modern numerical methods for large scale eigenvalue problems · introduction algorithms for linear...

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

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

A x = λ x

n× n matrix A

Some facts

A x = λ x

n× n matrix A

Some facts

A x = λ x

n× n matrix A

Some facts

A x = λ x

n× n matrix A

Some facts

A x = λ x

n× n matrix A

Some facts

A x = λ x

n× n matrix A

Some facts

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∑

λiAix = 0, e.g., (λ2M + λD + K )x = 0

Rational problems:

T (λ)x =

(A + λB +

p∑i=0

1λ−µi

)x = 0

Transcendent problems:T (λ)x =

[λ2M + λD + K − e−λ(U + λV )

]x = 0

Nonlinear eigenvalue problems

standard problems: (A− λI )x = 0

T (λ)x =p∑

λiAix = 0, e.g., (λ2M + λD + K )x = 0

Rational problems:

T (λ)x =

(A + λB +

p∑i=0

1λ−µi

)x = 0

[λ2M + λD + K − e−λ(U + λV )

]x = 0

T (λ)x =p∑

λiAix = 0, e.g., (λ2M + λD + K )x = 0

Rational problems:

T (λ)x =

(A + λB +

p∑i=0

1λ−µi

)x = 0

[λ2M + λD + K − e−λ(U + λV )

]x = 0

T (λ)x =p∑

λiAix = 0, e.g., (λ2M + λD + K )x = 0

Rational problems:

T (λ)x =

(A + λB +

p∑i=0

1λ−µi

)x = 0

[λ2M + λD + K − e−λ(U + λV )

]x = 0

T (λ)x =p∑

λiAix = 0, e.g., (λ2M + λD + K )x = 0

Rational problems:

T (λ)x =

(A + λB +

p∑i=0

1λ−µi

)x = 0

[λ2M + λD + K − e−λ(U + λV )

]x = 0

T (λ)x =p∑

λiAix = 0, e.g., (λ2M + λD + K )x = 0

Rational problems:

T (λ)x =

(A + λB +

p∑i=0

1λ−µi

)x = 0

[λ2M + λD + K − e−λ(U + λV )

]x = 0

T (λ)x =p∑

λiAix = 0, e.g., (λ2M + λD + K )x = 0

Rational problems:

T (λ)x =

(A + λB +

p∑i=0

1λ−µi

)x = 0

[λ2M + λD + K − e−λ(U + λV )

]x = 0

T (λ)x =p∑

λiAix = 0, e.g., (λ2M + λD + K )x = 0

Rational problems:

T (λ)x =

(A + λB +

p∑i=0

1λ−µi

)x = 0

[λ2M + λD + K − e−λ(U + λV )

]x = 0

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

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.

IntroductionApplications - Linear dynamical systems and model order reduction

M x + D x + K x = B u

y = C x

Original large-scale system

M x + D x + K x = B u

y = C x

Model-Order-Reduction

M x + D x + K x = B u

y = C x

W TM V ¨x + W T

D V ˙x + W TK V x = W T

y = C V x

with V = · · · W T = ...

right left

eigenvectors x , y of T (λ) = λ2M + λD + K

Projection of original system

M x + D x + K x = B u

y = C x

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

Reduced system

M x + D x + K x = B u

y = C x

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

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)

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

GaAs quantum dot

The Schrodinger equation

−∇ ·(

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).

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

GaAs quantum dot

The Schrodinger equation

−∇ ·(

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).

For this semiconductor setting the non-parabolic effective mass model

mj(λ)=

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]

For this semiconductor setting the non-parabolic effective mass model

mj(λ)=

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]

Overview

1 Introduction

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

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 =

Disadvantage

A TH A T =

Disadvantage

A TH A T =

Disadvantage

Algorithms for Linear ProblemsLarge-Scale Problems

General procedure for large-scale problems

Project A onto low-dimensional subspaceV = colspan(V ), V ∈ Rn×k , k n

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 .

If ‖r‖ is not sufficiently small?

YES: converged !NO: expand V by some appropriatenew vector t and goto 1.

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

II. Lanczos method for A = AT

• eigs in MATLAB

• ARnoldiPACKage (ARPACK), Fortran 77 library

• Scalable Library for Eigenvalue Problem Computations(SLEPc), C++ library for parallel computers

Available software

Algorithms for Linear ProblemsLarge-Scale Problems II: Jacobi–Davidson Methods

Jacobi-Davidson methods: Impose no special structure on 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)

Algorithms for Linear ProblemsLarge-Scale Problems II: Jacobi–Davidson Methods

Jacobi-Davidson methods: Impose no special structure on 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)

Overview

1 Introduction

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[

]− [∂F (v , θ)]−1 F (v , θ).

• Requires good initial approximations (θ, v)

• Matrix inversion infeasible for large problems

Drawbacks

F (x , λ) =

[T (λ)x

wHx − 1

], F : Cn+1 7→ Cn+1.

T (θ) T (θ)vwH 0

]−1 [T (λ)v

wHv − 1

Drawbacks

F (x , λ) =

[T (λ)x

wHx − 1

], F : Cn+1 7→ Cn+1.

T (θ) T (θ)vwH 0

]−1 [T (λ)v

wHv − 1

Drawbacks

Methods for Nonlinear Eigenvalue ProblemsLarge-Scale Problems - Nonlinear Jacobi-Davidson

Nonlinear Jacobi-Davidson: Project the operator T (λ) onto V

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

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 .

Advantage

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 .

Advantage

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

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

• 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!

• 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!

modern numerical methods for large scale eigenvalue problems · introduction algorithms for linear...

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