model reduction for dynamical systems lecture 7 · 2019-06-23 · v model reduction for dynamical...

v

Model Reduction for Dynamical Systems Lecture 7 Lihong Feng

Otto-von-Guericke Universität Magdeburg Faculty of Mathematics Summer term 2019

Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory

Magdeburg, Germany [email protected]

https://www.mpi-magdeburg.mpg.de/3668354/mor_ss19

mailto:[email protected]

Outline

• Method based on Pade approximation, explicit moment-matching (AWE)

• Method based on Pade, Pade-type approximation, implicit moment-matching

• Method based on rational interpolation

Transfer function

The transfer function is a function of s.

)(ˆ)( sHsH ≈)(ˆ sHDoes there exist a , such that , but can be computed fastly?

Motivation of AWE method

.0)0(),()(

),()()(

==

+=

xtCxty

tButAxdt

tdxE

Original large-scale system

BAsECsH 1)()( −−=

AWE method [Pillage, Rohrer’ 90] : Asymptotic waveform evaluation method.

)(ˆ sH

Padé approximation

• Rational function:

A rational function is the quotient of two polynomials PN(x) and QM(x) of degree N and M respectively:

bxaforxQxPxR

M

NMN ≤≤= ,

)()(

)(,

• Padé approximation:

Approximates a (analytic) function f(x) by a rational function, and requires that f(x) and its derivatives be continuous at x=0.

The transfer function can be approximated by Padé approximation!

• PN(x) and QM(x):

.1)(

)(2

21

2210

MMM

NNN

xqxqxqxQ

xpxpxppxP

+++=

++++=

• Notice that in QM(x), q0=1, which is without loss of generality. Because, RN,M(x) is not changed when both PN(x) and QM(x) are divided by the same constant.

)()()(, xQ

xPxRM

NMN =

Padé approximation

Padé approximation: .1)(

)(

21

2210

MMM

NNN

xqxqxqxQ

xpxpxppxP

+++=

++++=

Maclaurin expansion: f (x) = f0 + f1x + f2x2 +· · ·+fk x k +· · · ,

The coefficients in PN(x) and QM(x) can be computed by requiring : f(x) and RN,M (x) agree at x=0 and at their derivatives (at x=0 ) up to N+M degree.

This implicates:

∑∞

++=

==−1

, )(:)()(MNj

jjMN xexexfxR

Maclaurin expansion: RN,M(x) = r0 + r1x + r2x2 +· · ·+rk xk +· · · ,

)()()(, xQ

xPxRM

NMN =

How to compute Padé approximation

)(/)()(, xQxPxR MNMN =

∑∞

++=

=−1

, )()(MNj

jjMN xexfxR

∑∑∞

++=

∞

++=

==−11

~)()()()(MNj

jj

MNj

jjMMN xexexQxfxQxP

:0x 000 =− pf

:1x 01101 =−+ pffq

:Nx

011 =−+++ +−−− NNMNMMNM pffqfq

(1)


:1+Nx 011211 =++++ ++−−+− NNMNMMNM ffqfqfq

:2+Nx 0211312 =++++ +++−−+− NNMNMMNM ffqfqfq

:MNx + 01111 =++++ +−++− MNMNNMNM ffqfqfq

(2)

M unknowns and M equations in (2), qis can be obtained by solving (2),

pis can be immediately obtained from (1) without solving equations.


An example:

10,1)( ≤≤+= xxxf

xqxpp

xQxPxR

1

10

1

11,1 1)(

)()(++

==

10 000 =⇒=− ppf

02/11)2/1(0

1

1101

=+×−=⇒=−+

ppffq

2/1/0 121211 −=−=⇒=+ ffqffq

.48/3!3/)0( ;16/1!2/)0(

;2/1)0( ;1)0()3(

3"

2

'10

==−==

====

ffffffff

1)

2)

3) =)(1,1 xR

221

10

2

12,1 1)(

)()(xqxq

xppxQxPxR

+++

==

00

32112

21102

=++=++

ffqfqffqfq1)

=

3

2

2

1

12

01

ff

qq

ffff

−=

25.01

2

1

qq

000 =− pf01101 =−+ pffq

2)

5.02/11

1

1

0

−=+−=

=pp

3) =)(2,1 xR


0 0.2 0.4 0.6 0.8 11

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

x

R11f(x)R12

0 0.2 0.4 0.6 0.8 11

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

x

R11R12R22f(x)


0 0.5 1 1.5 2 2.5 30

5

10

15

20

25

30

35

40

x

f(x)=exp(x)R11R22R33

0 1 2 3 40

10

20

30

40

50

60

70

80

x

f(x)=exp(x)R33

Padé approximation is only accurate around x=0


N, M ?

MOR: AWE based on Padé approximation

MOR: AWE tries to find a Padé approximation of H(s). )(ˆ sH

)()(

)(, xQxPxR

M

NMN =

• How to choose N and M in PN(x) and QM(x)?:

Proposition*: For a fixed value of N+M, the error is smallest when PN(x) and QM(x) have the same degree: N=M or when PN(x) is one degree lower than QM(x), i.e.: N=M-1.

*from the book: John H. Mathews and Kurtis K. Fink, Numerical Methods Using Matlab, 4th Edition, Prentice-Hall Inc. Upper Saddle River, New Jersey, USA 2004.


),,,(,~21

1ndiagZZA λλλ =ΛΛ= −

bsIc

bZsIcZbAAsIcsH

~)(~)(

)()~()(

1

11

11

−

−−

−−

Λ−=

Λ−=

−−=

∑= −

=n

j j

jj

sbc

sH1 1

~~)(

λ

• H(s) is a rational function. • Numerator is a polynomial of degree at most n-1, denominator is a polynomial of degree n.

)()()()( 1111 bAIEsAcbAsEcsH −−−− −+−=−=

,:~ 1EAA −=

)(

)()(

tcxy

tButAxdtdxE

=

+=Given a system

The transfer function is


Therefore, it is natural to take M=r, and N=r-1

)()(

)()()( 1

,1 xQxP

xQxPxR

r

r

M

Nrr

−− ==

rr

rr

r

r

sqsqspspp

sQsPsH

++

+++==

−−−

1

11101

1)()()(ˆ

Computing the coefficients:

Pade approximation requires: The values of H(s) at s=0, and the derivatives of H(s) at s=0 till r-1+r degree should be the same as those of . )(ˆ sH


+++=== ∑∑∞

=

∞

=

2210

00

~~)( smsmmsmsbAcsHi

ii

i

i

i

Derivatives of H(s) at s=0 are the coefficients of Maclaurin series of H(s):

+++=

+++++

=−

− 2210

1

1110 ˆˆˆ

1)(ˆ smsmm

sqsqspsppsH r

r

rr

++=

+++++

−+++ ++

−− 12

122

21

11102

210 1)( r

rr

rrr

rr sesesqsq

spsppsmsmm

i

i

i sbAcbAEsAIcbAsEcsH ∑∞

=

−−−− =−−=−=0

1111 ~~)()()()(

,1,0,~~== ibAcm i

i are the moments of the transfer function.

A~ b~


011110 =++++ −− rrrr mmqmqmq

011211 =++++ +− rrrr mmqmqmq

01222111 =++++ −−−− rrrrrr mmqmqmq

As has been introduced before, qis can be obtained by solving :

pis can be immediately obtained from:

000 =− pm

01101 =−+ pmmq

0111201 =−+++ −−−− rrrr pmmqmq

(3)

rr

rr

r

r

sqsqspspp

sQsPsH

++

+++==

−−−

1

11101

1)()()(ˆ

Ok ?

(4)


Any other possible way? Yes!: Partial Fractions Decomposition:

rr

rr

r

r

sqsqspspp

sQsPsH

++

+++==

−−−

1

11101

1)()()(ˆ

)(ˆ sH is inaccurate at high frequency due to floating point overflow.

r

rr

r

rr

ask

ask

ask

sqsqspsppsH

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

1)(ˆ

2

2

1

11

1

1110

−++

−+

−=

+++++

= −

−−

If we know (approximate poles) and (approximate residues) Then is known and is easily computed.

raaa ˆ,,ˆ,ˆ 21 rkkk ˆ,,ˆ,ˆ21

)(ˆ sH

How to compute the poles and the residues?

and we have known how to compute qis !

raaa ˆ,,ˆ,ˆ 21 are nothing but the roots of rr sqsq +++ 11


What is left? computation of the residues:

11

22

21

11

1 )ˆ

1(ˆ

ˆ)

ˆ1(

ˆ

ˆ)

ˆ1(

ˆ

ˆ−−− −−−−−−−

rr

r

as

ak

as

ak

as

ak

From Pade approximation:

++=

+++++−−++−+−+

+

−−

−−−

1212

22

22

121210

1122

111 ][)ˆ(ˆ)ˆ(ˆ)ˆ(ˆ

rr

rr

rr

rrrr

sese

smsmsmmaskaskask

)ˆˆ

1(ˆ

ˆ)

ˆˆ1(

ˆ

ˆ2

2

21

2

11

1 +++−−+++−rrr

r

as

as

ak

as

as

ak

rkkk ˆ,,ˆ,ˆ21


02

2

1

1

ˆ

ˆ

ˆ

ˆ

ˆ

ˆm

ak

ak

ak

r

r −=+++

1221

221

1

ˆ

ˆ

ˆ

ˆ

ˆ

ˆm

ak

ak

ak

r

r −=+++

12

2

1

1

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ−−=+++ rr

r

rrr m

ak

ak

ak

−−−

−−−

−−−

rr

rr

r

r

aaa

aaaaaa

ˆˆˆ

ˆˆˆˆˆˆ

21

222

21

112

11

−

−−

=

−1

1

0

2

1

ˆ

ˆˆ

rr m

mm

k

kk

the residues can be obtained by solving the above euqations!

r

r

ask

ask

asksH

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ)(ˆ

2

2

1

1

−++

−+

−=

Once raaa ˆ,,ˆ,ˆ 21 and rkkk ˆ,,ˆ,ˆ21 are computed, we have:

(5)

• Why not approximation by *truncated* Taylor expansion?


• The Padé approximant often gives better approximation of the function than truncating its Taylor series, and it may still work where the Taylor series does not converge.

• Round off error or overflow in truncated Taylor expansion is avoided .

• Poles and residues of H(s) can be computed more easily by Pade approximation. • H(s) itself is a rational function.


Implementation of AWE:

2. Compute the roots of the polynomial:

1. Solve (3) to get qis.

rr sqsq +++ 11

3. Solve (5) to get rkkk ˆ,,ˆ,ˆ21

the roots are: (poles) raaa ˆ,,ˆ,ˆ 21

r

r

ask

ask

asksH

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ)(ˆ

2

2

1

1

−++

−+

−=

4. Form the reduced (simpler) transfer function: Much Easier to be computed than H(s)!

In MATLAB, step 2. and 3. are implemented in the function: residue.m


In MATLAB:


2. Get pis from (4).

3. Use *residue(p,q)* in matlab to get the poles and residues:

rkkk ˆ,,ˆ,ˆ21

4. Form the Pade approximation (approximate transfer function):

r

r

ask

ask

asksH

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ)(ˆ

2

2

1

1

−++

−+

−=

;ˆ,,ˆ,ˆ 21 raaa


How to compute the output response y(t) in time domain from ?

Definition of transfer function:

)(ˆ sH

)(/)()( sUsYsH =

If the input is the unit impulse function:

≠=

==0,00,1

)()(tt

ttu δ

1

t

)()(/)()( sYsUsYsH ==

then ∫∞ − ==

01)()( dtetsU stδ

Therefore with impulse input:

∫∫∞+

∞−

∞+

∞−==

j

j

ststj

jdsesH

jdsesY

jty

γ

γ

γ

γ ππ)(

21)(

21)(


r

r

ask

ask

asksH

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ)(ˆ

2

2

1

1

−++

−+

−=

We get:

Replace with: )(ˆ sH

We obtain:

∫∫∞+

∞−

∞+

∞−≈=

j

j

ststj

jdsesH

jdsesH

jty

γ

γ

γ

γ ππ)(ˆ

21)(

21)(

∑=

≈r

i

tai

iekty1

ˆˆ)(

Because:

tai

i

istj

ji

i iekas

kLdseas

kj

ˆ1 ˆ)ˆ

ˆ(

ˆ

ˆ

21

=−

=−

−∞+

∞−∫γ

γπ

Impulse output response in time domain

Numerical instability of AWE

011110 =++++ −− rrrr mmqmqmq


011211 =++++ +− rrrr mmqmqmq

01222111 =++++ −−−− rrrrrr mmqmqmq

(3)

−−

−

221

21

110

rrr

r

r

mmm

mmmmmm

−

−−

=

−

+−

12

1

1

1

r

r

r

r

r

m

mm

q

qq

−

−

−−−

−

−

bA

bA

bAbA

b

c

r

r

~

~~

~~~~

~

12

1

2

1

=

−

−

−−−

−12

2

1

0

r

r

m

m

mmm

bAbEAA 11 ~,~ −− −==

−

−

−−−

−12

2

1

0

r

r

m

m

mmm

−

−

−−−

− bA

bA

bAbA

b

r

r

~~

~~

~~~~

~

12

2

1


)~( 1ξλξ

=A

ξ

bAx ~~1 =

bAx ~~22 =

bAx ~~66 = bAx ~~7

7 =

bAx ~~88 =

bA ~~9

bA ~~10

, i=1,2,... run parallel to soon ! ξ

Usually, after i=8, all will in the same direction with .

bAx ii

~~=

bAx ii

~=

ξ


.~ of rseigenvecto are ,~2211 Ab inn ξξαξαξα +++=

nm

nmmm AAAbA ξαξαξα ~~~~~

2211 +++=

nmnn

mm ξλαξλαξλα +++= 222111

++

+= n

m

nn

mm ξ

λλ

ααξ

λλ

ααξλα

112

1

2

1

2111

∞→m

1111 :)(~~ σξξλα =→ mmbA

)( 21 nλλλ >>>

121 to~~ changesslowly error off-round close,not are and If σζλλ bAm


1~~ σξ→bAm

1ξbAm ~~

1ξ

bAm ~~

0>σ

0<σ


Notice: bAcm ii

~~=

.~ , , ofn informatio econtain thonly ~~ means this

, with line same on thealmost be will~~ all ,8 when examples,many For

11111

1

ξλξξλ

ξ

=

>

AbA

bAii

i

.inaccurate are residues and poles computed thesuch that errors, off-round todue accurately computed becannot 8, Therefore >imi

moments! more usingby improved becannot )(ˆ

ofaccuracy they,numericall ),(ˆ accurate more tolead willand moments, more

match will more compute tomoments more employing ly,heoreticalAlthough t

sH

sH

qi

Conclusion:


L. T. Pillage, R. A. Rohrer, Asymptotic Waveform Evaluation for Timing Analysis, IEEE Transactions on computer-aided design, Vol. 9, No. 4, 1990.

Outline

• Method based on Pade, Pade-type approximation, implicit moment-matching

• Method based on rational interpolation

Angle between two vectors:

Dot product of two vectors u, v is defined as:

Inner products in : uvvu T=⋅

Preliminaries

Orthogonality of two vectors: 00||||||||

0)cos(22

=⇔=⇔= vuvu

vu TT

θ

22 ||||||||)cos( vuvu θ=⋅

nR

ab

Pb

acc ⊥:

• Orthogonalization of two vectors

If Pb is the projection of b onto a, then c=b-Pb is orthogonal to a.

How to compute c?

maPb = (m is a scalar)

amabPbbc ⊥−=−=

0)( =−mabaT

aabam T

T=

Finally: aaababc T

T−=

nRba ∈,

},{},{ acspanabspan =An important information:

Preliminaries

• Orthogonalization of a vector b to the subspace spanned by a group of orthogonal vectors laaa ,,, 21

b

Pb

la2a 1a

c=b-Pb

c=b-Pb

ll amamamPb +++= 2211 iac ⊥

0)( 2211 =−−−− llTi amamamba

0=jTi aa

and

iTi

Ti

i aabam =

ll

Tl

Tl

T

T

T

Ta

aabaa

aabaa

aababc −−−−= 2

22

21

11

1

},,,,{},,,,{ 2121 ll aaacspanaaabspan =

Preliminaries

Step 1:

used to orthogonalize a group of vectors mbbb ,,, 21

• Gram-Schmidt process:

21, bb 111

2122

~b

bbbbbb T

T−=

direction in b1

Step i:

321 ,~

, bbb2

22

321

11

3133

~~~

~~

bbbbbb

bbbbbb T

T

T

T−−= direction in

Step 2:

ibbbb ,~

,~

, 321

2~b

111

12

22

21

11

1 ~~~

~~

~~~

~−

−−

−−−−−= ii

Ti

iT

iT

iT

Ti

T

ii bbbbbb

bbbbb

bbbbbb

direction in

1~−ib

Preliminaries

Gram-Schmidt process:

for i=2,3, ..., m

111

12

22

21

11

1 ~~~

~~

~~~

~−

−−

−−−−−= ii

Ti

iT

iT

iT

Ti

T

ii bbbbbb

bbbbb

bbbbbb

end

What is the relation between and ? mbbb ,,, 21 mbbb ~,,~, 21

}~,,~,{},,,{ 2121 mm bbbspanbbbspan =

Preliminaries

Gram-Schmidt process:

for i=2,3, ..., m

111

12

22

21

11

1 ~~~

~~~~

~~−

−−

−−−−−= ii

Ti

iT

iT

iT

Ti

T

ii bbbbbb

bbbbb

bbbbbb

end

It is accurate in accurate arithmetic, brings errors in finite arithmetic, not exactly orthogonal

Computation with computers is finite arithmetic!

for i=2,3, ..., m

for j=1,2, ..., i-1

jj

Tj

iTj

ii bbbbb

bb ~~~~~

~~−=

end end

Any difference, and what difference?

Numerically stable.

Preliminaries

Modified Gram-Schmidt process:

ii bb =~

Usually the vectors are required to be orthonormalized, so that there will be no overflow in the computation with computers.

Modified Gram-Schmidt process :

for i=2,3, ..., m

for j=1,2, ..., i-1

jj

Tj

iTj

ii bbb

bbbb −=

end

end

|||| 1

11 b

bb =

|||| i

ii b

bb =

What if is zero or close to zero? What does it mean?

|||| ib

Preliminaries

Modified Gram-Schmidt process with deflation :

||||/ 111 bbb =

for i=2,3, ..., m

for j=1,2, ..., i-1

jj

Tj

iTj

ii bbb

bbbb −=

end

end

bii bb ε/=If

|||| ib b=ε

else delete ib

end

deflation

Preliminaries

tolb ≥ε

},,,,{span),( 12 rArAArrrAK pp

−=

• Arnoldi algorithm:

It computes an orthonormal basis of the Krylov subspace: qvvv ,, 21

The core in Arnoldi algorithm is the Modified Gram-Schmidt process.

i.e. ),(},,,{span 21 rAKvvv pq =

Preliminaries

Arnoldi algorithm

||||/1 rrv =

},,,,{),( 12 rArAArrrAK pp

−=

for i=2,3, ..., p

for j=1,2, ..., i-1

jj

Tj

Tj vvv

wvww −=

end

1−= iAvw

|||| ww =εIf

wi wv ε/=else

stop

end end

Why?

pqrAKvvv pq ≤= ),,(},,,{span 21

It is clear:

Preliminaries

tolw ≥ε

Arnoldi algorithm

21 ||||/ rrv =for i=2,3, ..., p

for j=1,2, ..., i-1

1)( )( 1,

1,

=−=

=

−

−

jTjjijii

iTjij

vvvhvv

vvh

end

1−= ii Avv

21, |||| iii vh =−

If

1,/ −= iiii hvvelse

stop

end end

Preliminaries

tolh ii ≥−1,

iii

iii

i

HVAV

pivvRH~

,,, and ,~matrix Hessenberg :output

1

1,1

+

+

=

<∈

Generalized minimal residual method (GMRES)

Preliminaries

GMRES is an iterative method of solving a nonsymmetric system of linear equations. The method approximates the solution by the vector in a Krylov subspace with minimal residual. The Arnoldi algorithm is used to find this vector.

00

0

residual initial guess initial

Axbrx

bAx

−=

=

0) when ,( },,,,,{),( 000001

02

000 ==−== − xbrAxbrrArAArrrAK pp

2

0

||||minarg , :residual theof norm-2 theminimizes such that ~,~ Find

rxAybrxKxxxx

yi

iiiii

=−=∈+=

200100 ||||/ i.e. , is lizedorthonorma be or tofirst vect The rrvAxbr =−=

)(ˆ)()(/

tCVzytBuWtAVzWdtEVdzW TTT

=+=Vzx ≈

)(/)()(/)()( susCxsusysH ==

i

i

i sssBsAC

BAEsIEAEsssCBAEsEssEC

BAsECsH

)()(~))(~(

)()))(((

)( )()(

00

00

10

1100

100

1

−=

−+−−=

−+−=

−=

∑∞

=

−−−

−

−

Implicit moment-matching(Pade, Pade-type approximation)

)(

)()(

tCxy

tButAxdtdxE

=

+=

Recall that for Petrov-Galerkin projection based MOR:

By definition:

Taylor expansion at s0:

BAEssBEAEssA 100

100 )()(~ ,)()(~ where −− −=−−=

Definition ( moments are defined for any expansion point )

function. transfer theof moments thecalled are ,...,0 ), system SISOfor )(~)(~ ( )(~)(~)( 00000 == isbsAcsBsACsM ii

i

∞<0s


Recall that in AWE method, the moments

are computed explicitly. It is numerically unstable.

12 ,...,0 ,)0(~)0(~−== ribAcM i

i

:intoeither divided becan ,...,0 ,)(~)(~ that Observe 00 == tsbsAcM tt

(5) ,...1,0 ),(~ and

(4) ...1,0 ),(~)(~

0

00

=

=

jsAc

isbsAj

i

Instead of explicitly computing the terms in (4)(5) or in (6)(7) as is done by AWE method, we compute the basis of

intoor

(7) and(6) ...1,0 ),(~)(~

00

ctsbsAt =

(9) }))(~(,,)(~,span{)W~( range

(8) })(~)(~,),(~)(~),(~span{range(V)1

00

001

000

TpTTTT

p

csAcsAc

sbsAsbsAsb−

−

=

=

Implicit moment-matching(Pade approximation)

How to compute W, V? [Feldmann, Freund ’95]

∆== :),,(diag~1 r

T ddVW

(9) )),(~(}))(~(,,)(~,span{)~( range

(8) ))(~ ,)(~(})(~)(~,),(~)(~),(~span{)range(

01

00

00001

000

TTp

TpTTTT

pp

csAKcsAcsAcW

sbsAKsbsAsbsAsbV

==

==−

−

W, V span two Krylov subspaces, so that they can be simultaneously computed by (Band) Lanczos algorithm, such that .

Recall that

TTT

T

T

cVbEsAW

EVEsAWT

VW

−

−−

−−

∆=

−∆=

−∆=

η

ρ 10

1

10

1

)(~)(~

,~][Freund'03 are algorithm Lanczos Band of outputs The

Implicit moment-matching (Pade approximation)

Applying Petrov-Galerkin projection (using ) to the transformed system

(10) ),(

),()()()()( 10

10

10

tcxy

tbuEsAtAxEsAdtdxEEsA

=

−+−=− −−−

One gets the reduced order model (ROM)

(11) ),(

),()(~)()(~)(~ 10

10

10

tcVzy

tbuEsAWtAVzEsAWdtdzEVEsAW TTT

=

−+−=− −−−

By studying the ROM in (11) [Freund ’03]

. and ,~)( with system original

the toprojectionGalerkin -Petrov applying toequivalent is (11)-(10) that seecan one

0 VWEsAW T−−=

VW ,~

Implicit moment-matching (Pade approximation)

.)( ofion approximat Pade a is )(ˆ Therefore,

.12,...,1,0 ),(ˆ)(

i.e. , )( of moments 2first match the (11)in ROM theoffunction transfer theof moments 2first the(10),in system SISO afor then ,),,diag(~satisfy

and (8)(9),in subspace theof basis theare RV ,W~ of coulumns theIf

00

1

sHsH

pisMsM

sHppddVW

ii

rT

r

−==

=

∈

Theorem 1 [Feldmann, Freund ’95]

Drawbacks of Lanczos method of computing the projection matrices: • The ROM computed by W, V may be unstable, there are eigenvalues with positive real parts.

Actually the ROM in (11) can be implicitly derived using the outputs of the Band Lanczos algorithm[Freund ’03] : (It can be easily proved that (12) is exactly (11))

(12) ),(

),()()( 0

tzy

tutzTsIdtdzT

T∆=

++=

η

ρ

outputs. and inputs are There .or in columns theofnumber thely,equivalentor model, reduced theoforder theis Here .)( oft approximan Padematrix

a isit and ,)( of moments r/n r/nfirst least theat matches )(ˆthen

,~satisfy and ,algorithm) Lanczosby generatedy necessarilnot are(they

(8)(9)in subspace theof basisany are V ,W~ if shown that isit '00], [FreundIn

IO

OI

T

nnWVrsH

sHsH

IVW

+

=

Number of moments matched for MIMO systems

Implicit moment-matching(Pade approximation)

It is immediately seen from the above statement that for SISO systems, there are at least 2r moments matched. It is in agreement with Theorem 1.

Implicit moment-matching(Pade-type approximation)

A passive (therefore stable) ROM can be obtained by using W=V.

))(~ ,)(~(})(~)(~,),(~)(~),(~span{)range( 00001

000 sbsAKsbsAsbsAsbV pp == −

algorithm. Arnoldiby computed becan V

.1,...,1,0 ),(ˆ)(

,)(ˆby matched are )( of moments first the(10),in system SISO afor then

, (8),in subspace Krylov theof basis orthogonalan constitute in columns theIf

00 −==

=

pisMsM

sHsHp

IVVV

ii

T

Theorem 2 [Odabasioglu, et.al ’97, Freund ‘03]

).( ofion approximat type-Pade a is (s)ˆ sHH

condition. meet this simulationcircuit from modelsMany ’97]. et.al lu,[Odabasiog definite-semi negative is and

definite,-semi positive is if passive, is model reduced resulting TheAA

EET

T

+

+

Recall:

∑∑∞

=

∞

=

− =−=⇒−=0

000

0001 )(~)(~)()())((~)(~)()()()(

i

ii

i

ii sbsAsgsusssbsAsxsbuAsEsx

))()(:)(( )(~)(~)()( 0

1

000 susssgsbsAsgsx i

i

p

i

ii −=≈∑

−

=

Vzx ≈


?n rather tha , using why ,projectionGalerkin -Petrov with matching-momentFor WzxVzx ≈≈

i

isssbsAcbAsEcsH )()(~)(~ )()( 0

000

1 −=−= ∑∞

=

−

))(~ ,)(~(} )(~)(~,),(~)(~),(~ span{)range( 00100000 sbsAKsbsAsbsAsbV pp

−==

The ROM is obtained by Galerkin projection onto the original system.

(11) )(ˆ)()(/

tcVzytbuVtAVzVdtEVdzV TTT

=+=

nonzero.or zero aschosen becan :point expansion The 00 σ+= sss


Expansion point

preferred. is 0spointexpansion nonzero a then wide,is rangefrequency ginterestin theIf

Implicit moment-matching(rational interpolation)

• A single expansion point is used in the method based on Pade approximation. Multiple expansion points are used in rational interpolation method.

• Rational interpolation views computing the transfer function as solving linear systems.

bTc xAsExbAsEAsEAsEcbAsEcsH )())(()()()( 111 −=−−−=−= −−−

where bxAsEcxAsE b

Tc

T =−=− )( ,)(

Applying a preconditioner to each of the linear systems,

bEsAxAsEAEscEsAxAsEAEs bTT

cTT 1

01

000 )()()( ,)()()( −−−− −=−−−=−−

If using Krylov-subspace iterative methods to solve the preconditioned linear systems, we have (this is the property of Krylov-subpace iterative methods, e.g. CG, GMRES, MINRES. etc..) [Grimme ’97] :


))(),()(( 10

10

0 bAEsAsEAEsKxxx qbqbb

−− −−−+∈≈

))(,)()(( 000 TTTT

qcqcc cAEsAsEAEsKxxx −− −−−+∈≈

))(,)(())(),()(( 10

10

10

10 bAEsEAEsKbAEsAsEAEsK qq

−−−− −−=−−−

IEAEsssAsEAEs +−−=−− −− 100

10 ))(()()( Since

).,()),(( , nonzero and g vector G,matrix any For rGKrIGK qq =+η

η

))(,)(())(,)()(( 0000TTTT

qTTTT

q cAEsEAEsKcAEsAsEAEsK −−−− −−=−−−

Lemma 2.2 [Grimme ’97] Krylov subspace is shift-invariant


(12) ))(,)(()range(such that , Compute

10

10 bEsAEEsAKV

V

p−− −−=

(13) ))(,)(()range(such that , Compute

00TTTT

p cEsAEEsAKWW

−− −−=

Then cccbbb WzxxVzxx =≈=≈ ˆ ,ˆ

bTTT

cbTT

cbTc zAVWEVsWzVzAsEWzxAsExsHsH )()(ˆ)(ˆ)(ˆ)( −=−=−=≈

Therefore the reduced matrices are: ,ˆ ,ˆ AVWAEVWE TT ==

.12,,1,0 ),(ˆ)( i.e.),(ˆby matched are )( of moments 2then

(12)(13),satisfy , matrices projection theif systems, MIMO and SISOboth For

00 −== pisMsMsHsHp

WV

ii

Theorem ([Grimme ’97])


Instead of using a single expansion point, multiple expansion points are used in rational interpolation method, such that

(14) ))(,)(()range( 11

0bEsAEEsAKV iip

k

i i

−−

=−−=

(15) ))(,)(()range(0

TTi

TTip

k

icEsAEEsAKW

i

−−

=−−=

.,,0;12,,1,0 ,)(ˆ)(

i.e.,point expansion each at )(ˆby matched are )( of moments 2then

(14)(15),satisfy , matrices projection theif systems, MIMO and SISOboth For

kjpisMsM

ssHsHp

WV

jjiji

jj

=−==

Theorem ([Grimme ’97])

. that requirednot isit method,ion interpolat rational theofproperty matching-moment For the

IVW T =

Remark


Computation of V, W in (14)(15)

• Rational Arnoldi method or rational Lanczos method. [Grimme ’97]

• Repeated modified Gram-Schmidt algorithm (Repeated Arnoldi algorithm).

How to decide the expansion points?

• Some heuristic methods • Using error estimation and a greedy algorithm. • Locally optimal algorithm: IRKA.



Using error estimation and a greedy algorithm

Error estimation, e.g. , :

|ˆ|)(/||||||||||ˆ|| :ˆ and between error :errorOutput

)(/||||||)(ˆ)(|| :ˆ and between error :error vector State||)(ˆ)(|||||| Residual

min222

min22

22

prdudupr rxAsErryyyy

AsErsxsxxxsxAsEBr

+−≤−

−≤−−−=

σ

σ

)(/||||||||||)(||

||ˆ)(()(||

||ˆ)()()(||||ˆ)(||||ˆ||

min2221

21

211

21

2

AsErrAsExAsEBAsE

xAsEAsEBAsExBAsExx

−=−≤

−−−=

−−−−=−−=−

−

−

−−−

σ

)(s∆

[Feng, Antoulas,Benner ’17])



A greedy algorithm: Selection of expansion points

WHILEEND);ˆ(

);(maxargˆ];,[orth ];,[orth

))(,)(()range( );)(,)(()range(

;ˆ;1 WHILE

interest of range thecovering of samples ofset a :[]; []; ;1 ;ˆ :pointexpansion Initial

11

0

s

ssWWWVVV

CAEsEAEsKWBAEsEAEsKVss

ii

sWViss

trains

ii

TTi

TTipiiipi

i

tol

train

∆=

∆===

−−=−−=

=+=

>Ξ

==−==

Ξ∈

−−−−

ε

εε Arnoldi algorithm is used.

orth: M-G-S


.,,1for )ˆ()ˆ(ˆ and )ˆ()ˆ(ˆ

:,,1,ˆat derivativefirst its and

)(both esinterpolat )(ˆThen .,,1,ˆat poles simple has )(ˆ that suppose and

||~||min||ˆ||

problemreduction model optimal for the dimension ofminimizer local a beˆ)ˆ(ˆ)(ˆlet ,)()( system SISO stable aGiven '08] alet [Gugercin 3.4. Theorem

''

)~dim(

21

1

2

stable :~2

riHHHH

ri

sHsHrisH

HHHH

HrbAsIcsH

bAsIcsH

iiii

i

i

HrHHH

=−=−−=−

=

=

−=−

−=

−=

=

−

−

λλλλ

λ

λ



How to decide the expansion points? • Locally optimal algorithm: IRKA for both SISO and MIMO system:

rT

rrT

rrT

r

rT

rrr

rT

rT

r

rrrr

rrTT

rii

ii

rT

rrT

r

j

oldjj

rT

rrrrT

rT

r

rrrr

rr

i

CVCBWBAVWAEVWE

WVWW

cCAEcCAEWbBAEbBAEVWV

ccCbbBYCCYBB

yyYri

riyAE

AVWAEVWE

WVWWcCAEcCAEWbBAEbBAEVWV

ccbb

tolri

====

=

−−=

−−=

====

==−=

==−

==

>

−

=−−=

−−=

=

−

−

=

−

ˆ,ˆ,ˆˆ .4

.)( (f)

.}~)(,,~)span{()Ran(

},~)( , ,~)span{()Ran( so and Update(e)

)~,,~(~),~,~(~,ˆ~,ˆ~ (d)

),,( ;,,1for ˆLet (c)

,,1,0)ˆˆˆ( Solve (b)

ˆ,ˆ (a)

tol)max( WHILE.3

.)( .}~)(,,~)span{()Ran(

},~)( , ,~)span{()Ran( that so and Choose 2.

.~,~,~,~ directions initial Choose

. fixed afor under closed,,,1for , ofselection initialan Make 1.(IRKA) Algorithm Krylov Rational IterativeAn

1

T-1

T-1

1-r1

1-1

11

1

r,1,j

1T-1

T-1

1-r1

1-1

11

σσ

σσ

λσ

λ

σσσ

σσ

σσ

σ

conjugation


}~)(,],~)Im[(],~)Re[(,,~)span{(

}~)(,,~)(,~)(,,~)span{(

thatSo]}.~)Im[(],~)span{Re[(}~)(,~)span{(

Therefore.)( :matrix complex any for since

~)(~)(~)(

have we, riablecomplex vaany For matrices as computed becan , then n,conjugatiounder closed are If

11111

11111

1111

1111-

111

bAEbAEbAEbAE

bAEbAEbAEbAE

bAEbAEbAEbAE

MMMMMMIM

bAEbAEbAE

.why?realVW

rii

rii

iiii

iii

i

i

−−−−

−−−−

−−−−

−−−

−−−

−−−−

=−−−−

−−=−−

=⇒⋅==

−=−=−

σσσσ

σσσσ

σσσσ

σσσ

σσ


• Locally optimal algorithm: IRKA for SISO system

• From Theorem 3.4, IRKA obtains a reduced model satisfies the local optimal necessary conditions in Theorem 3.4 in [Gugercin et al. ’08].

Upon convergence, Algorithm IRKA leads to:

.,,1for )ˆ()ˆ(ˆ and )ˆ()ˆ(ˆ ' riHHHH iiii =−=−−=− λλλλ


References Pade- and Pade-type approximation; [Pillage, Rohrer’ 90] L. T. Pillage and R. A. Rohrer, “Asymptotic waveform evaluation for timing analysis,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst., vol. 9, no. 4, pp. 352–366, Apr. 1990. [Freund ‘00] R. W. Freund, Krylov-subspace methods for reduced-order modeling in circuit simulation, Journal of Computational and Applied Mathematics, 123: 395-421, 2000. [Freund ‘03] R. W. Freund, Model reduction methods based on Krylov subspaces, Acta Numerica, 12: 267-319, 2003. [Feldmann, Freund ‘95] P. Feldmann and R. W. Freund, Efficient linear circuit analysis by Pade approximation via the Lanczos process. IEEE Transactions on Computer-aided design of Integrated Circuits and Systems. 14(5), 1995. [Odabasioglu et al ‘97] Odabasioglu, M. Celik, L. T. Pileggi, PRIMA: passive reduced-order interconnect macromodeling algorithm. IEEE/ACM ICCAD, 58-65, 1997. Rational interpolation: [Grimme ‘97] E. J. Grimme, Krylov projection methods for model reduction, PhD thesis, University of Illinois at Urbana-Champaign, 1994. IRKA: [Gugercin et al ‘08] S. Gugercin, A. C. Antoulas, C. Beattie, H_2 Model reduction for large-scale linear dynamical systems. SIAM J. Matrix Analysis, 30(2): 609-638, 2008. Error estimation: [Feng, Benner, Antoulas ’14] L. Feng, A. C. Antoulas, and P. Benner, Some a Posteriori Error Bounds for Reduced Order Modelling of (Non-)Parametrized Linear Systems, Preprint MPIMD/15-17, Max Planck Institute Magdeburg, Germany, 2015.

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