑
i=0
λiAix = 0, e.g., (λ2M + λD + K )x = 0
Rational problems:
T (λ)x =
(A + λB +
p∑i=0
1λ−µi
Ci
)x = 0
Transcendent problems:T (λ)x =
[λ2M + λD + K − e−λ(U + λV )
]x = 0
Nonlinear eigenvalue problems
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 4/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
IntroductionApplications - Linear dynamical systems
Important application: Vibration analysis of mechanical structures
(a) Mass-spring-damper-chain (b) Structural FE-model of a crankshaft
Description by second order control system
M x(t) + D x(t) + K x(t) = B u(t)
y(t) = C x(t)Mass Damping Stiffness
large n × n matrices
input / control,e.g., excitationby external forces
output equation
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 5/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
IntroductionApplications - Linear dynamical systems
Using the Laplace transformation yields the transfer function of thecontrol system
H(s) = C (s2M + sD + K )−1B, s ∈ C.
Its poles are the eigenvalues of the quadratic eigenvalue problem
T (λ)x = (λ2M + λD + K )x = 0.
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 6/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
IntroductionApplications - Linear dynamical systems and model order reduction
M x + D x + K x = B u
y = C x
Original large-scale system
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 7/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
IntroductionApplications - Linear dynamical systems and model order reduction
M x + D x + K x = B u
y = C x
Original large-scale system
Model-Order-Reduction
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 7/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
IntroductionApplications - Linear dynamical systems and model order reduction
M x + D x + K x = B u
y = C x
Original large-scale system
Model-Order-Reduction
W TM V ¨x + W T
D V ˙x + W TK V x = W T
B u
y = C V x
with V = · · · W T = ...
right left
eigenvectors x , y of T (λ) = λ2M + λD + K
Projection of original system
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 7/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
IntroductionApplications - Linear dynamical systems and model order reduction
M x + D x + K x = B u
y = C x
Original large-scale system
Model-Order-Reduction
M ¨x + D ˙x + K x = B u
y = C x
with M, D, K ∈ Rk×k,B ∈ Rk×m, C ∈ Rp×k
and k n.
with V = · · · W T = ...
right left
eigenvectors x , y of T (λ) = λ2M + λD + K
Reduced system
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 7/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
IntroductionApplications - Linear dynamical systems and model order reduction
M x + D x + K x = B u
y = C x
Original large-scale system
Model-Order-Reduction
M ¨x + D ˙x + K x = B u
y = C x
with M, D, K ∈ Rk×k,B ∈ Rk×m, C ∈ Rp×k
and k n.
with V = · · · W T = ...
right left
eigenvectors x , y of T (λ) = λ2M + λD + K
Reduced system
• see also our poster
Modal Approximation of Large–Scale DynamicalSystems using Jacobi-Davidson Methods
• other MOR techniques in the talks of
→ Peter Benner (yesterday)→ Lihong Feng (1000)→ Tobias Breiten (1030)
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 7/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
IntroductionApplications – Quantum Mechanics
Another application: semiconductor devices in microelectronicsConsider a so called quantum dot, an extremely small semiconductornanostructure, i.e. its dimensions are smaller than the electronwavelength (≈ 10−12 m).Hence, its electronic states are quantized at discrete energy levels.
Ω1 - InAs matrix
Ω2
GaAs quantum dot
The Schrodinger equation
−∇ ·(
~2
2mj(λ)∇ψ)
+ Vjψ = λψ
for j ∈ 1, 2 reveals the relevant energylevels λ and wave functions ψ of the elec-trons.
~ - reduced Planck constant,
Vj - (constant) confinement potential in Ωj ,
mj(λ) - effective electron mass in Ωj (j ∈ 1, 2).
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 8/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
IntroductionApplications – Quantum Mechanics
Another application: semiconductor devices in microelectronicsConsider a so called quantum dot, an extremely small semiconductornanostructure, i.e. its dimensions are smaller than the electronwavelength (≈ 10−12 m).Hence, its electronic states are quantized at discrete energy levels.
Ω1 - InAs matrix
Ω2
GaAs quantum dot
The Schrodinger equation
−∇ ·(
~2
2mj(λ)∇ψ)
+ Vjψ = λψ
for j ∈ 1, 2 reveals the relevant energylevels λ and wave functions ψ of the elec-trons.
~ - reduced Planck constant,
Vj - (constant) confinement potential in Ωj ,
mj(λ) - effective electron mass in Ωj (j ∈ 1, 2).
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 8/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
IntroductionApplications – Quantum Mechanics
For this semiconductor setting the non-parabolic effective mass model
1
mj(λ)=
p2j
~2
(2
λ+ Vj − τj− 1
λ+ Vj − µj
), τj 6= µj ∈ R, j ∈ 1, 2
leads with a discretization of the Schrodinger equation, e.g. by FEM, . . .,to a rational eigenvalue problem T (λ)x = 0, where
T (λ) = λ M − 1m1(λ) N − 1
m2(λ) P − Q
large, sparse n × n FE - matrices
Solution with nonlinear Jacobi-Davidson algorithm:
multiply by greatest common denominator and solve quinticpolynomial problem T (λ) = λ5A5 +λ4A4 +λ3A3 +λ2A2 +λA1 + A0
[Hwang, Lin, Wang, Wang ’04/’05]
directly as rational eigenproblem[Voss ’06]
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 9/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
IntroductionApplications – Quantum Mechanics
For this semiconductor setting the non-parabolic effective mass model
1
mj(λ)=
p2j
~2
(2
λ+ Vj − τj− 1
λ+ Vj − µj
), τj 6= µj ∈ R, j ∈ 1, 2
leads with a discretization of the Schrodinger equation, e.g. by FEM, . . .,to a rational eigenvalue problem T (λ)x = 0, where
T (λ) = λ M − 1m1(λ) N − 1
m2(λ) P − Q
large, sparse n × n FE - matrices
Solution with nonlinear Jacobi-Davidson algorithm:
multiply by greatest common denominator and solve quinticpolynomial problem T (λ) = λ5A5 +λ4A4 +λ3A3 +λ2A2 +λA1 + A0
[Hwang, Lin, Wang, Wang ’04/’05]
directly as rational eigenproblem[Voss ’06]
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 9/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
Overview
1 Introduction
2 Algorithms for Linear Problems
3 Methods for Nonlinear Eigenvalue Problems
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 10/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
Algorithms for Linear ProblemsSmall Problems
Back to Ax = λx where A is of small or moderate size.General procedure:Transform A via a similarity transformation T to an easier form:
A T−1 A T =
such that the eigenvalues appear on the diagonal.
Francis’ QR algorithm, QZ method, Divide & Conquer, . . .MATLAB R© (eig, schur), LAPACK
Algorithms & available software
Similarity / unitary transformations are very expensive(complexity O(n3))⇒ not feasible for large-scale problems
Disadvantage
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 11/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
Algorithms for Linear ProblemsSmall Problems
Back to Ax = λx where A is of small or moderate size.General procedure:Transform A via a unitary transformation T to an easier form:
A TH A T =
such that the eigenvalues appear on the diagonal.
Francis’ QR algorithm, QZ method, Divide & Conquer, . . .MATLAB R© (eig, schur), LAPACK
Algorithms & available software
Similarity / unitary transformations are very expensive(complexity O(n3))⇒ not feasible for large-scale problems
Disadvantage
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 11/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
Algorithms for Linear ProblemsSmall Problems
Back to Ax = λx where A is of small or moderate size.General procedure:Transform A via a unitary transformation T to an easier form:
A TH A T =
such that the eigenvalues appear on the diagonal.
Francis’ QR algorithm, QZ method, Divide & Conquer, . . .MATLAB R© (eig, schur), LAPACK
Algorithms & available software
Similarity / unitary transformations are very expensive(complexity O(n3))⇒ not feasible for large-scale problems
Disadvantage
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 11/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
Algorithms for Linear ProblemsSmall Problems
Back to Ax = λx where A is of small or moderate size.General procedure:Transform A via a unitary transformation T to an easier form:
A TH A T =
such that the eigenvalues appear on the diagonal.
Francis’ QR algorithm, QZ method, Divide & Conquer, . . .MATLAB R© (eig, schur), LAPACK
Algorithms & available software
Similarity / unitary transformations are very expensive(complexity O(n3))⇒ not feasible for large-scale problems
Disadvantage
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 11/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
Algorithms for Linear ProblemsLarge-Scale Problems
General procedure for large-scale problems
1.
Project A onto low-dimensional subspaceV = colspan(V ), V ∈ Rn×k , k n
V T
A V = HSimilar approach asin MOR with W = V .
2.Compute eigenvalues θ and eigenvectors q of H usingmethods for small matrices.
3.Approximate eigenpairs of A are (θ, v := Vq),residual r := Av − λv .
4.
If ‖r‖ is not sufficiently small?
YES: converged !NO: expand V by some appropriatenew vector t and goto 1.
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 12/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
Algorithms for Linear ProblemsLarge-Scale Problems I: Krylov–Subspace Methods
Generate V as Krylov subspaceKk(A, v) = spanv , Av , A2v , . . . , Ak−1v.
I. Arnoldi method
V T
A V =
II. Lanczos method for A = AT
V T
A V =
• eigs in MATLAB
• ARnoldiPACKage (ARPACK), Fortran 77 library
• Scalable Library for Eigenvalue Problem Computations(SLEPc), C++ library for parallel computers
Available software
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 13/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
Algorithms for Linear ProblemsLarge-Scale Problems II: Jacobi–Davidson Methods
Jacobi-Davidson methods: Impose no special structure on V
V T
A V =
Expand V orthogonally by t ⊥ v , obtained from the (inexact) solution ofthe Jacobi-Davidson correction equation
(I − vvT )(A− θI )(I − vvT )t = −r .
[Sleijpen, Van der Vorst, et al ’96/’98/’00]
• MATLAB routines (JDQR, JDQZ, . . .)[Sleijpen, Van der Vorst, Fokkema ’98]
• SLEPc (since August 2010)
• PRIMME (SLEPc add-on for A = AT ) [Stathopoulos ’07]
Available software
Preconditioned eigensolver, similar toPINVIT in the talk by
→ Thomas Mach (900)
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 14/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
Algorithms for Linear ProblemsLarge-Scale Problems II: Jacobi–Davidson Methods
Jacobi-Davidson methods: Impose no special structure on V
V T
A V =
Expand V orthogonally by t ⊥ v , obtained from the (inexact) solution ofthe Jacobi-Davidson correction equation
(I − vvT )(A− θI )(I − vvT )t = −r .
[Sleijpen, Van der Vorst, et al ’96/’98/’00]
• MATLAB routines (JDQR, JDQZ, . . .)[Sleijpen, Van der Vorst, Fokkema ’98]
• SLEPc (since August 2010)
• PRIMME (SLEPc add-on for A = AT ) [Stathopoulos ’07]
Available softwarePreconditioned eigensolver, similar toPINVIT in the talk by
→ Thomas Mach (900)
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 14/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
Overview
1 Introduction
2 Algorithms for Linear Problems
3 Methods for Nonlinear Eigenvalue Problems
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 15/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
Methods for Nonlinear Eigenvalue ProblemsSmall Problems – Newton’s Method
Note that an eigenpair (λ, x) of T (λ) is a root of the function
F (x , λ) =
[T (λ)x
wHx − 1
], F : Cn+1 7→ Cn+1.
First idea: apply Newton’s method.Initial approximation (θ, v) ≈ (λ, x), Newton system for the next(hopefully better) approximation (θ+, v+) is[
v+
θ+
]=
[vθ
]− [∂F (v , θ)]−1 F (v , θ).
• Requires good initial approximations (θ, v)
• Matrix inversion infeasible for large problems
Drawbacks
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 16/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
Methods for Nonlinear Eigenvalue ProblemsSmall Problems – Newton’s Method
Note that an eigenpair (λ, x) of T (λ) is a root of the function
F (x , λ) =
[T (λ)x
wHx − 1
], F : Cn+1 7→ Cn+1.
First idea: apply Newton’s method.Initial approximation (θ, v) ≈ (λ, x), Newton system for the next(hopefully better) approximation (θ+, v+) is[
v+
θ+
]=
[vθ
]−[
T (θ) T (θ)vwH 0
]−1 [T (λ)v
wHv − 1
].
• Requires good initial approximations (θ, v)
• Matrix inversion infeasible for large problems
Drawbacks
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 16/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
Methods for Nonlinear Eigenvalue ProblemsSmall Problems – Newton’s Method
Note that an eigenpair (λ, x) of T (λ) is a root of the function
F (x , λ) =
[T (λ)x
wHx − 1
], F : Cn+1 7→ Cn+1.
First idea: apply Newton’s method.Initial approximation (θ, v) ≈ (λ, x), Newton system for the next(hopefully better) approximation (θ+, v+) is[
v+
θ+
]=
[vθ
]−[
T (θ) T (θ)vwH 0
]−1 [T (λ)v
wHv − 1
].
• Requires good initial approximations (θ, v)
• Matrix inversion infeasible for large problems
Drawbacks
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 16/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
Methods for Nonlinear Eigenvalue ProblemsLarge-Scale Problems - Nonlinear Jacobi-Davidson
Nonlinear Jacobi-Davidson: Project the operator T (λ) onto V
V T
T (λ) V = H(λ)
Solve the small problem H(θ)q = 0, e.g., using Newton’s method
Expand V orthogonally by t ⊥ v , obtained from the (inexact) solution ofthe Jacobi-Davidson correction equation for the nonlinear EVP(
I − T (θ)vvT
vT T (θ)v
)T (θ)
(I − vvT
)t = −r = −T (θ)v .
[Betcke, Voss ’04, Schreiber, Schwetlick ’07/’08]
Applies only cheap operations compared to Newton’s method.
Advantage
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 17/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
Methods for Nonlinear Eigenvalue ProblemsLarge-Scale Problems - Nonlinear Jacobi-Davidson
Nonlinear Jacobi-Davidson: Project the operator T (λ) onto V
V T
T (λ) V = H(λ)
Solve the small problem H(θ)q = 0, e.g., using Newton’s methodExpand V orthogonally by t ⊥ v , obtained from the (inexact) solution ofthe Jacobi-Davidson correction equation for the nonlinear EVP(
I − T (θ)vvT
vT T (θ)v
)T (θ)
(I − vvT
)t = −r = −T (θ)v .
[Betcke, Voss ’04, Schreiber, Schwetlick ’07/’08]
Applies only cheap operations compared to Newton’s method.
Advantage
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 17/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
Methods for Nonlinear Eigenvalue ProblemsLarge-Scale Problems - Nonlinear Jacobi-Davidson
Nonlinear Jacobi-Davidson: Project the operator T (λ) onto V
V T
T (λ) V = H(λ)
Solve the small problem H(θ)q = 0, e.g., using Newton’s methodExpand V orthogonally by t ⊥ v , obtained from the (inexact) solution ofthe Jacobi-Davidson correction equation for the nonlinear EVP(
I − T (θ)vvT
vT T (θ)v
)T (θ)
(I − vvT
)t = −r = −T (θ)v .
[Betcke, Voss ’04, Schreiber, Schwetlick ’07/’08]
Applies only cheap operations compared to Newton’s method.
Advantage
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 17/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
Methods for Nonlinear Eigenvalue ProblemsLarge-Scale Problems - Nonlinear Jacobi-Davidson
Still highly dependent on good initial approximations!
Disadvantage
• nonlinear Arnoldi
• Rayleigh functional iteration, inverse iteration
• safeguarded iteration, method of successive linear problems
• homotopy methods
• . . .
Other methods
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 18/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
Methods for Nonlinear Eigenvalue ProblemsLarge-Scale Problems - Nonlinear Jacobi-Davidson
Still highly dependent on good initial approximations!
Disadvantage
• nonlinear Arnoldi
• Rayleigh functional iteration, inverse iteration
• safeguarded iteration, method of successive linear problems
• homotopy methods
• . . .
Other methods
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 18/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
Methods for Nonlinear Eigenvalue ProblemsLarge-Scale Problems - Nonlinear Jacobi-Davidson
• efficient treatment of involved linear systems
• stable computation of several eigenvalues and correspondingeigenvectors
• computation of left eigenvectors
• application within MOR for nonlinear systems
• . . .
Further Challenges
Thank you for your attention!
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 19/19
Introduction Algorithms for Linear Problems Methods for Nonlinear Eigenvalue Problems
Methods for Nonlinear Eigenvalue ProblemsLarge-Scale Problems - Nonlinear Jacobi-Davidson
• efficient treatment of involved linear systems
• stable computation of several eigenvalues and correspondingeigenvectors
• computation of left eigenvectors
• application within MOR for nonlinear systems
• . . .
Further Challenges
Thank you for your attention!
Max Planck Institute Magdeburg Patrick Kurschner, Modern Numerical Methods for Large–Scale Eigenvalue Problems 19/19