NXP PowerPoint template (Title) Template for presentations (Subtitle) Name Eigenvalue problems and model order reduction in the electronics industry Joost Rommes [[email protected]] NXP Semiconductors/Corp. I&T/DTF/A&M/PDM/Mathematics O-MOORE-NICE! COMSON Autumn School on MOR, Terschelling September 21–25, 2009
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NXP PowerPoint template (Title)Template for presentations (Subtitle)
Name
Subject
Project
MMMM dd, yyyy
Eigenvalue problems andmodel order reductionin the electronics industry
51/64NXP Semiconductors Corp. I&T/DTF, Joost Rommes, September 21–25, 2009
Subject/Department, Author, MMMM dd, yyyy
CONFIDENTIAL 3
−3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.50
2
4
6
8
10
12
14
real (1/s)
imag
(rad
/s)
Figure: Multi-parameter RL plot of sensitive poles of BIPS/97 bySASPA. Various poles become more damped as PSS gains increase, twocritical rightmost poles crossing the imag axis and the 5% damping ratioboundary. Squares: poles when three PSSs have zero gain.
52/64NXP Semiconductors Corp. I&T/DTF, Joost Rommes, September 21–25, 2009
Subject/Department, Author, MMMM dd, yyyy
CONFIDENTIAL 3
−3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.50
2
4
6
8
10
12
14
real (1/s)
imag
(ra
d/s)
Figure: Multi-parameter RL plot of poles of BIPS/97 by QR. Variouspoles become more damped as PSS gains increase, two critical rightmostpoles crossing the imag axis and 5% damping ratio boundary. Squares:poles when three PSSs have zero gain.
53/64NXP Semiconductors Corp. I&T/DTF, Joost Rommes, September 21–25, 2009
Subject/Department, Author, MMMM dd, yyyy
CONFIDENTIAL 3
Running times SASPA vs. QR
Table: CPU times (seconds) for computation of root-locus plots withSASPA and the QR method for BIPS/97 and BIPS/07. Notation:S=single parameter, M=multiple parameters, n=number of states,N=dimension of descriptor realization, steps=number of steps in theroot-locus.
BIPS S/M n N steps SASPA(#poles) QR
97 S 1664 13250 60 1450 (10) 780097 M 1664 13250 30 1400 (10) 360097 M 1664 13250 30 3000 (20) 360007 M 3172 21476 30 4664 (20) 24000
54/64NXP Semiconductors Corp. I&T/DTF, Joost Rommes, September 21–25, 2009
Subject/Department, Author, MMMM dd, yyyy
CONFIDENTIAL 3
Applications of DPA and SPA
I Pole-zero analysis in circuit simulation
I Stability analysis
I Behavioral modeling
I Inverse problems?I Applications in model order reduction
I Modal approximationI Combinations with rational Krylov methodsI MOR for parameterized systems?
55/64NXP Semiconductors Corp. I&T/DTF, Joost Rommes, September 21–25, 2009
Subject/Department, Author, MMMM dd, yyyy
CONFIDENTIAL 3
Modal approximation and 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).
56/64NXP Semiconductors Corp. I&T/DTF, Joost Rommes, September 21–25, 2009
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).
57/64NXP Semiconductors Corp. I&T/DTF, Joost Rommes, September 21–25, 2009
Subject/Department, Author, MMMM dd, yyyy
CONFIDENTIAL 3
Dominant poles: location in complex plane (zoom)Dominant poles not necessarily at outside of spectrum
Figure: Pole spectrum (zoom) of complete system (n = 66), modalapproximation (k = 12), and dual Arnoldi model (k = 30).
58/64NXP Semiconductors Corp. I&T/DTF, Joost Rommes, September 21–25, 2009
Subject/Department, Author, MMMM dd, yyyy
CONFIDENTIAL 3
Inverse problems
Q1: compute eigenvalues most sensitive to parameterchanges?
1. Define parameters of interest (and their ranges)
2. Apply SPA to compute most sensitive eigenvalues
Q0: which parameters must be tuned, and how, to moveeigenvalue(s)?
I Brute force uses SPA for every parameter and rankssensitivities
I More elegant solutions to this inverse problem?
59/64NXP Semiconductors Corp. I&T/DTF, Joost Rommes, September 21–25, 2009
Subject/Department, Author, MMMM dd, yyyy
CONFIDENTIAL 3
Modal approximation vs. moment matching
I Modal approximation (project on eigenspace)+ sparse ROMs, preservation of poles– may be expensive, requires decay in residues
I Moment matching (project on Krylov space)+ cheap and robust implementations– dense ROMs, requires decay in moments
I Best results usually obtained by combining both: usedominant poles for improving Krylov models and/ordetermining shifts in rational interpolation
Q Can we combine both methods for parameterizedMOR?
60/64NXP Semiconductors Corp. I&T/DTF, Joost Rommes, September 21–25, 2009
Subject/Department, Author, MMMM dd, yyyy
CONFIDENTIAL 3
MOR for parameterized systems
I Multi-parameter moment matching [Bai, Daniel, Ioan]
I Parameterized modal approximation:
H(s, p) =n∑
i=1
Ri (p)
s − λi (p)
1. Compute dominant and sensitive poles and eigenvectors,including the sensitivities
2. Construct parameterized reduced order model byparameterizing
I dominant and sensitive poles (λ(p))I eigenspaces X (p) and Y (p)
I Combine moment matching with parameterized sensitiveeigenspace
61/64NXP Semiconductors Corp. I&T/DTF, Joost Rommes, September 21–25, 2009
Subject/Department, Author, MMMM dd, yyyy
CONFIDENTIAL 3
Concluding remarks
I Mathematical problems occur in several applications:I ESD analysisI Reduction of large parasitic networksI Modeling of handle wafers and thermal effectsI Circuit analysis and optimization
I State-of-the-art mathematics helps inI getting insight in problemsI speeding up design cycleI obtaining first-time-right
I Combinations of techniques are required
The challenges are:
I Large-scale and parameterized systems
I Nonlinearity
I Many inputs, outputs, and parameters
62/64NXP Semiconductors Corp. I&T/DTF, Joost Rommes, September 21–25, 2009
63/64NXP Semiconductors Corp. I&T/DTF, Joost Rommes, September 21–25, 2009
Subject/Department, Author, MMMM dd, yyyy
CONFIDENTIAL 3
1 3
mathematics in industry 13
ISBN 978-3-540-78840-9
Wilhelmus H. A. SchildersHenk A. van der Vorst · Joost Rommes
Model Order ReductionTheory, Research Aspects and Applications
Model Order Reduction
Schilders Van der Vorst · Rom
mes
1The goal of this book is three-fold: it describes the basics of model order reduction and related aspects in numerical linear algebra, it covers both general and more specialized model order reduction techniques for linear and nonlinear systems, and it discusses the use of model order reduction techniques in a variety of practical applications. The book contains many recent advances in model order reduction, and presents several open problems for which techniques are still in development. It will serve as a source of inspiration for its readers, who will discover that model order reduction is a very exciting and lively field.
W. H. A. Schilders · H. A. van der Vorst · J. RommesModel Order Reduction
mii
13
THE EUROPEAN CONSOR TIUM FOR MATHEMATICS IN INDUSTRY
Available now at Springer!
64/64NXP Semiconductors Corp. I&T/DTF, Joost Rommes, September 21–25, 2009