1 introduction to model order reduction thanks to jacob white, peter feldmann ii.1 – reducing...
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1
Introduction to Model Order Reduction
Thanks to Jacob White, Peter Feldmann
II.1 – Reducing Linear Time Invariant Systems
Luca Daniel
2
Model Order Reduction
Linear Time Invariant Systems
• II.1.a via Modal Analysis
• II.1.b via Rational Function Fitting (point matching)
• II.1.c. via Quasi Convex Optimization
• II.1.d via Pade’ approximation and AWE
3
Introduction to Model Order Reduction
Thanks to Jacob White, Peter Feldmann
II.1.a – Reduction using Modal Analyis
Luca Daniel
4
State-Space Description
Dynamic Linear case
1 1
( )NxN Nx
scalarinp
T
Nxscalarouu tt put
y tdx t
A x t b u t c x tdt
• Original Dynamical System - Single Input/Output
• Reduced Dynamical System
• q << N, but input/output behavior preserved
1 1
( )rr r
qxq qx scalarinp
Tr r r
scalar qxoutputut
dx tA x yt b u t
dt x
tc t
5
Defining Accuracy Defining Accuracy
• Time-domain response should be “close” Time-domain response should be “close” – For which possible inputs? For which possible inputs?
• Frequency response should match Frequency response should match – At what frequencies? At what frequencies?
6
Matching Frequency Response Matching Frequency Response
• Ensure accuracy for only some inputs? Ensure accuracy for only some inputs? • Example: Example:
– low frequency inputs, low frequency inputs,
– or some band, or some band,
– or some points in the frequency responseor some points in the frequency response
Original matching some
part of the frequency
response
)(log H
log
7
Reminder about Eigenanalysis
1Change of variab (le ) ( ) ( ) (s ): Ew t x t w t E x t
1
1 2 1 2
10 0
0 0
0 0
Eigendecomposition: N N
N
E
A E E E E E E
Substituting: ( )
( ) , (0) 0dEw t
AEw t bu t Ewdt
11 1Multiply by : ( )
( )dw t
E AEw t E bu tdt
E
Consider an ODE( )
( ) , (0) 0: dx t
Ax t bu t xdt
8
Reminder about Eigenanalysis Cont.
1
11 1 1
1
0 0
0 0
0 0N N NN
E bw wd
u tdt
w w E b
b
Decoupled Equations
TT T Ty t c x t c Ew t E c w t
c
Output Equation
9
Reminder about Eigenanalysis Cont.
0
i
tt
i iw t e b u d
Solving Decoupled Equations
1
N
i ii
y t c w t
Output Equation
Assuming Zero Initial Conditions
10
Reduced models via mode truncationDynamic Linear Case
1 1 1 10 0
0 0
0 0q q q q
w w b
u t
w w b
1
q
i ii
y t c w t
Output Equation
11
Reduced models via mode TruncationDynamic Linear Case
Why?
• Certain modes are not affected by the input
• Certain modes do not affect the output
• Keep least negative eigenvalues (slowest modes)– Look at response to a constant input
1, , are all smallk Nb b
1, , are all smallk Nc c
0
1
Small if large
i i
tt t
i i i ii
i
w t e b ud b u b ue
12
Reduced models via mode truncationDynamic Linear Case
Heat Conducting bar Results
N=100
Exact
q=1
q=3
q=10
Keep qth slowest modes
13
Another way to look at Reduction by Modal Analysis
1TH s c sI A b
Transfer Function
Apply Eigendecomposition
1 1TH s c E sI E b
1
10 0
0 0
10 0
T
N
s
c b
s
1
Ni i
i i
c bH s
s
elimitate eachmode for whichthis term is small
14
Model Order Reduction Model Order Reduction via Eigenmode Analysis via Eigenmode Analysis
n
i i
ii
s
bcsH
1
~~)(
)(
)()(
1
1
1
i
n
i
i
n
i
s
ssH
Pole-Residue FormPole-Residue FormPole-Zero Form (SISO)Pole-Zero Form (SISO)
• Ideas for reducing order:Ideas for reducing order:– Drop terms with small residues Drop terms with small residues
– Drop terms with large negative (“fast” modes)Drop terms with large negative (“fast” modes)
– Remove pole/zero near-cancellations Remove pole/zero near-cancellations
– Cluster poles that are “together”Cluster poles that are “together”
iRe
n
i
tii
iebcth1
~~)(
iibc~~
15
Eigenmode Analysis Based Reduction SummaryEigenmode Analysis Based Reduction Summary
• Advantages Advantages – Conceptually familiar Conceptually familiar
– Simple physical interpretation : retains dominant Simple physical interpretation : retains dominant system modes/poles system modes/poles
• Drawbacks Drawbacks – Relatively expensive : Relatively expensive : have to find the eigenvalues firsthave to find the eigenvalues first
– Relatively inefficient. For a given model size, many Relatively inefficient. For a given model size, many other approaches can provide better accuracy for the other approaches can provide better accuracy for the same computational costsame computational cost
• e.g. Hankel Model Order Reduction e.g. Hankel Model Order Reduction • e.g. Truncated Balance Realizatione.g. Truncated Balance Realization
O(n3)
16
Model Order Reduction
Linear Time Invariant Systems
• II.1.a via Modal Analysis
• II.1.b via Rational Function Fitting (point matching)
• II.1.c. via Quasi Convex Optimization
• II.1.d via Pade’ approximation and AWE
17
Introduction to Model Order Reduction
Thanks to Jacob White
II.1.b – Reduction using Fitting
Luca Daniel
18
A canonical form for model order reductionA canonical form for model order reduction
( ) ( )
( ) ( )T
dxE x t b u t
dt
y t c t
Assuming A is non-Assuming A is non-singular we can cast the singular we can cast the dynamical linear system dynamical linear system into one canonical form into one canonical form for model order for model order reductionreduction
Note: not necessarily Note: not necessarily always the best, but the always the best, but the simplest for educational simplest for educational purposespurposes
bAb
EAE1
1
)()(
)()(
txcty
tbutAxdt
dxE
T
19
Original System Transfer Function:Original System Transfer Function:
1
0 1 1
11
NN
NN
b b s b sH s
a s a s
Model Reduction = Find a low order (q << N) Model Reduction = Find a low order (q << N) rational function matchingrational function matching
Model Order Reduction Model Order Reduction via Rational Transfer Function Fittingvia Rational Transfer Function Fitting
rational functionrational function
1
0 1 1
1
ˆ ˆ ˆˆ
ˆ ˆ1
b b s b sH s
a s a s
reduced orderreduced orderrational functionrational function
20
Reduced Model Dynamical SystemReduced Model Dynamical System
1
1
ˆ ˆˆ ˆ
ˆ ˆ ˆ( )
qxq qxscalarinp
T
qxscalaroutput
ut
dxE x t b u t
d
y c x t
t
t
Reduced Model Transfer FunctionReduced Model Transfer Function
2 2q qcoefficientscoefficients
2qcoefficientscoefficients
Rational Transfer Function Fitting: Rational Transfer Function Fitting: Degrees of FreedomDegrees of Freedom
1
0 1 1
1
ˆ ˆ ˆˆ
ˆ ˆ1
b b s b sH s
a s a s
21
Reduced Model Transfer FunctionReduced Model Transfer Function
ˆ ˆˆ ˆ
ˆ ˆ ˆ( ) T
dxE x t b u t
d
y t c x t
t
Apply any invertible change of variables to the stateApply any invertible change of variables to the state
1 ˆˆˆTH s c sE I b
1 1ˆ ˆˆ ˆ
ˆ ˆ ˆ( ) T
dwU EU w t U b
y t c Uw t
u tdt
11 1
11 1
ˆˆˆ
ˆˆˆ
T
T
H s c U sU EU I U b
c UU sE I UU b
Many Dynamical Systems have the same transfer function!!
ˆ ˆ( ) ( )x t U w t
Rational Transfer Function Fitting: Rational Transfer Function Fitting: Degrees of Freedom (cont.)Degrees of Freedom (cont.)
I I
22
H s
Rational Transfer Function Fitting: Rational Transfer Function Fitting: via Point Matchingvia Point Matching
H s
11 0 1 1
ˆ ˆ ˆˆ ˆ1 0q qi q i i i q ia s a s H s b b s b s
For i = 1 to 2q
• cross multiplying generates a linear systemcross multiplying generates a linear system
• Can match 2q pointsCan match 2q points 1
0 1 1
1
ˆ ˆ ˆ
ˆ ˆ1
qi q i
i qi q i
b b s b sH s
a s a s
23
• Columns contain progressively higher powers of the test Columns contain progressively higher powers of the test frequencies: problem is numerically ill-conditionedfrequencies: problem is numerically ill-conditioned
• also... missing data can cause severe accuracy problemsalso... missing data can cause severe accuracy problems
2 111 1 1 1 1 1
122 2
2 11 22 2 2 2 2
q
q
qq qq q q q q
H ss H s s H s s a
H ss a
b H ss H s s H s s
Rational Transfer Function Fitting: Rational Transfer Function Fitting: Point Matching matrix can be ill-conditionedPoint Matching matrix can be ill-conditioned
SMA 2005 MIT 24
Hard to Solve Systems
Fitting Example
Polynomial InterpolationTable of Data
t0 f (t0)t1 f (t1)
tN f (tN)
f
tt0 t1 t2 tN
f (t0)
Problem fit data with an Nth order polynomial2
0 1 2( ) NNf t t t t
SMA 2005 MIT 25
Hard to Solve Systems
Example Problem
Matrix Form2
0 0 0 0 0
21 11 1 1
2
interp
1 ( )
( )1
( )1
N
N
NN NN N N
t t t f t
f tt t t
f tt t t
M
SMA 2005 MIT 26
Hard to Solve Systems
Fitting Example
CoefficientValue
Coefficient number
Fitting f(t) = t
f
t
SMA 2005 MIT 27
Hard to Solve Systems
Perturbation Analysis
Geometric Approach is clearer
1 2 1 1 2 2[ ], Solving is finding M M M M x b x M x M b
2
2|| ||
M
M
1
1|| ||
M
M
2
2|| ||
M
M
1
1|| ||
M
M
When vectors are nearly aligned, difficult to determinehow much of versus how much of 1M
2M
Case 16
1 0
0 10
Case 16
6
1 1 10
1 10 1
Columns orthogonal Columns nearly aligned
1x
2x
1x
2x
SMA 2005 MIT 28
Hard to Solve Systems
Geometric Analysis
Polynomial Interpolation
4 8 16 32
1010
1015
1020
~314~106
~1013
~1020
log(cond(M))
n
The power series polynomialsare nearly linearly dependent
21 1
22 2
2
1
1
1 N N
t t
t t
t t
1
11
t
t 2
t
29
Course Outline
Numerical Simulation Quick intro to PDE Solvers Quick intro to ODE SolversModel Order reduction Linear systems Common engineering practice Optimal techniques in terms of model accuracy Efficient techniques in terms of time and memory Non-Linear SystemsParameterized Model Order Reduction Linear Systems Non-Linear Systems
Yesterday
Today
FridayThursday
Tomorrow
30
Introduction to Model Order ReductionIntroduction to Model Order Reduction
Luca Daniel
Massachusetts Institute of Technology
http://onigo.mit.edu/~dluca/2006PisaMOR
www.rle.mit.edu/cpg
31
Course Outline
Numerical Simulation Quick intro to PDE Solvers Quick intro to ODE SolversModel Order reduction Linear systems Common engineering practice Optimal techniques in terms of model accuracy Efficient techniques in terms of time and memory Non-Linear SystemsParameterized Model Order Reduction Linear Systems Non-Linear Systems
Monday
Yesterday
FridayTomorrow
Today
32
Model Order Reduction
Linear Time Invariant Systems
• II.1.a via Modal Analysis
• II.1.b via Ratianal Function Fitting (point matching)
• II.1.c. via Quasi Convex Optimization
• II.1.d via Pade’ approximation and AWE
33
Introduction to Model Order Reduction
Thanks to Kin C. Sou, Alexander Megretski
II.1.c – Reduction using Optimization
Luca Daniel
34
Overview
• Optimization based reduction
• Quasi-convex optimization MOR setup
• Solving the MOR setup
• Application examples
• Conclusions
35
H s
Recall Recall Rational Transfer Function Fitting via Point MatchingRational Transfer Function Fitting via Point Matching
H s
11 0 1 1
ˆ ˆ ˆˆ ˆ1 0q qi q i i i q ia s a s H s b b s b s
For i = 1 to 2q
• cross multiplying generates a linear systemcross multiplying generates a linear system
• Can match 2q pointsCan match 2q points 1
0 1 1
1
ˆ ˆ ˆ
ˆ ˆ1
qi q i
i qi q i
b b s b sH s
a s a s
36
Optimization based rational fit Model Order Reduction Setup
p(s),q(s)
( )minimize ( )
( )
p sH s
q s
From field solverOr measurements
Small stable and passivereduced order model
Least Square method• Cast as nonlinear least
squares (solved by Gauss-Newton)
Quasi-convex method• Cast as quasi-convex
program (solved by convex optimization algorithm)
• Do not consider stability or passivity while finding poles (need post-processing)
• Explicitly take care of stability and passivity while finding poles
37
Change of variables• To make our program tractable, we introduce a change offrequency variables (bilinear transform)
Laplace frequency variablez frequency variable
11
zzs
[s] [z]
38
• Desirable MOR setup to solve• Feasible set is not convex if m 3 For example, but
• Problem has not been proved to be NP complete either
Modified optimal H-inf norm MOR setup
331 5q z z 33
2 5q z z
,
( )minimize ( )
( )
deg ,subject to
deg ,
p q
p zH z
q z
q m
p m
31 2( ) ( ) 27
2 2 25
q z q zz z
Stability: q(z) Schur polynomial (roots inside unit circle)
Passivity, and possibly other constraints
39
Overview
• Optimization based reduction
• Quasi-convex optimization MOR setup
• Solving the MOR setup
• Application examples
• Conclusions
40
Relaxation
• Original problem is difficult• Made easier if some constraints are dropped (relaxed)• Solve the relaxed problem • Construct original solution from relaxation• For example, LP relaxation (polynomial time) of IP problems (exponential time).
General idea
-c
feasible set…
optimal solution-c
feasible set
optimal relaxed solution
nearest rounding
41
Relaxation of the H-inf norm MOR setup
1
1, ,
( )minimize ( )
( )
deg , deg ,subject to
deg
p q r
r z
q
p zH z
q z z
q m p m
r m
Benefit: Relaxation equivalent to a quasi-convex program.Drawback: May obtain suboptimal solutions
Anti-stableterm
Stability: q(z) Schur polynomial (roots inside unit circle)
Passivity, and possibly other constraints
42
How bad is this relaxation?
,
1
1,( , , ) arg min
q p r
p zq p r H z
q
r
q zz
z
1m
p zH z m H
q z
Let
such that deg(q) = m, q(z) is Schur polynomial
Then
m+1th Hankel singular value
THEOREM:
43
Change of variables
ˆ , 0,
j j j j
j
j j j
p e r e b e jc eH e
q e q e a e
where a(z) b(z) and c(z) are trigonometric polynomials:
1 11 0( )m m m m
ma z z z a z z a
2cos m whenjz e
0ja e 0, Prop: Stability
44
Passivity
0, 0,jb e
• For SISO systems, passivity means1. H(z) is analytic for |z|>=1 2. H(z)*=H(z*) 3. Re(H(z))>0 for |z|=1 for impedance,
Conclusion: Stability and passivity = positivity of trigonometric polynomials
for all frequencies!
45
Equivalent quasi-convex setup
This is a quasi-convex program, because
1 02cos( ) 2cos(( 1) ) 0jma e m m a a
defines an intersection of halfspaces and sub-level set is
Re j j j j jH e a e b e jc e a e is again intersection of halfspaces parameterized by and
convex set0
1
, ,
( )minimize ( )
( )
deg , deg
deg ,subject to
0, [0, ]
0, [0, ]
j j
jja b c
j
j
b e jc eH e
a e
a m b m
c m
a e
b e
=0
=1
=2
=3
quasi-convex function
convex set
0 0 1 02cos 1 0 1 0mm m a a 1 1 1 02cos 1 0 1 0mm m a a 2 2 1 02cos 1 0 1 0mm m a a 3 3 1 02cos 1 0 1 0mm m a a
46
Additional constraints
• Can model additional constraints such as
• Bounded real passivity (for scatter parameters)• Explicit minimization of quality factor error (for inductors)• Weighting of frequency responses• Point-wise transfer function (and/or derivatives) matching
47
Overview
• Optimization based reduction
• Quasi-convex optimization MOR setup
• Algorithm Summary
• Application examples
• Conclusions
48
Summary of QCO algorithm
Step 2: Compute coefficients of q(z) using the relation
1q z q z a z and q(z) being a Schur polynomial
Step 3: Compute coefficients of p(z) by solving
minimize
subject to deg
p zH z
q z
p m
Solved for example by the ellipsoid algorithm
( )( ) min
( )
j j
jj
b e jc eH e
a e
Step 1: Compute optimal solution a(z),b(z),c(z) of the relaxation
subject to stability, passivity…
,stability, passivity…
Solved for example by the ellipsoid algorithm
49
Stability?
Solving quasi-convex programs(a,b,c,) current iteratelocalization set
(e.g. ellipsoid)
?b jc
Ha
Passivity?
…
Generate cut
N
N
N
N
Y
Y
Y
Decrease
All Yes
?c
Qb
Termination?
N
target set
localization set
center
cut
min volume covering ellipsoid
new center
new cut
and so on
Updatelocalization set
Objective oracle, stabilityoracle, passivity oracle…
50
Overview
• Optimization based reduction
• Quasi-convex optimization MOR setup
• Algorithm summary
• Application examples
• Conclusions
51
Example 1: RLC line (MNA)
“PRIMA” (Moment Matching) Model Order Reduction
Quasi Convex OptimizationModel Order Reduction
• RLC line full model 20th order [Vasilyev 2004]• Open circuit terminal• 10th order reduced model by existing PRIMA and our QCO
0 2000 4000 6000 8000 10000 12000 14000 160000
0.5
1
1.5
2
2.5
3
3.5
4
frequency (Hz)
mag
nitu
de
Full model
QCOROM
0 2000 4000 6000 8000 10000 12000 14000 160000
0.5
1
1.5
2
2.5
3
3.5
4
frequency (Hz)
mag
nitu
de
Full model
MMROM
4
4
2
52
Example 2: RF inductor with substrate(from field solver)
0 0.5 1 1.5 2 2.5 3
x 109
-2
-1
0
1
2
3
4
5
6
7
8
frequency (Hz)
qual
ity f
acto
r
training data
test pointsROM
• RF inductor with substrate effect captured by layered Green’s function [Hu Dac 05]• System matrices are frequency dependent• Full model has infinite order• Reduced model has order 6
53
Example 3: RF inductor model (from measurement)
0 1 2 3 4 5 6 7 8 9 10
x 109
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
frequency (Hz)
real
par
t
Fabricated 7 turn spiral inductorBlue: measurementRed: 10th order reduced model (positive real part constraint imposed)
0 0.5 1 1.5 2 2.5 3 3.5
x 109
-5
0
5
10
15
20
25
30
35
40
frequency (Hz)
qual
ity f
acto
r
54
Example 4: Model of graphic card package (from measurement)
• Industry example of a multi-port device (390 frequency samples)• 12th order SISO reduced models are constructed• Bounded realness constraint is imposed• Frequency weight is employed
0 1 2 3 4 5 6
0.4
0.5
0.6
0.7
0.8
0.9
1
mag
nitu
de
frequency (GHz)0 1 2 3 4 5 6
0
0.02
0.04
0.06
0.08
0.1
0.12
mag
nitu
de
frequency (GHz)
S11 S13
Solid: ROMDot: measurement
Solid: ROMDot: measurement
55
Example 5: Large IC power distribution grid(from field solver)
• Power distribution grid (dimension size = 7mm, wire width = 2 µm)• Blue: full model (order 2046)• Green: PRIMA 40th order reduced model• Red: QCO 40th order reduced model (positive real)
0 10 20 30 40 50 600
500
1000
1500
2000
2500
frequency (GHz)
mag
nitu
de
0 10 20 30 40 50 60-2
-1.5
-1
-0.5
0
0.5
1
1.5
frequency (GHz)
phas
e
3 curves on top of each other
3 curves on top of each other
56
Conclusion
• QCO competes reasonably well in terms of accuracy with moment matching (e.g. PRIMA) for reducing large systems
• But in addition: QCO can reduce models with frequency dependent matrices
• QCO is very flexible in imposing constraints such as stability and passivity
• QCO can be extended to parameterized MOR problems (see IV.2)
57
Model Order Reduction
Linear Time Invariant Systems
• II.1.a via Modal Analysis
• II.1.b via Ratianal Function Fitting (point matching)
• II.1.c. via Quasi Convex Optimization
• II.1.d via Pade’ approximation and AWE
58
Point matching vs. Moment MatchingPoint matching vs. Moment Matching
Point matching:Point matching:can be very inaccurate can be very inaccurate in between pointsin between points
H s
H s
H s
H sMoment (derivatives)Moment (derivatives)matching:matching:accurate around accurate around expansion point, expansion point, but inaccurate on wide but inaccurate on wide frequency bandfrequency band
59
1
0
( ) ( )T T k k
k
H s c I sE b c E b s
1 2 2
01 20
( ) T T T kk
km mmH s c b c E b s c E b s m s
The Taylor coef. = frequency domain moments = The Taylor coef. = frequency domain moments = = derivatives of the transfer function (up to a constant)= derivatives of the transfer function (up to a constant)
Frequency Domain "Moments" (or Taylor Frequency Domain "Moments" (or Taylor coefficients) of the transfer functioncoefficients) of the transfer function
Taylor Series Expansion of the original transfer Taylor Series Expansion of the original transfer function around s=0 function around s=0
2 32 3
2 30 0 0
1 1( ) (0)
2! 3!s s s
dH d H d HH s H s s s
ds ds ds
60
Time domain moments Time domain moments of the impulse responseof the impulse response
Definition:Definition:
0
0
22
0
1
0
0
)(ˆ
)(ˆ
)(ˆ
)(ˆ
dtthtm
dtthtm
dttthm
dtthm
61
Connection to the time-domain moments of the Connection to the time-domain moments of the circuit responsecircuit response
0
2 12 2 2 1 2 1 2
0 0 0 0
( ) ( )
1 ( 1)( ) ( ) ( ) ( ) ( )
2! (2 1)!
st
qq q q
H s e h t dt
h t dt th t dt s t h t dt s t h t dt s O sq
Time-domain momentsTime-domain moments12210 ˆ,,ˆ,ˆ,ˆ qmmmm
2 2 1 20 1 2 2 1( ) ( )q q
qH s m m s m s m s O s
Compare:Compare:
Hence the the Taylor coeff. Hence the the Taylor coeff. are, up to a constant, the are, up to a constant, the time-domain moments of the time-domain moments of the circuit response.circuit response.
( 1)ˆ
!
k
k km mk
62
Rational function fitting via moment matching: Rational function fitting via moment matching: Pade Approximation (AWE)Pade Approximation (AWE)
2 2 1 20 1 2 2 1( ) ( )q q
qH s m m s m s m s O s
1
0 1 1
1
ˆ ˆ ˆˆ
ˆ ˆ1
b b s b sH s
a s a s
0 1 1 1 2
Choose the 2q rational function coefficients ˆ ˆ ˆ ˆ ˆ ˆ, , , , , , ,so that the reduced rational functionmatches the first 2q moments of the original transfer function
q qb b b a a a
10 1 1 2 2 1
0 1 2 2 11
ˆ ˆ ˆ
ˆ ˆ1
qq q
qqq
b b s b sm m s m s m s
a s a s
63
Rational function fitting via moment matching: Rational function fitting via moment matching: Pade Approximation (AWE)Pade Approximation (AWE)
– Step 1:Step 1: calculate the first 2q moments of H(s) calculate the first 2q moments of H(s)
– Step 2:Step 2: calculate the 2q coeff. of the Pade’ approx, calculate the 2q coeff. of the Pade’ approx, matching the first 2q moments of H(s)matching the first 2q moments of H(s)
12210 ,,,, qmmmm
qq aabbb ,,,,,, 1110
0,1, 2 1T kkm c E b k q
64
Step 1: calculation of moments Step 1: calculation of moments simulating equivalent circuits (AWE)simulating equivalent circuits (AWE)
• Historical note Historical note – Electrical engineers calculated freq. domain Taylor coef. by Electrical engineers calculated freq. domain Taylor coef. by
calculating time domain moments, calculating time domain moments,
– synthesizing and simulating circuit networks. synthesizing and simulating circuit networks.
– Specifically the momets can be calculated evaluating the Specifically the momets can be calculated evaluating the asymptotic behaviors of the circuit waveforms, asymptotic behaviors of the circuit waveforms,
– Hence the name AWE (Asymptotic Waveform Evaluation)Hence the name AWE (Asymptotic Waveform Evaluation)
65
Step 1: Calculation of moments (algebraically)Step 1: Calculation of moments (algebraically)
122122
1101
000
ˆ~~
ˆ~~
ˆ
qT
qqq
T
T
wcmwEwA
wcmwEwA
wcmbw
bEAc
bEcmkT
kTk
~~ 1
• For sparse system For sparse system – can use one initial LU decomposition on A can use one initial LU decomposition on A
– then solve 2q linear triangular systems for the 2q moments then solve 2q linear triangular systems for the 2q moments
• For dense systems For dense systems – can use iterative methods and matrix implicit matrix-vector can use iterative methods and matrix implicit matrix-vector
productsproducts
66
Step 2: Calculation of Pade’ coeff. (AWE)Step 2: Calculation of Pade’ coeff. (AWE)
1212
2210
1
1110
1
q
qqq
qq smsmsmm
sasa
sbsbb
012111
0111
00
mamamb
mamb
mb
qqqq
For coeff. a’s For coeff. a’s solve the solve the following following linear system:linear system:
12
2
1
1
2
1
221
2
21
1210
q
q
q
q
q
q
q
q
m
m
m
m
a
a
a
a
mm
m
mm
mmmm
For coeff. b’s For coeff. b’s simply simply calculate:calculate:
67
Heat Conducting BarDemonstration Example State-Space Description
endT
x
1
2
N
h x
h x
h x
b
2 1 0 0
1 2
0 0
2 1
0 0 1 1
A
0
0
1
c
1 1
( )NxN Nx
scalarinp
T
Nxscalarouu tt put
y tdx t
A x t b u t c x tdt
Given the right scaling
Heat In0 0T
A
N=100
Exact
q=1q=3
q=10
Keep qth slowest eigenmodes
Exact
q=1
q=3
Matches q moments
Keeping Eigenmodes versus matching momentsDynamic Linear Case
Heat Flow Results
69
b
Eb
2E b3E b
Vectors will line up with dominant eigenspace!Vectors will line up with dominant eigenspace!
Numerical problem for q >20 (cannot get accuracy)Numerical problem for q >20 (cannot get accuracy)
matrix powers converge to the eigenvector corresponding to the largest eigenvalue.
T kkm c E b 0
kkm m
70
Pade matrix can be very ill-conditionedPade matrix can be very ill-conditioned
• matrix powers converge to the eigenvector matrix powers converge to the eigenvector corresponding to the largest eigenvalue.corresponding to the largest eigenvalue.
T kkm c E b 0
kkm m
12
2
1
1
2
1
221
2
21
1210
q
q
q
q
q
q
q
q
m
m
m
m
a
a
a
a
mm
m
mm
mmmm
Columns become linearly dependent for large qColumns become linearly dependent for large qthe problem is numerically very ill-conditioned!the problem is numerically very ill-conditioned!
71
Pade matrix can be very ill-conditionedPade matrix can be very ill-conditioned
• matrix powers converge to the eigenvector matrix powers converge to the eigenvector corresponding to the largest eigenvalue.corresponding to the largest eigenvalue.
T kkm c E b 0
kkm m
12
2
1
1
2
1
022
012
1
001
1
01
02
10
q
q
q
q
q
q
q
qqq
m
m
m
m
a
a
a
a
mmm
mmm
mmmm
Columns become linearly dependent for large qColumns become linearly dependent for large qthe problem is numerically very ill-conditioned!the problem is numerically very ill-conditioned!
72
Example: simulation of voltage gain of a filter with Example: simulation of voltage gain of a filter with Pade via AWEPade via AWE
73
Model Order Reduction
Summary. Linear Time Invariant Systems
• II.1.a via Modal Analysis
• II.1.b via Rational Function Fitting (point matching)
• II.1.c. via Quasi Convex Optimization
• II.1.d via Pade’ approximation and AWE