A

Second-OrderKrylovSubspace

anditsApplicationsfortheSolutionoftheQuadraticEigenvalueProblem

andModelReductionoftheSecond-OrderSystem

ZhaojunBaiandYangfengSu

Mini-WorkshoponDimensionalReductionofLarge-ScaleSystems

Oberwolfach,October19{24,2003

Introduction

�

Themostwidelyusedmethodsforsolvinglarge-scalestan-

darde-problem

Ax=

�xarethemethodsbasedonKrylov

subspace

Kn(A;u)=spanfu;Au;A2u;:::;An�

1ug:

�

The

generalized

e-problem

Cx

=

�Gx

istransformed

to

\Ax=�x",suchasA=G�

1C

�

Thequadratice-problem

(QEP)

��2M

+�D+K

�x=0

istypicallyprocessedintwostages:

(1)transform

theQEPto\Cx=�Gx"viaalinearization

(2)reduce\Cx=�Gx"to\Ax=�x".

�

Modelreductionofadescriptor(DAE)linearsystem,anda

second-ordersystem

(SOS)areprocessedsimilarly.

Goalofthiswork

developamethodologywhichappliesdirectlytotheQEP

andthedimensionalreductionofSOSinauniform

way

andwithoutlossofperformance.

Outlineoftodaytalk

I.A

second-orderKrylovsubspace

{A

generalizedKrylovsubspacebasedonapairofmatrices

andavector

{Generationofanorthonormalbasis:SOAR

procedure

{De ationandbreakdown

II.Quadraticeigenvalueproblem

(keepitshort)

{A

projectionmethodappliedtotheQEPdirectly

{Numericalexamples

III.ModelreductionoftheSOS

{A

projectionmethodappliedtotheSOSdirectly

{Momentmatching

{Examples

IV.Discussionandfuturework

PartI:A

subspacebasedonapairofmatricesandavector

De�nition.LetAandBbesquarematricesoforderN

,andu6=0

beanN

vector.Thenthesequence

r0;r1;r2;:::;rn�

1

where

r0

=

u;

r1

=

Ar0;

rj

=

Arj�

1+Brj�

2

forj�2

iscalledasecond-orderKrylovsequence.Thespace

Gn(A;B;u)=spanfr0;r1;r2;:::;rn�

1 g;

iscalledasecond-orderKrylovsubspace.

Remarks:

�

WhenB=0,Gn(A;0;u)=Kn(A;u).

�

rj=pj (A;B)u,pj (�;�)arepolynomialsin�

and�.

�

...

Rationale

�

TheQEPisequivalentto\Cx=�Gx"

�

TheSOSisequivalentto

8>><>>:C_x(t)+Gx(t)=

Bu(t);

y(t)=

LTx(t)

�

ThenaturalKrylovsubspaceis

Kn(H;v)=span

�v;Hv;H2v;:::;Hn�

1v �

with

H=G�

1C�

2664A

BI

03775

�

Thenwehave

2664rj

rj�

1 3775=Hjvwithv=

�uT

0�T

.

�

Gn(A;B;u)=

spanfr0;r1;r2;:::;rn�

1 gshouldprovidesamein-

formationasKn(H;v)=

8>><>>: 2664r0

03775

; 2664r1

r0 3775;:::; 2664rn�

1

rn�

2 3775 9>>=>>; .

SOAR:Second-OrderARnoldiProcedure(�

version)

Inputs:A;B;u6=0;n�1

Output:anorthonormalbasisofGn(A;B;u):q1;q2;:::;qn

1.

q1=u=kuk

basisvector

2.

p1=0

auxiliaryvector

3.

forj=1;2;:::;n

do

4.

r=Aqj+Bpj

5.

s=qj

6.

fori=1;2;:::;jdo

orthogonalwrtq-vectors

7.

tij=qTir

8.

r:=r�qi tij

9.

s:=s�pi tij

10.

tj+1;j=krk2

11.

iftj+1;j=0,stop

de ationorbreakdown

12.

qj+1=r=tj+1;j

basisvector

13.

pj+1=s=tj+1;j

auxiliaryvector

SOAR

vs.Arnoldi

SOAR

inmatrixnotation:

AQn+BPn

=

Qn Tn+qn+1 eTtn+1;n

Qn

=

Pn Tn+pn+1 eTtn+1;n

or

2664A

BI

03775 2664

Qn

Pn 3775=

2664Qn+1

Pn+1 3775 dTn

ArnoldiprocedureforgeneratinganorthonormalbasisofKn(H;v):

2664A

BI

03775

Vn=Vn+1 dHn

Di�erence:

SOAR: dTn

{enforceorthonormalityofq1;q2;:::;qn2RN

Arnoldi:dH

n{enforceorthonormalityofv1;v2;:::;vn2R2N

Theorem

1

Iftherearenode ationandbreakdown,thevectorse-

quencefq1;q2;:::;qj gformsanorthonormalbasisofthe

second-orderKrylovsubspaceGj(A;B;u)forj�1.

De ationandBreakdown

WhenSOAR

stops(i.e.,tj+1;j=

krk=

0forsomej),thereare

twopossiblecases:

1.de ation:

fr0;r1;:::;rj�

1 gislinearlydependent,but

8>><>>: 2664r0

03775

; 2664r1

r0 3775;:::; 2664rj�

1

rj�

2 3775 9>>=>>;islinearlyindependent.

)

de ationisanadvantageofSOAR!

2.breakdown:

bothfr0;r1;:::;rj�

1 gand

8>><>>: 2664r0

03775

; 2664r1

r0 3775;:::; 2664rj�

1

rj�

2 3775 9>>=>>;arelinearlyde-

pendent.

)

SOAR

breaksdowni�Arnoldibreaksdown!

SOAR

withde ation(�

version)

...3.

forj=1;2;:::;n

do

4.

r=Aqj+Bpj

5.

s=qj

6.

fori=1;2;:::;jdo

7.

tij=qTir

8.

r:=r�qi tij

9.

s:=s�pi tij

10.

tj+1j=krk

11.

iftj+1j=0

12.

ifs2spanfpiji:qi=0g,breakdown

13.

else(de ation)

14.

resettj+1j=1

15.

qj+1=0

16.

pj+1=s

17.

else(normalcase)

18.

qj+1=r=tj+1j

19.

pj+1=s=tj+1j

Theorem

2

SOAR

with(A;B;u)breaksdownatstepj

ifandonlyif

Arnoldiwith(H;v)breaksdownatthesamestepj.

Understandingthebreakdown

�

WhenArnoldibreaksdown,spanfVj gisan(orthonormal)

invariantsubspaceofH.

�

WhenSOARbreaksdown,span

8>><>>: 2664Qj

Pj 3775 9>>=>>;isalsoan(non-orthonormal)

invariantsubspaceofH.

�

But,whatdoesQjmeantotheQEP?

�

Openquestion:theoryofaninvariantsubspaceofQEP?

SOAR

withde ationandmemorysaving(

version)

...3.

forj=1;2;:::;n

do

4.

r=Aqj+Bf

5.

fori=1;2;:::;jdo

6.

tij=qTir

7.

r:=r�qi hij

8.

tj+1j=krk2

9.

iftj+1j6=0,

10.

qj+1:=r=tj+1j

11.

f=Qj dT(2:j+1;1:j)�

1ej

12.

else

13.

resettj+1j=1

14.

qj+1=0

15.

f=Qj dT(2:j+1;1:j)�

1ej

16.

savefandcheckde ationandbreakdown

OpsandMemory

�

Thenumberofmatrix-vectorproductsAqandBf

SOAR

=

Arnoldi

�

Others:

SOAR

Arnoldi

Flopcounts

3n2N

+3nN

4n2N

+10nN

Memoryrequirements

(n+3)N

2(n+2)N

Example:De ationandBreakdown

�

LetM

=�I,D=�IandK=T=TT.

�

De ationinSOAR:

fr0

r1

r2

r3

r4

r5

:::g)

fq1

0

q3

0

q5

0

:::g

dim(Gn )=n=2:

�

Arnoldigeneratesfullbasis:

8><>: 264r0

0375264

r1

r0 375

264r2

r1 375

264r3

r2 375

264r4

r3 375

264r5

r4 375::: 9>=>;)

fv1

v2

v3

v4

v5

v6

:::g

dim(Kn )=n:

�

Ifu=r0

isalinearcombinationofjeigenvectorsofT,thenSOAR

and

Arnoldibreakdownatthesamej.

�

SOAR

andArnoldideliverthesameapproximateeigenvalues

)

De ationisanadvantageofSOAR

(PartII).

PartII:A

projectionmethodapplieddirectlytotheQEP

�

TheQEP

��2M

+�D+K

�x=0;

isequivalentto

��2I��A�B

�x=0;

whereA=�M�

1D

andB=�M�

1K.

�

Rayleigh-RitzprojectiontechniquebasedonGn�Gn(A;B;u):

seekanapproximateeigenpair(�;z),where�

2

C

and

z2Gn ,byimposingthefollowingGalerkincondition

��2M

+�D+K

�z

?

Gn;

orequivalently,

vT ��2M

+�D+K

�z=0;

8v2Gn:

A

projectionmethodapplieddirectlytotheQEP,cont'd

�

Sincez2Gn ,

z=Qk g

wherespanfQk g=spanfqi1 ;qi2 ;:::;qik g=Gn

(k<

n

whende a-

tionhappens).

�

Hence,�andgmustsatisfythereducedQEP:

��2M

k+�Dk+Kk �g=0

where

Mk=QTkMQk;

Dk=QTkDQk;

Kk=QTkKQk:

�

Ritzpairs(approximateeigenpairs):(�;z)=(�;Qk g).

�

Residuals:r=(�2M

+�D+K)z.

A

SOAR-basedmethodforsolvingQEPdirectly

1.UsetheSOARtogenerateanorthonormalbasisQkofGn(A;B;u)

2.ComputeMk ,Dk

andKk

explicitly

3.SolvetheQEPoforderk:

��2M

k+�Dk+Kk �g=0

4.ComputetheRitzpairs

(�;z)=

0BB@�;

Qk g

kQk gk2 1CCA:

5.Testforconvergencebytherelativeresidualnorms:

k(�2M

+�D+K)zk2

j�j 2kMk1+j�jkDk1+kKk1

Recall:basicArnoldimethodtosolveQEPwithlinearization

1.Transform

theQEPtoanequivalentgeneralizede-problem

2664�D

�K

I

0

3775

|

{z

}

C

2664�xx

3775=

2664M

0

0

I 3775

|{z}

G

2664�xx

3775:

2.ApplytheArnolditogenerateanorthonormalbasisVnofthe

KrylovsubspaceKn(H;v),where

H=G�

1C=

2664�M�

1D

�M�

1K

I

0

3775;

v=

2664u0

3775

3.Solvethereducede-problem:(VTnHVn)

|

{z

}

\free00

t=�t:

4.ExtractapproximateeigenpairsoftheQEP:

(�;z)=

0BB@�;

y(N

+1:2N;:)

ky(N

+1:2N;:)k2 1CCA;

wherey=Vn t:

5.Testforconvergencebytherelativeresidualnorms.

NumericalExamples

Dataorigin

OrderN

Remark

1

random

(general)

200

2

random

(gyroscopic)

200

3

soundwaveswithabsorbing

1331(300,000)[Sleijpenetal.]

4

uid-structurecoupling

3600

NASTRAN

5

acoustics uid

9148

[Tisseuretal,R.C.Lietal]

6

micromirror(MEMS)

11706(846)

SUGAR

NumericalExperiment:WhattoExpect?

1.TheconvergencebehaviorsofSOARmethodisgenerallysim-

ilartoArnoldimethodforlinearizedQEP.Speci�cally,

�

Theeigenvalueswiththelargestmagnitudeconverge�rst.

�

Therateofconvergenceisaboutthesame.

2.TheSOAR-basedmethodpreservesthestructualproperties

oftheQEP.

3.De ationisanadvantageofSOAR.

)

withoutlossofperformance

E1: random nonsymmetric M, D and K. N = 200, n = 20

−10 −5 0 5 10−10

−8

−6

−4

−2

0

2

4

6

8

10

real part

imag

inar

y pa

rt

Approximate Eigenvalues

ExactSOAR (Alg.5)Arnoldi (Alg.6)Hybrid (Alg.7)

0 5 10 15 20 25 30 35 4010

−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

Relative Residual Norms of Approximate Eigenpairs

eigenvalue indexre

sidu

al n

orm

SOAR (Alg.5)Arnoldi (Alg.6)Hybrid (Alg.7)

E2: gyroscopic random system, N = 200, n = 20

−150 −100 −50 0 50 100 150−6

−4

−2

0

2

4

6

real part

imag

inar

y pa

rt

Approximate Eigenvalues

ExactSOAR (Alg.5)Arnoldi (Alg.6)Hybrid (Alg.7)

0 5 10 15 20 25 30 35 4010

−20

10−15

10−10

10−5

100

Relative Residual Norms of Approximate Eigenpairs

eigenvalue indexre

sidu

al n

orm

SOAR (Alg.5)Arnoldi (Alg.6)Hybrid (Alg.7)

E3: sound waves with absorbing walls, N = 1331, n = 20

small simulation of [Sleijpen and van der Vorst and van Gijzen, SIAM News, 1996]

−300 −200 −100 0 100 200 300 400 500−5000

−4000

−3000

−2000

−1000

0

1000

2000

3000

4000

real part

imag

inar

y pa

rt

Approximate Eigenvalues

ExactSOAR (Alg.5)Arnoldi (Alg.6)Hybrid (Alg.7)

�max Rel. Residual

\Exact" �1:952652244810165� 102 � 4:314162072894026e� 103i

SOAR �1:952652244809287� 102 � 4:314162072894454� 103i 2:64� 10�12

Arnoldi �1:952652250694968� 102 � 4:314162072541710� 103i 1:95� 10�8

Hybrid �1:952652244681936� 102 � 4:314162072901392� 103i 2:17� 10�12

E4: uid-structure coupling (NASTRAN), N = 3600, n = 100

nonzeros symmetry pos.def. 1-norm cond.est

M 5521 yes no 36.00 Inf

D 19570 yes no 1.025 Inf

K 59062 yes no 2:19� 1012 8:42� 1016

0 50 100 150 20010

0

102

104

106

108

1010

1012

Scaled Relative Residual Norms of Approximate Eigenpairs

eigenvalue index

resi

dual

nor

m

SOAR (Alg.5)Arnoldi (Alg.6)Hybrid (Alg.7)

E5: acoustic uid (FEM), N = 9148, n = 20; 40

example used in [Ho�nung and Li and Ye, submitted 2003]

0 5 10 15 20 25 30 35 4010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Relative Residual Norms of Approximate Eigenpairs

eigenvalue index

resi

dual

nor

m

SOAR (Alg.5)Arnoldi (Alg.6)Hybrid (Alg.7)

0 20 40 60 80 10010

−14

10−12

10−10

10−8

10−6

10−4

10−2

100

Relative Residual Norms of Approximate Eigenpairs

eigenvalue index

resi

dual

nor

m

SOAR (Alg.5)Arnoldi (Alg.6)Hybrid (Alg.7)

E6:SUGAR

simulationofamicromirror(MEMS)

E6:naturalmodesoflumpedmicromirror,N

=846(11706),

n=20

\Exact"Eigenvalues

Approx.EigenvaluesbySOAR

�3:86956991�101�1:77069907�1010i

�3:86957019�101�1:77069882�1010i

�3:86956892�101�1:77058295�1010i

�3:86995776�101�1:77042123�1010i

�3:87127747�101�1:71833665�1010i

�3:87127684�101�1:71833659�1010i

�3:86956523�101�1:71357793�1010i

�3:86961207�101�1:71354925�1010i

Remarks:

�

VeryhardQEP,O(1020)scaledi�erenceinelementsofM,D

andK

�

Arnoldifailstodelivertheseeigenvaluesevenforn=80.

PartIII:ModelReductionoftheSecond-OrderSystem

�

Thesecond-ordersystem:

8>><>>:M�q(t)+D_q(t)+Kq(t)=

bu(t)

y(t)=

l Tq(t)

�

Anequivalentlinearsystem

8>><>>:C_x(t)+Gx(t)=

cbu(t);

y(t)=

bl Tx(t);

where

x(t)=

2664q(t)

_q(t) 3775;C=

2664D

M

�I0

3775;G=

2664K

0

0

I 3775; cb=

2664b0

3775; bl=

2664b0

3775

TransferFunctionandMoments

�

Transferfunction

h(s)=

l T(s2M

+sD+K)�

1b

=

bl T(sC+G)�

1 cb

=

m0+m1s+m2s2+���

�

fmj garecalledmoments

mj=(�1)j bl T �G�

1C�jG�

1 cb

�

Inpractice,fmj garede�nedbasedonanexpansionpoints0 .

ModelreductionoftheSOS

�

Notethat

H=�G�

1C=

2664�K�

1D

�K�

1M

I

0

3775�

2664A

BI

03775

�

UsetheSOARtogenerateanorthonormalbasisQkofGn(A;B;K�

1b),

i.e.,

spanfQk g=Gn(A;B;K�

1b):

�

A

reduced-ordermodeloftheSOS

8>><>>:Mk�z(t)+Dk_z(t)+Kk z(t)=

bk u(t)

cy(t)=

l Tkz(t);

where

Mk=QTkMQk;Dk=QTkDQk;Kk=QTkKQk;bk=QTkb;lk=QTkl:

�

Basicstructures(andproperties)arepreserved.

ReducedSOS:TransferFunctionandMoments

�

Transferfunctionh

k (s)=

l Tk(s2M

k+sDk+Kk )�

1bk

=

bl Tk(sCk+Gk )�

1 cbk

=

m(k)

0

+m(k)

1

s+m(k)

2

s2+���

�

fm(k)

j

garecalledmomentsofthereducedSOS

m(k)

j

=(�1)j bl Tk �G�

1k

Ck �

jG�

1k

cbk

where

Ck=

2664Dk

Mk

�Ik

0

3775;

Gk=

2664Kk

0

0

Ik 3775;

Theorem

3

�

Ingeneral,the�rstn

momentsarematched;

mj=m(k)

j

forj=0;1;:::n�1

�

IftheSOSissymmetric(M

=MT,D=DT,K=KT

andb=l),

thenthe�rst2n

momentsarematched;

mj=m(k)

j

forj=0;1;:::2n�1

)

withoutlossofperformance

NumericalExamples

dataorigin

OrderN

Remark

1

resonator

63

SUGAR

2

clampedbeam

model

174

SLICOT

3

shaftonabearingsupportwithadamper400

NASTRAN

4

lumpedmicromirror

846

SUGAR

5

lumpedmicromirror

11706

SUGAR

E1: Linear-drive multi-mode resonator

n = 5

102

103

104

105

106

−14

−12

−10

−8

−6

−4

Frequency (Hz)

Log1

0(m

agni

tude

)

102

103

104

105

106

−200

−100

0

100

200

Frequency (Hz)

phas

e(de

gree

)

exactsoararnoldi

exactsoararnoldi

n = 10

102

103

104

105

106

−14

−12

−10

−8

−6

−4

Frequency (Hz)

Log1

0(m

agni

tude

)

102

103

104

105

106

−200

−100

0

100

200

Frequency (Hz)

phas

e(de

gree

)

exactsoararnoldi

exactsoararnoldi

n = 20

102

103

104

105

106

−14

−12

−10

−8

−6

−4

Frequency (Hz)

Log1

0(m

agni

tude

)

102

103

104

105

106

−200

−100

0

100

200

Frequency (Hz)

phas

e(de

gree

)

exactsoararnoldi

exactsoararnoldi

n = 30

102

103

104

105

106

−14

−12

−10

−8

−6

−4

Frequency (Hz)

Log1

0(m

agni

tude

)

102

103

104

105

106

−200

−100

0

100

200

Frequency (Hz)

phas

e(de

gree

)

exactsoararnoldi

exactsoararnoldi

E2: clamped beam model

n = 20

10−2

10−1

100

101

102

10−4

10−3

10−2

10−1

100

101

102

103

Frequency (Hz)

Log1

0(m

agni

tude

)exactsoararnoldi

n = 50

10−2

10−1

100

101

102

10−4

10−3

10−2

10−1

100

101

102

103

Frequency (Hz)

Log1

0(m

agni

tude

)

exactsoararnoldi

n = 100

10−2

10−1

100

101

102

10−4

10−3

10−2

10−1

100

101

102

103

Frequency (Hz)

Log1

0(m

agni

tude

)

exactsoararnoldi

E3: a shaft on bearing support with a damper

n = 10

0 100 200 300 400 500 600 700 800 900 1000−9

−8

−7

−6

−5

−4

−3

Frequency (Hz)

Log1

0(m

agni

tude

)

0 100 200 300 400 500 600 700 800 900 1000−200

−100

0

100

200

Frequency (Hz)

phas

e(de

gree

)

exactsoararnoldi

exactsoararnoldi

n = 20

0 100 200 300 400 500 600 700 800 900 1000−7.5

−7

−6.5

−6

−5.5

−5

Frequency (Hz)

Log1

0(m

agni

tude

)

0 100 200 300 400 500 600 700 800 900 1000−200

−150

−100

−50

0

Frequency (Hz)

phas

e(de

gree

)

exactsoararnoldi

exactsoararnoldi

E4: Lumped micromirror, N = 846

n = 10

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000−1

0

1

2

3

4

5

Frequency (Hz)

Log1

0(m

agni

tude

)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000−200

−100

0

100

200

Frequency (Hz)

phas

e(de

gree

)

exactsoararnoldi

exactsoararnoldi

n = 20

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000−1

0

1

2

3

4

5

Frequency (Hz)

Log1

0(m

agni

tude

)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000−200

−100

0

100

200

Frequency (Hz)

phas

e(de

gree

)

exactsoararnoldi

exactsoararnoldi

E5: micromirror, N = 11706

n = 10

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

1

2

3

4

5

Frequency (Hz)

Log1

0(m

agni

tude

)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000−200

−100

0

100

200

Frequency (Hz)

phas

e(de

gree

)

soararnoldi

soararnoldi

n = 20

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

1

2

3

4

5

Frequency (Hz)

Log1

0(m

agni

tude

)

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000−200

−100

0

100

200

Frequency (Hz)

phas

e(de

gree

)

soararnoldi

soararnoldi

PartIV:DiscussionandFutureWork

�

Recallthat

Gn(A;B;u)=spanfr0;r1;r2;:::;rn�

1 g=spanfQn g�RN

andK

n(H;v)=

8>><>>: 2664r0

03775

; 2664r1

r0 3775; 2664r2

r1 3775;:::; 2664rn�

1

rn�

2 3775 9>>=>>;=spanfVn g�R2N;

�

Itcanbeshownthat

spanfVn g�span

8>><>>: 2664Qn

0

0

Qn 3775 9>>=>>;:

Hence,KrylovsubspaceKn

ofR2N

isembeddedintotheS.-

O.KrylovsubspaceGn

ofRN

�

QEPandSOSarereallynoproblem?

[SleijpenandvanderVorstandvanGijzen,SIAM

News,1996]

DiscussionandFutureWork,cont'd

�

Weseem

togetthe\right"projectionsubspacetoworkwith

forsolvingtheQEPandmodelreductionofSOS,directly.

�

Weonlypresentedbasicideas,no�netuning.Thereismuch

worktodo,theoreticalandpractical,suchase�ectsoflossof

orthogonality,restarting,errorestimation,convergencethe-

ory,...

�

ExtensionstotheMIMO

systems,andhigher-ordersystems

arereadilyavailable.