Eigenvalue problems NXP PowerPoint template (Title) and ... › casa › meetings › colloquium ›...

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Subject

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MMMM dd, yyyy

Eigenvalue problemsand model order reductionin industry

Joost Rommes [[email protected]]NXP Semiconductors/Corp. I&T/DTF/MathematicsJoint work with Nelson Martins (CEPEL), Gerard Sleijpen (UU)

CASA Colloquium, TU/eJanuary 9, 2008

Subject/Department, Author, MMMM dd, yyyy

CONFIDENTIAL 3

Part I: Introduction and motivation

Part II: Eigenvalue problems and purification

Part III: Dominant poles

Part IV: Model order reduction

Conclusions

2/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008


CONFIDENTIAL 3

Part I: Introduction and motivation

I Large-scale eigenvalue problems arise inI electrical circuit simulationI structural engineeringI power system engineeringI . . .

I Eigenvalues and eigenvectors are needed forI stability analysisI controller designI behavioral modeling

I Relatively few eigenvalues of practical importanceI Practical questions:

I Which eigenvalues are important?I How to compute these eigenvalues efficiently?I How to use these eigenvalues in model order reduction?



CONFIDENTIAL 3

Motivating example I: Pole-zero analysisFrequency response of regulator IC (1000 unknowns). Is the ICstable? Which pole causes peak around 6MHz?

1.010.0

100.01.0k

10.0k100.0k

1.0M10.0M

100.0M1.0G

10.0G

(LOG)

-60.0

-50.0

-40.0

-30.0

-20.0

-10.0

0.0

10.0

20.0(LIN)

Oct 17, 200711:35:07

Bode Plot

Analysis: AC

User: nlv18077 Simulation date: 17-10-2007, 10:21:28

File: /home/nlv18077/test/pstar/stability_ne.sdif

F

DB(VN(VREG))

Note dB(x)= 20 ·10 log(x), e.g. -60 dB = 10−3



CONFIDENTIAL 3

Motivating example II: Model order reduction

Transfer function of power system (66 unknowns). How tocompute reduced order model?

0 5 10 15 20 25 30 35−120

−110

−100

−90

−80

−70

−60

−50

−40

−30

Frequency (rad/s)

Gai

n (d

B)

ExactReduced (series) (k=11)Error (orig − reduced)



CONFIDENTIAL 3

Design of large ICsIterative design process:

1. System design

2. Functional design

3. Circuit design

4. Layout

5. Fabrication

Typical difficulties:

I Increased complexity of systems

I Reduced time-to-market

I Large-scale circuit and layoutsimulations



CONFIDENTIAL 3

Circuit equations

s s

s

s��

sss s

sj1

3

+

−

1

0

2

c2c1

r1l1

I Kirchoff’s CurrentLaw:

∑k ink = 0

I Kirchoff’s VoltageLaw:

∑k∈loop vk = 0

I Branch constitutiveequations:

I Resistor: i = v/RI Capacitor: i = C dv

dtI Inductor: v = L di

dt

Leads to system of Differential Algebraic Equations:

d

dtq(t, x) + j(t, x) = bu(t)



CONFIDENTIAL 3

Types of analysis

d

dtq(t, x) + j(t, x) = bu(t)

I DC analysis: solve j(xDC ) = 0

I Transient analysis: time-domain simulation

I Pole-zero analysis: linearize around xDC

I AC, Periodic Steady State, Harmonic Balance, . . .

Focus on Pole-zero analysis



CONFIDENTIAL 3

LinearizationLet xDC be steady-state solution and

E =∂q

∂x

∣∣∣∣xDC

and A = − ∂j

∂x

∣∣∣∣xDC

Linearization around steady-state gives dynamical system{E x(t) = Ax(t) + bu(t)y(t) = c∗x(t),

where

u(t), y(t) ∈ R, input, output

x(t),b, c ∈ Rn, state, input-to-, -to-output

E ∈ Rn×n capacitance matrix

A ∈ Rn×n conductance matrix



CONFIDENTIAL 3

Frequency domain

Time-domain formulation:{E x(t) = Ax(t) + bu(t)y(t) = c∗x(t)

Laplace transform gives frequency domain formulation:{(sE − A)X (s) = bU(s)Y(s) = c∗X (s)

Transfer function

H(s) =Y(s)

U(s)= c∗(sE − A)−1b

Note that H : C −→ C



CONFIDENTIAL 3

Transfer functionFirst-order SISO dynamical system:{

E x(t) = Ax(t) + bu(t)y(t) = c∗x(t)

with transfer function

H(s) = c∗(sE − A)−1b

Poles are λ ∈ C for which

lims→λ|H(s)| =∞,

or, equivalently,det(λE − A) = 0,

i.e. the eigenvalues of (A,E )



CONFIDENTIAL 3

Pole-zero analysis

Given first-order SISO dynamical system:{E x(t) = Ax(t) + bu(t)y(t) = c∗x(t)

with transfer function

H(s) = c∗(sE − A)−1b

compute

I Poles λ ∈ C: lims→λ |H(s)| =∞ or det(λE − A) = 0

I Zeros z ∈ C: H(z) = 0

Both are (large) eigenvalue problems



CONFIDENTIAL 3

Bode plot

−4 −2 0 2 4 6 8 10

x 105

−6

−4

−2

0

2

4

6

x 106

real

imag

100

102

104

106

108

1010

−80

−70

−60

−50

−40

−30

−20

−10

0

10

20

Frequency (Hz)

Gai

n (d

B)

Poles ( Λ(A,E ) ) Bode plot (ω, |H(iω)|)

I poles λ with real(λ)> 0: unstable solution

I dominant poles cause peaks



CONFIDENTIAL 3

Computational problems

Large-scale eigenvalue problem Ax = λEx

Brute force approach:

1. Compute all eigenvalues (and left and right eigenvectors)

2. Select eigenvalues of interest (positive real part, dominant)

Computational complications:

I Matrices can become very large: n of O(103) up to O(106)

I Dense methods QR/QZ too expensive (O(n3) CPU, memory)

In practice:

I Only few (k � n) specific eigenvalues of practical interest

I How to compute specifically these eigenvalues?

Similar eigenproblems arise in many other areas:

I Fluid dynamics, structural engineering, power systems



CONFIDENTIAL 3

Sparse matrix

0 200 400 600 800 1000

0

100

200

300

400

500

600

700

800

900

1000

nz = 4816

I Few (O(1)) entries per row/column

I Cheap Ax and LU = A



CONFIDENTIAL 3

Part II: Eigenvalue problems and purification

I Generalized eigenvalue problems Ax = λEx

I The Arnoldi method

I B can be singular

I Eigenvalues at infinity

I Purification



CONFIDENTIAL 3

Pole-zero stability analysis

I Generalized eigenproblem

Ax = λEx

I Wanted: eigenvalues with largest real part

Re(λ) > 0→ unstable

I A, E are large, sparse matrices

I E may be singular

I Few (k � n) specific eigenvalues are wanted

I Full space methods like QR and QZ too expensive (O(n3))



CONFIDENTIAL 3

Shift-and-InvertGeneralized eigenproblem

Ax = λEx

Choose shift σ ∈ C:

(A− σE )x = (λ− σ)Ex

and invert:(A− σE )−1Ex = (λ− σ)−1x

With S = (A− σE )−1E :

Ax = λEx ⇐⇒ Sx = λx , λ = (λ− σ)−1

λ(A,E ) near σ are transformed to outside of spectrum Λ(S)



CONFIDENTIAL 3

The Arnoldi method [Arnoldi 1951]

Given S , construct orthonormal basis v1, . . . , vk+1 for

Krylov space Kk+1(S , v1) = span(v1,Sv1, . . . ,Skv1)

1. choose v1 with ‖v1‖2 = 1

2. For i = 1 to k do

2.1 compute w = Svi

2.2 compute hj,i = v∗j w for j = 1, . . . , i2.3 compute w = w − Vh2.4 compute hi+1,i = ‖w‖22.5 set vi+1 = w/hi+1,i



CONFIDENTIAL 3

Orthonormal basis v1, . . . , vk+1 for Kk+1(S , v1):

Vk = [v1, . . . , vk ] ∈ Cn×k

V ∗k Vk = I ,

SVk = VkHk + hk+1,kvk+1eTk

Require for approximate eigenpair (θ, Vky)

S(Vky)− θ(Vky) ⊥ Vk (Ritz-Galerkin)

1. Compute eigenpairs (θi , yi ) of Hk = V ∗k SVk ∈ Ck×k

Hkyi = θiyi

2. Compute Ritz pairs (θi ,Vkyi ) of S and select wanted

3. Check residual norm ‖r‖2 = ‖SVkyi − θiVkyi‖2 = |hk+1,kyi(k)|20/47

NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008


CONFIDENTIAL 3

Eigenvalues at infinity

I One finite, one infinite eigenvalue

A = A−1 =

[1 00 1

],E =

[1 00 0

]⇒ λ(A,E ) = {1,∞}

I Defective, infinite eigenvalue

A = A−1 =

[1 00 1

],E =

[0 10 0

]⇒ λ(A,E ) = {∞}

I Note λ(A,E ) =∞ becomes λ(A−1E ) = 0

I Eigenvalues at ∞ are not of interest



CONFIDENTIAL 3

Numerical problem

I Start Arnoldi with v1 = S21 ∈ range(S2)

I PN : projection on N = ker(S)

I PG : projection on G = ker(S2)\ ker(S)

j ||PN vj ||2 ||PGvj ||21 3.5 · 10−11 7.6 · 10−12

2 7.5 · 10−9 1.2 · 10−10

3 2.1 · 10−7 2.5 · 10−9

4 5.5 · 10−7 5.1 · 10−8

5 1.5 · 10−4 1.1 · 10−6

15 3.1 · 10+7 3.0 · 10−4

One spurious eigenvalue θ = 6.4 · 1010



CONFIDENTIAL 3

Numerical problemI Recall V∞ = N (S) = N (E ) = {x ∈ Rn | Ex = 0}I In exact arithmetic: v1 ∈ R ⇒ vj = Svj−1 ∈ R

However, in finite arithmetic

I Rounding errors (Svj , orth) lead to components in N +G in vj

I Arnoldi can find approximations θi to λ = 0:

(V ∗k SVk)yi = θiyi

I Back transformation λ = θ−1i + σ leads to spurious

eigenvalues

Purification:

1. Remove/prevent spurious eigenvalue approximations

2. Improve wanted eigenpair approximations by removingcomponents in N + G from vj



CONFIDENTIAL 3

Exploiting structure [Bomhof (2000), R. (2007)]

Consider block structured generalized eigenvalue problem[K CCT 0

] [up

]= λ

[M 00 0

] [up

],

with C ∈ Rm×k , and K ,M ∈ Rm×m (n = m + k)Corresponding ordinary eigenproblem is[

S1 0S2 0

] [up

]= λ

[up

], S1 ∈ Rm×m, S2 ∈ Rk×m,

Reduced problem

S1u = λu←→[S1 0S2 0

] [u

λ−1S2u

]= λ

[u

λ−1S2u

]



CONFIDENTIAL 3

Exploiting structure

S1u = λu←→[S1 0S2 0

] [u

λ−1S2u

]= λ

[u

λ−1S2u

]I S , and in particular S1, not available explicitly in generalI but MVs Sv = (A− σE )−1Ev are available:

I LU = A− σE (once)

1. w = Ev2. x = L−1w3. y = U−1x

Use projectors to compute S1u

S1u = [Im 0]

[S1 0S2 0

] [Im0

]u

I dim(gen ker(S)) = k vs. dim(gen ker(S1)) = 0



CONFIDENTIAL 3

0 2 4 6 8 10 12 14 16 18 20−18

−16

−14

−12

−10

−8

−6

−4

Iteration k

log10(Ψ

k+

1)

SS

1

Figure: The size of ‖Ψk+1‖2 = ‖Vk+1Hk − SVk‖2 for Arnoldi applied toS = (A− 60E )−1E , and Arnoldi applied to S1.



CONFIDENTIAL 3

Further improvements

I Implicit restarts [Sorensen 1992]:I Additional purification [Meerbergen/Spence 1995]I Control convergence [R. 05/07]

I Find missed eigenvalues:I Clever shifts [Cliffe/Garratt/Spence 1994, R. 05/07]I Cayley transformations [Cliffe/Garratt/Spence 1994, R. 05/07]

I Very large problems (LU = (A− σB) not feasible):I Jacobi-Davidson methods [Sleijpen/Van der Vorst 1996, R.

05/07]



CONFIDENTIAL 3

Part III: Dominant poles

I Dominant poles

I Dominant Pole Algorithm

I Applications in pole-zero analysis



CONFIDENTIAL 3

Transfer function H(s) = c∗(sE − A)−1b

Can be expressed as

H(s) =n∑

i=1

Ri

s − λi,

where residues Ri are

Ri = (c∗xi )(y∗i b),

and (λi , xi , yi ) are eigentriplets (i = 1, . . . , n)

Axi = λiExi , right eigenpairs

y∗i A = λiy∗i E , left eigenpairs

y∗i Exi = 1, normalization

y∗j Exi = 0 (i 6= j), E -orthogonality



CONFIDENTIAL 3

Dominant poles cause peaks in Bode-plot

H(s) = c∗(sE − A)−1b =n∑

i=1

Ri

s − λiwith Ri = (c∗xi )(y

∗i b)

Bode-plot is graph of (ω, |H(iω)|)I frequency ω ∈ RI magnitude |H(iω)| usually in dB (note dB(x)= 20 ·10 log(x))

Consider pole λ = α + βi with residue R, then

limω→β

H(iω) = limω→β

R

iω − (α + βi)+

n−1∑j=1

Rj

iω − λj

=R

α+ Hn−1(iβ)

Pole λ with large∣∣∣ RRe(λ)

∣∣∣ is dominant and causes peak



CONFIDENTIAL 3

Dominant Pole Algorithm [Martins (1996)]

H(s) = c∗(sE − A)−1b

I Pole λ: lims→λ |H(s)| =∞, or lims→λ1

H(s) = 0

Apply Newton’s Method to 1/H(s):

sk+1 = sk +1

H(sk)

H2(sk)

H ′(sk)

= sk −c∗(skE − A)−1b

c∗(skE − A)−1E (skE − A)−1b

Note dHds = −c∗(skE − A)−1E (skE − A)−1b



CONFIDENTIAL 3

Dominant Pole Algorithm

1: Initial pole estimate s1, tolerance ε� 12: for k = 1, 2, . . . do3: Solve vk ∈ Cn from (skE − A)vk = b4: Solve wk ∈ Cn from (skE − A)∗wk = c5: Compute the new pole estimate

sk+1 = sk −c∗vk

w∗kEvk

6: The pole λ = sk+1 with x = vk/‖vk‖2 and y = wk/‖wk‖has converged if

‖(sk+1E − A)x‖2 < ε

7: end for



CONFIDENTIAL 3

Extensions of DPA

I DPA is a single pole algorithm

I May have very local behavior

I In practice more than one pole neededI Subspace Accelerated DPA [R., Martins (2006)]

I Subspace AccelerationI Several pole selection strategiesI Deflation techniques

I Dominant poles of MIMO systems [R., Martins (2006)]

I Dominant zeros [Pellanda, Martins, R. (2007)]

I Convergence analysis [R., Sleijpen (2006)]



CONFIDENTIAL 3

Pole-zero analysis

I Large nonlinear Regulator IC (n = 1000)

I Designed to deliver constant output voltage

I Turns unstable for certain loads

I Interested in positive poles and dominant poles

I Linearization around DC solution

Results:

Method Time (s) Poles

QR 450 allDPA 41 994 · 103 ± i5.6 · 106

−8.0 · 106 ± i4 · 106

−337 · 103



CONFIDENTIAL 3

Pole-zero analysisFrequency response of circuit (1000 unknowns). Pole994 · 103 ± i5.6 · 106 causes peak around 6MHz.

1.010.0

100.01.0k

10.0k100.0k

1.0M10.0M

100.0M1.0G

10.0G

(LOG)

-60.0

-50.0

-40.0

-30.0

-20.0

-10.0

0.0

10.0

20.0 (LIN)

Oct 17, 200716:38:18

names: A_* --> 3 stability_ne.qr.sdif (AC)Bode Plot + B_* --> 1 stability_ne_dpa_3.cgap (AC)Bode Plot

Analysis: AC

User: nlv18077 Simulation date: 17-10-2007, 10:21:28

File: /home/nlv18077/test/pstar/stability_ne.qr.sdif

F

- y1-axis -

A_DB(VN(VREG))

B_DB



CONFIDENTIAL 3

Part IV: Model order reduction

Given large-scale dynamical system{E x(t) = Ax(t) + bu(t)y(t) = c∗x(t) + du(t)

where x(t),b, c ∈ Rn and E ,A ∈ Rn×n, find{Ek xk(t) = Akxk(t) + bku(t)yk(t) = c∗kxk(t) + du(t)

where xk(t),bk , ck ∈ Rk , Ek ,Ak ∈ Rk×k and

I k � n

I approximation error ‖y − yk‖ small



CONFIDENTIAL 3

Model order reduction

Model order reduction via projection:

1. Construct matrices V ,W ∈ Rn×k whose columns form a basisfor the dominant dynamics

2. Project using V and W :

Ek = W ∗EV , Ak = W ∗AV , bk = W ∗b, ck = V ∗c

Various projection based methods:

I Modal truncation: columns V , W are eigenvectors of (A,E )

I Moment matching: columns V , W are bases for Krylov spaces

I Balanced truncation: V , W part of balancing transformation



CONFIDENTIAL 3

Modal approximationGeneral framework for modal approximation:

H(s) =n∑

i=1

Ri

s − λi=

n∑i=1

(c∗xi )(y∗i b)

s − λi

1. Sort (λi ,Ri ) in decreasing |Ri |/Re(λi ) order

2. Truncate at |Ri |/Re(λi ) < Rmin

3. Project with Yk = [y1, . . . , yk ] and Xk = [x1, . . . , xk ]{˙x = Λk x(t) + bu(t)y(t) = c∗x(t)

Hk(s) =k∑

i=1

Ri

s − λi

Use SADPA [R.,Martins (2006)] to compute dominant poles



CONFIDENTIAL 3

Moment matchingSeries expansion of H(s) = c∗(sE − A)−1b around s0 is

H(s) =∞∑i=0

mi (s − s0)i

with moments mi = c∗G i (s0E − A)−1b and G = (s0E − A)−1EModel order reduction: Match only 2k � n moments:

1. Compute bases V ∈ Rn×k and W ∈ Rn×k for (Arnoldi)

Kk((s0E − A)−1E ,b) and Kk((s0E − A)−∗E ∗, c)

2. Petrov-Galerkin projection gives k-th order system:E x = Ax(t)

+ bu(t)y(t) = c∗x(t)

⇒

(W ∗EV ) ˙x = (W ∗AV )x(t)+ (W ∗b)u(t)

y(t) = (c∗V )x(t)



CONFIDENTIAL 3

Modal approximation vs. moment matching

0 2 4 6 8 10 12 14 16 18 20−90

−80

−70

−60

−50

−40

−30

−20

Frequency (rad/s)

Gai

n (d

B)

SADPA (k=12)Dual Arnoldi (k=30)Orig (n=66)

Figure: Frequency response of complete system (n = 66), modalapproximation (k = 12), and dual Arnoldi model (k = 30).



CONFIDENTIAL 3

Dominant poles: location in complex plane

−16 −14 −12 −10 −8 −6 −4 −2 0 2−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

real

imag

exact polesSADPA (k=12)Dual Arnoldi (k=30)

region of interest

Figure: Pole spectrum of complete system (n = 66), modalapproximation (k = 12), and dual Arnoldi model (k = 30).



CONFIDENTIAL 3

Dominant poles: location in complex plane (zoom)Dominant poles not necessarily at outside of spectrum

−1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

real

imag

exact polesSADPA (k=12)Dual Arnoldi (k=30)

Figure: Pole spectrum (zoom) of complete system (n = 66), modalapproximation (k = 12), and dual Arnoldi model (k = 30).



CONFIDENTIAL 3

Rational Krylov methods [Ruhe (1998)]

General approach:

1. Choose m interpolation points si

2. Construct Vi ,Wi ∈ Cn×ki such that

colspan(Vi ) = Kki ((siE − A)−1E , (siE − A)−1Eb)

colspan(Wi ) = Kki ((siE − A)−∗E ∗, (siE − A)−∗E ∗c)

3. Project with V = [V1, . . . ,Vm] and W = [W1, . . . ,Wm]

Open question:

I How to choose interpolation points si?

I Use imaginary parts of poles as interpolation points

I See also PhD thesis Grimme (1997)



CONFIDENTIAL 3

10−1

100

101

−250

−200

−150

−100

−50

0

50

Frequency (rad/sec)

Gai

n (d

B)

k=70 (RKA)ExactRel Error

Figure: Breathing sphere (n = 17611). Exact transfer function (solid),70th order SOAR [Bai/Su 2005] RKA model (dash) using interpolationpoints based on dominant poles, and relative error (dash-dot).



CONFIDENTIAL 3

Concluding remarks

Robust and effective methods for large eigenproblems:

I Computation of rightmost eigenvalues

I Computation of dominant poles

Applications in model order reduction:

I Stability analysis

I Construction of modal approximations

I Interpolation points for rational Krylov

Generalizations:

I Higher-order systems∑n

i=1 Andn

dtn x = bu

I Multiple input - multiple output (MIMO) systems

I Computation of dominant zeros z : H(z) = 0



CONFIDENTIAL 3

Thank you!

[email protected]

MOOREN I C E!


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