NXP PowerPoint template (Title) Template for presentations (Subtitle) Name Model order reduction via dominant poles Joost Rommes [[email protected]] NXP Semiconductors/Corp. I&T/DTF/Mathematics Joint work with Nelson Martins (CEPEL), Gerard Sleijpen (UU) Symposium on recent advances in MOR November 23, 2007
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NXP PowerPoint template (Title)Template for presentations (Subtitle)
Name
Subject
Project
MMMM dd, yyyy
Model order reductionvia dominant poles
Joost Rommes [[email protected]]NXP Semiconductors/Corp. I&T/DTF/MathematicsJoint work with Nelson Martins (CEPEL), Gerard Sleijpen (UU)
Symposium on recent advances in MORNovember 23, 2007
Subject/Department, Author, MMMM dd, yyyy
CONFIDENTIAL 3
Introduction
Eigenvalue problems and applications
Dynamical systems and transfer functions
Dominant poles
Dominant Pole Algorithm
Applications
Conclusions
2/38NXP Semiconductors Corp. I&T/DTF, Joost Rommes, November 23, 2007
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Introduction
I Large-scale dynamical systems arise inI electrical circuit simulationI structural engineeringI power system engineeringI . . .
I Transfer function and properties are used forI simulationI behavioral modelingI stability analysisI controller design
I Relatively few transfer function poles of practical importanceI Three key questions:
I Which poles are important (dominant)?I How to compute these poles efficiently?I How to use these poles in model order reduction?
3/38NXP Semiconductors Corp. I&T/DTF, Joost Rommes, November 23, 2007
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Motivating example I: Pole-zero analysisFrequency response of regulator IC (1000 unknowns). Which polecauses peak around 6MHz?
15/38NXP Semiconductors Corp. I&T/DTF, Joost Rommes, November 23, 2007
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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
16/38NXP Semiconductors Corp. I&T/DTF, Joost Rommes, November 23, 2007
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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
17/38NXP Semiconductors Corp. I&T/DTF, Joost Rommes, November 23, 2007
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CONFIDENTIAL 3
Twosided Rayleigh quotient iteration
Note that with v ≡ vk and w ≡ wk
sk+1 = sk −c∗(skE − A)−1b
w∗Ev
= skw∗Ev
w∗Ev− c∗(skE − A)−1(skE − A)(skE − A)−1b
w∗Ev
=w∗Av
w∗Ev
Step DPA Twosided RQI
3 solve (skE − A)vk = b solve (skE − A)vk = Evk−1
4 solve (skE − A)∗wk = c solve (skE − A)∗wk = E ∗wk−1
Original work on twosided RQI [Ostrowski (1958), Parlett (1974)]
18/38NXP Semiconductors Corp. I&T/DTF, Joost Rommes, November 23, 2007
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CONFIDENTIAL 3
Convergence behavior: DPA vs. RQI
Figure: λ = −0.47 + 8.9i : DPA: red + yellow, RQI: red + light blue.
19/38NXP Semiconductors Corp. I&T/DTF, Joost Rommes, November 23, 2007
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Convergence behavior: DPA vs. RQI
Typically, with initial pole guess s0,I DPA converges to dominant pole closest to s0
I with ∠(c, x) and ∠(b, y) smallI i.e., large |R| with R = (c∗x)(y∗b)
I Quadratic rate of convergence
I See also [R., Sleijpen (2006)]
while
I RQI converges to pole closest to s0I Originally intended for refinement of eigenpairs
I Cubic rate of convergence
I See also [Ostrowski (1958), Parlett (1974)]
20/38NXP Semiconductors Corp. I&T/DTF, Joost Rommes, November 23, 2007
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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
21/38NXP Semiconductors Corp. I&T/DTF, Joost Rommes, November 23, 2007
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CONFIDENTIAL 3
Subspace acceleration and selectionI Keep approximations vk and wk in search spaces V and W
I Petrov-Galerkin leads to projected eigenproblem
Ax = θE x,
y∗E = θy∗A
where E = W ∗EV ∈ Ck×k and A = W ∗AV ∈ Ck×k
I Gives k approximations (θi , xi = V xi , yi = W yi ) in iter k
I Select approximation with largest residue as next shift:
sk+1 = argmaxi
∣∣∣∣(c∗xi )(y∗i b)
Re(θi )
∣∣∣∣I Similarities with twosided Jacobi-Davidson ([Hochstenbach
(2003), Stathopoulos (2002)])
22/38NXP Semiconductors Corp. I&T/DTF, Joost Rommes, November 23, 2007
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Deflation for H(s) = c∗(sE − A)−1b
I Triplet (λ, x, y): Ax = λEx and y∗A = λy∗E
I New search spaces: V ⊥ E ∗y and W ⊥ Ex
I Usual deflation (every iteration):
vk ← (I − xy∗E )vk
wk ← (I − yx∗E ∗)wk
I More efficient: deflate only once
bd ← (I − Exy∗)b ⇒ vk = (skE − A)−1bd ⊥ E ∗y
cd ← (I − E ∗yx∗)c ⇒ wk = (skE − A)−∗cd ⊥ Ex
I Note that y∗bd = c∗dx = 0
23/38NXP Semiconductors Corp. I&T/DTF, Joost Rommes, November 23, 2007
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Deflation of dominant poles removes peaks
0 2 4 6 8 10 12 14 16 18 20−90
−80
−70
−60
−50
−40
−30
Frequency (rad/sec)
Gain
(dB
)
Bodeplot
Exact
Modal Equiv.
Figure: Exact transfer function (solid) with removal of dominant poles:−0.467± 8.96i (square), −0.297± 6.96i (asterisk), −0.0649 (diamond),and −0.249± 3.69i (circle).
24/38NXP Semiconductors Corp. I&T/DTF, Joost Rommes, November 23, 2007
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Applications of DPA
I Pole-zero analysis in circuit simulationI Applications in model order reduction
I Modal approximationI Dominant poles may lie anywhere in complex planeI Combinations with rational Krylov methods
25/38NXP Semiconductors Corp. I&T/DTF, Joost Rommes, November 23, 2007
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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
26/38NXP Semiconductors Corp. I&T/DTF, Joost Rommes, November 23, 2007
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Pole-zero analysisFrequency response of circuit (1000 unknowns). Pole994 · 103 ± i5.6 · 106 causes peak around 6MHz.
3. Project with V = [V1, . . . ,Vm] and W = [W1, . . . ,Wm]
Open question:
I How to choose interpolation points si?
I See also PhD thesis Grimme (1997)
33/38NXP Semiconductors Corp. I&T/DTF, Joost Rommes, November 23, 2007
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10−1
100
101
−60
−40
−20
0
20
40
60
Frequency (rad/sec)
Gai
n (d
B)
k=40 (RKA)k=10 (QDPA)k=50 (RKA+QDPA10)Exact
Figure: Breathing sphere (n = 17611). Exact transfer function (solid),40th order SOAR RKA model (dot), 10th (dash-dot) order modalequivalent, and 50th order hybrid RKA+QDPA (dash).
34/38NXP Semiconductors Corp. I&T/DTF, Joost Rommes, November 23, 2007
I Two-sided rational SOAR [Bai and Su (2005)] model (k = 40,shifts 0.1, 0.5, 1, 5) misses peaks
I Small QDPA model (k = 10) matches some peaks, missesglobal response
I 500 s (SOAR, k = 2 · 40) vs. 2800 s (QDPA, 108 iters)
I Hybrid: Y = [YQDPA,YSOAR] and X = [XQDPA,XSOAR]
I Larger SOAR models: no improvement
I More poles with QDPA: expensive
I Use imaginary parts of poles as shifts for SOAR!
I Shifts σ1 = 0.65i , σ2 = 0.78i , σ3 = 0.93i , and σ4 = 0.1
I Two-sided rational SOAR, 10-dimensional bases
35/38NXP Semiconductors Corp. I&T/DTF, Joost Rommes, November 23, 2007
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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 RKA model (dash) using interpolation points based ondominant poles, and relative error (dash-dot).
36/38NXP Semiconductors Corp. I&T/DTF, Joost Rommes, November 23, 2007
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Concluding remarks
I DPA for computation of dominant poles
I Subspace acceleration, selection, and efficient deflation
I Straightforward implementationI Applications:
I Various specialized eigenvalue problemsI Model order reduction:
I Construction of modal approximationsI Interpolation points for rational KrylovI Behavioral modeling
Generalizations:
I Second and higher-order systems
I MIMO systems
I Computation of dominant zeros z : H(z) = 0
37/38NXP Semiconductors Corp. I&T/DTF, Joost Rommes, November 23, 2007