Research center Matheon - math.tu-berlin.de · Introduction Second order model reduction Ilinear...

Quadratification for second order model reduction

Christian Schroder

Research center Matheon, TU Berlin

AbsolventenseminarBerlin, October 27th 2016

Work in progress!

Introduction

Introduction

Background on MOR for 1st and 2nd order

Linearization and quadratification

Counter arguments

The EndC. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 2 / 15

Introduction

Second order model reductionI linear time invariant (LTI) dynamical system of second order

x = A0x + A1x + B0u + B1u, (x(0) = x0, x(0) = x1)

y = C0x + C1x + Du

I x(t): state ∈ Rn, u(t): input ∈ Rm, y(t): output ∈ Rp

I arises in mechanical systemsI often n� m, p (semi discretization of PDE)⇒ simulation and control are expensive

I wanted: reduced order model of state dimension r � n

xr = Ar 0xr + Ar 1xr + Br 0u + Br 1u

yr = Cr 0xr + Cr 1xr + Dru (notation: subr= reduced to size r)

I such that y ≈ yr , i.e., ‖G − Gr‖ is small, usually H2- or H∞-norm

G (s) = (C0 + sC1)(s2I − sA1 − A0)−1(B0 + sB1) + D

I here: use linearization and quadratification

C. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 2 / 15

Introduction


x = A0x + A1x + B0u + B1u, (x(0) = x0, x(0) = x1)

y = C0x + C1x + Du






I such that y ≈ yr

, i.e., ‖G − Gr‖ is small, usually H2- or H∞-norm

G (s) = (C0 + sC1)(s2I − sA1 − A0)−1(B0 + sB1) + D



Introduction


x = A0x + A1x + B0u + B1u, (x(0) = x0, x(0) = x1)

y = C0x + C1x + Du






I such that y ≈ yr , i.e., ‖G − Gr‖ is small, usually H2- or H∞-norm

G (s) = (C0 + sC1)(s2I − sA1 − A0)−1(B0 + sB1) + D




Introduction



Counter arguments



1st order MOR

χ = Aχ+ Bu χr = Arχr + Bruy = Cχ+Du yr = Crχr +Dru

G (s) = C(sI −A)−1B +D (curly = 1st order)

I numerous good/trusted/well-understood methods existsI most are projection methods: Vr ∈ Rn,r ,Wr ∈ Rr ,n,WrVr = Ir

Ar = WrAVr , Br = WrB, Cr = CVr , Dr = DI e.g. partial realization, imW T

r = Kr (AT , cT ), Vr = similarmatches 2r + 1 moments at ∞

I interpolation, imW Tr = span{(σi I −A)−T cT}ri=1 ⇒ G (σi )

2r=

i = 1Gr (σi )

I IRKA, if {σ1, . . . , σr} = eig(−Ar ), then Gr is H2-optimalI balanced truncation, Wr ,Vr from balancing transformation,

sequence of ROMs with H∞error bound



1st order MOR




Ar = WrAVr , Br = WrB, Cr = CVr , Dr = DI e.g.

partial realization, imW Tr = Kr (AT , cT ), Vr = similar

matches 2r + 1 moments at ∞I interpolation, imW T

r = span{(σi I −A)−T cT}ri=1 ⇒ G (σi )2r=

i = 1Gr (σi )





1st order MOR







2r=

i = 1Gr (σi )





1st order MOR







2r=

i = 1Gr (σi )

I IRKA, if {σ1, . . . , σr} = eig(−Ar ), then Gr is H2-optimal

I balanced truncation, Wr ,Vr from balancing transformation,sequence of ROMs with H∞error bound



1st order MOR







2r=

i = 1Gr (σi )


sequence of ROMs with H∞error boundC. Schroder (Matheon and TU Berlin) Quadratification for 2nd order model reduction Abs Sem, Oct 2016 3 / 15


Projection methods for 2nd order

Vr ∈ Rn,r ,Wr ∈ Rr ,n,WrVr = Ir

Ai r = WrAiVr , Bi r = WrBi , Ci r = CiVr , Dr = D

I most methods have been adapted, work often well. . .

I but have some sort of problem

I partial realization, SOAR (second order Arnoldi method), but nolucky breakdown

I interpolation, imW Tr = span{(σ2

i I − σiA1 − A0)−T (c0 + σic1)T}ri=1

I 2r interp. points, but no optimal points [Beattie and Benner, 2014]

I balanced truncation, choice of positions or velocities [Stykel and Reis]

The other standard approach for 2nd order systems is. . .



Introduction



Counter arguments



Linearizationthe 2nd order system

x = A0x + A1x + B0u + B1u

y = C0x + C1x + Du

can be linearized (= reformulated in 1st order) to[xz

]=

[0 IA0 A1

] [xz

]+

[B1

B0 + A1B1

]u

y =[C0 C1

] [xz

]+ (D + C1B1)u

I χ = Aχ+ Bu, y = Cχ+Du with A in companion form

I transfer functions G coincide

I (A0,A1,B0,B1,C0,C1,D)linearization�

quadratification(A,B, C,D)



Algorithm

1. given (A0,A1,B0,B1,C0,C1,D) with state dim n

2. linearize, state dim 2n

A =

[0 IA0 A1

], B =

[B1

B0 + A1B1

], C =

[C0 C1

], D = D+C1B1

3. reduce to state dimension 2r

A2r = W2rAV2r , B2r = W2rB, C2r = CV2r , D2r = D

4. quadratify to state dim r , i.e., partition and read off blocks

A2r =

[0 IAr 0 Ar 1

],B2r =

[Br 1

Br 0 + Ar 1Br 1

], C2r =

[Cr 0 Cr 1

],D2r =Dr+Cr 1Br 1

I but A2r is not in companion form!

. . . or is it?

I replace W = [W1,W2]→[W1 00 W2

],V =

[V1

V2

]→[V1 00 V2

]this often works, but doubles the dimension and changes Gr→ bad!



Algorithm



A =

[0 IA0 A1

], B =

[B1

B0 + A1B1

], C =

[C0 C1

], D = D+C1B1




A2r =

[0 IAr 0 Ar 1

],B2r =

[Br 1

Br 0 + Ar 1Br 1

], C2r =

[Cr 0 Cr 1

],D2r =Dr+Cr 1Br 1


. . . or is it?


],V =

[V1

V2

]→[V1 00 V2

]this often works, but doubles the dimension and changes Gr

→ bad!



Algorithm



A =

[0 IA0 A1

], B =

[B1

B0 + A1B1

], C =

[C0 C1

], D = D+C1B1




A2r =

[0 IAr 0 Ar 1

],B2r =

[Br 1

Br 0 + Ar 1Br 1

], C2r =

[Cr 0 Cr 1

],D2r =Dr+Cr 1Br 1


. . . or is it?


],V =

[V1

V2

]→[V1 00 V2




Algorithm



A =

[0 IA0 A1

], B =

[B1

B0 + A1B1

], C =

[C0 C1

], D = D+C1B1




A2r =

[0 IAr 0 Ar 1

],B2r =

[Br 1

Br 0 + Ar 1Br 1

], C2r =

[Cr 0 Cr 1

],D2r =Dr+Cr 1Br 1

I but A2r is not in companion form! . . . or is it?


],V =

[V1

V2

]→[V1 00 V2




When is Ar in companion form?

I Theorem For A ∈ Rn,n and V2r ,WT2r ∈ Rn,2r with W2rV2r = I let

A2r = W2rAV2r . Then

W2r =

[Wup

WupA

]⇒ A2r =

[0 Ir∗ ∗

].

I partial realization: imW T

→ quadratifying Arnoldi method [Li et al., 2012]

I interpolation (SISO only)span{(A− σ1I )

−T cT , (A− σ2I )−T cT} = span{w ,ATw} with

w = (A− σ1I )−T (A− σ2I )

−T cT

I does not hold for MIMO systems

I does not hold for balanced truncation

I what to do then?






W2r =

[Wup

WupA

]⇒ A2r =

[0 Ir∗ ∗

].

I partial realization: imW T = span[cT ,AT cT ,A2T cT ,A3T cT , . . .]

→ quadratifying Arnoldi method [Li et al., 2012]



w = (A− σ1I )−T (A− σ2I )

−T cT



I what to do then?






W2r =

[Wup

WupA

]⇒ A2r =

[0 Ir∗ ∗

].

I partial realization: imW T = span[cT ,A2T cT , ..,AT cT ,A3T cT , ..]→ quadratifying Arnoldi method [Li et al., 2012]



w = (A− σ1I )−T (A− σ2I )

−T cT



I what to do then?



Post processing

I Theorem For A2r ∈ R2r ,2r and T ∈ R2r ,2r invertible we have

T =

[Tup

TupA2r

]⇔ TA2rT

−1 =

[0 Ir∗ ∗

].

I Is there such an invertible T? e.g. A2r = 0, I

[De Teran et al., 2015]No, iff A2r has an eigenvalue of geometric multiplicity > r .Yes, for almost all Tup, otherwise.

I How to get T?I proof is constructiveI run quadratifying Arnoldi method to the endI random Tup

I T means update: V ← VT−1,W ← TW

I so, A2r can be brought to companion form



Post processing


T =

[Tup

TupA2r

]⇔ TA2rT

−1 =

[0 Ir∗ ∗

].

I Is there such an invertible T? [De Teran et al., 2015]No, iff A2r has an eigenvalue of geometric multiplicity > r .Yes, for almost all Tup, otherwise.

I How to get T?

I proof is constructiveI run quadratifying Arnoldi method to the endI random Tup





Post processing


T =

[Tup

TupA2r

]⇔ TA2rT

−1 =

[0 Ir∗ ∗

].


I How to get T?I proof is constructive

I run quadratifying Arnoldi method to the endI random Tup





Post processing


T =

[Tup

TupA2r

]⇔ TA2rT

−1 =

[0 Ir∗ ∗

].


I How to get T?I proof is constructiveI run quadratifying Arnoldi method to the end

I random Tup





Post processing


T =

[Tup

TupA2r

]⇔ TA2rT

−1 =

[0 Ir∗ ∗

].


I How to get T?I proof is constructiveI run quadratifying Arnoldi method to the endI random Tup





Remarks

about the linearize-reduce-quadratizize method:

I (Ar 0,Ar 1,Br 0,Br 1,Cr 0,Cr 1,Dr ) inherits any properties of Gr

I linearization + partial realization: match 4r + 1 moments at ∞I linearization + IRKA: H2 optimal; interpolates 4r conditions

I linearization + BT: H∞ bound

I this is no projection method, in particular

X

{= 0

small rank6⇒ Xr

{= 0

small rank,X ∈ {Ai r ,Bi r ,Ci r ,Dr}

I D changes (difference to projection approach)

I initial conditions can be dealt with

I A does not have to be in companion form


Counter arguments

Introduction



Counter arguments


Counter arguments

“Only a dead B1 is a good B1”B1 6= 0 is non standard

x = A0x + A1x + B0u + 0u ⇒ xr = Ar 0xr + Ar 1xr + Br 0u + Br 1u

y = C0x + C1x + Du yr = Cr 0xr + Cr 1xr + Dru

I “My u is not differentiable!”→ rewrite state eqn. to

ddt z = d

dt (xr − Br 1u) = Ar 0xr + Ar 1xr + Br 0u

I “My code can’t handle B1!”→ reformulate system as

x1 = A0x1 + A1x1 + B0u

x2 = A0x2 + A1x2 + B1u

y = C0x1 + C1x1 + C1A0x2 + (C0 + C1A1)x2 + (D + C1B1)u

I “I still don’t like it!”→ in random algo. choose Tup s.t. TupB2r = 0Lemma [Meyer and Srinivasan, 1996]Such Tup exists ⇔ rank[B2r ,A2rB2r ] = 2 rank B2r .


Counter arguments




I “My u is not differentiable!”

→ rewrite state eqn. to

ddt z = d



x1 = A0x1 + A1x1 + B0u

x2 = A0x2 + A1x2 + B1u




Counter arguments





ddt z = d



x1 = A0x1 + A1x1 + B0u

x2 = A0x2 + A1x2 + B1u




Counter arguments





ddt z = d


I “My code can’t handle B1!”

→ reformulate system as

x1 = A0x1 + A1x1 + B0u

x2 = A0x2 + A1x2 + B1u




Counter arguments





ddt z = d



x1 = A0x1 + A1x1 + B0u

x2 = A0x2 + A1x2 + B1u




Counter arguments





ddt z = d



x1 = A0x1 + A1x1 + B0u

x2 = A0x2 + A1x2 + B1u


I “I still don’t like it!”

→ in random algo. choose Tup s.t. TupB2r = 0Lemma [Meyer and Srinivasan, 1996]Such Tup exists ⇔ rank[B2r ,A2rB2r ] = 2 rank B2r .


Counter arguments





ddt z = d



x1 = A0x1 + A1x1 + B0u

x2 = A0x2 + A1x2 + B1u




Counter arguments

important directions, twice as large

I “What are the important directions?”I for projection we have x(t) ≈ Vxr (t) and x(t) ≈ V xr (t)I the columns of V are the important directionsI with linearization: similar relation

x(t) ≈ V1

[xr (t)

xr (t)− Br 1u(t)

], x(t) ≈ [V2,B1]

xr (t)xr (t)− Br 1u(t)

u(t)

,V =

[V1

V2

]I So, different important directions for x and x

I “The linearization is twice as large!”→ operations with companion form matrices are cheaper


The End

Introduction



Counter arguments


The End

Conclusions

I discussed the linearize-reduce-quadratizize approach to 2nd ordermodel reduction

I A2r needs to be in companion formsometimes it is naturally,otherwise can be forced (almost always)

I enabled ROMs with H2 optimality or with H∞ error bound

I not a projection method

I generalization to DAE and higher order straight forward for the mostpart

I open: structure preservation

Thanks for your attention!


The End

GeneralizationsI DAE: linearization needs im B1 ⊂ im A2; for quadratification:[

Wup

Wlow

](E ,A)

[V1,V2

]=

([I 0∗ ∗

],

[0 I∗ ∗

])⇔[WupEWupA

] [V1,V2

]= I

I higher order...x = A0x + A1x + A2x + B0u + B1u + B2u

y = C0x + C1x + C2x + D0u + D1u

can be linearized (= reformulated in 1st order) toxzv

=

0 I 00 0 IA0 A1 A2

xzv

+

B2

B1 + A2B2

B0 + (A1 + A22)B2 + A2B1

u

y =[C0 C1 C2

] xzv

+ (D0 + C1B2 + C2B1 + C2A2B2)u + (D1 + C2B2)u

T =

Tup

TupA3r

TupA23r


The End

References

Beattie, C. and Benner, P. (2014).

H2-optimality conditions for structured dynamicalsystems.preprint MPIMD 14/18, Max-Planck-Institut furDynamik komplexer technischer Systeme Magdeburg.

De Teran, F., Dopico, F. M., and Van Dooren, P.

(2015).Matrix polynomials with completely prescribedeigenstructure.SIAM J. Matrix Anal. Appl., 36(1):302–328.

De Teran, F., Dopico, F. M., and Van Dooren, P.

(2016).Constructing strong `-ifications from dual minimalbases.Linear Algebra Appl., 495:344–372.

Lawrence, P. W., Van Barel, M., and Van Dooren, P.

(2016).Backward error analysis of polynomial eigenvalueproblems solved by linearization.SIAM J. Matrix Anal. Appl., 37(1):123–144.

Li, Y.-T., Bai, Z., Lin, W.-W., and Su, Y. (2012).

A structured quasi-Arnoldi procedure for model orderreduction of second-order systems.Linear Algebra Appl., 436(8):2780–2794.

Meyer, D. G. and Srinivasan, S. (1996).

Balancing and model reduction for second-order formlinear systems.IEEE Trans. Automat. Control, 41(11):1632–1644.

Tern, F. D., Dopico, F. M., and Mackey, D. S. (2014).

Spectral equivalence of matrix polynomials and theindex sum theorem.Linear Algebra and its Applications, 459:264 – 333.

Tisseur, F. and Zaballa, I. (2013).

Triangularizing quadratic matrix polynomials.SIAM J. Matrix Anal. Appl., 34(2):312–337.

Wyatt, S. (2012).

Issues in Interpolatory Model Reduction: InexactSolves, Second-order Systems and DAEs.Phd thesis, Virginia Tech, Blacksburg, VA.


The End

papers to cite

I DeTeranDopicaMackeyVandooren, quadratization foreigenvalues[Tern et al., 2014, De Teran et al., 2015,De Teran et al., 2016]

I VanDooren, backward error, lineariztaion[Lawrence et al., 2016]

I quadratifying arnoldi[Li et al., 2012]

I conditions for 2nd order formulation (ref byserkan)[Meyer and Srinivasan, 1996, no pdf]

I 2nd order Schur (Tisseur, Zaballa, ...)[Tisseur and Zaballa, 2013]

I PhDthesis of serkans student[Wyatt, 2012]

I Benner, conditions for 2nd order optimal interpolationpoints[Beattie and Benner, 2014]


Research center Matheon - math.tu-berlin.de · Introduction Second order model reduction Ilinear...

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