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Name
Subject
Project
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
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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?
3/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
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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
4/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
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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)
5/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
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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
6/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
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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)
7/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
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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
8/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
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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
9/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
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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
10/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
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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 )
11/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
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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
12/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
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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
13/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
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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
14/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
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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
15/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
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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
16/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
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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))
17/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
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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)
18/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
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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
19/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
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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
Subject/Department, Author, MMMM dd, yyyy
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
21/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
Subject/Department, Author, MMMM dd, yyyy
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
22/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
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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
23/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
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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
]
24/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
Subject/Department, Author, MMMM dd, yyyy
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
25/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
Subject/Department, Author, MMMM dd, yyyy
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.
26/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
Subject/Department, Author, MMMM dd, yyyy
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]
27/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
Subject/Department, Author, MMMM dd, yyyy
CONFIDENTIAL 3
Part III: Dominant poles
I Dominant poles
I Dominant Pole Algorithm
I Applications in pole-zero analysis
28/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
Subject/Department, Author, MMMM dd, yyyy
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
29/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
Subject/Department, Author, MMMM dd, yyyy
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
30/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
Subject/Department, Author, MMMM dd, yyyy
CONFIDENTIAL 3
0 2 4 6 8 10 12 14 16 18 20−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Frequency (rad/s)
Gai
n (d
B)
ResponseDominant poles
Figure: Bode plot (ω, |H(iω)|). Pole λj dominant if|Rj |
|Re(λj )|large.
31/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
Subject/Department, Author, MMMM dd, yyyy
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
32/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
Subject/Department, Author, MMMM dd, yyyy
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
33/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
Subject/Department, Author, MMMM dd, yyyy
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)]
34/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
Subject/Department, Author, MMMM dd, yyyy
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
35/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
Subject/Department, Author, MMMM dd, yyyy
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
36/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
Subject/Department, Author, MMMM dd, yyyy
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
37/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
Subject/Department, Author, MMMM dd, yyyy
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
38/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
Subject/Department, Author, MMMM dd, yyyy
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
39/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
Subject/Department, Author, MMMM dd, yyyy
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)
40/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
Subject/Department, Author, MMMM dd, yyyy
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).
41/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
Subject/Department, Author, MMMM dd, yyyy
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).
42/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
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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).
43/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
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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)
44/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
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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).
45/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
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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
46/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008
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Thank you!
MOOREN I C E!
47/47NXP Semiconductors Corp. I&T/DTF, Joost Rommes, January 9, 2008