View
1
Download
0
Category
Preview:
Citation preview
v
Model Reduction for Dynamical Systems Lecture 8 Lihong Feng
Otto-von-Guericke Universität Magdeburg Faculty of Mathematics Summer term 2019
Max Planck Institute for Dynamics of Complex Technical Systems Computational Methods in Systems and Control Theory
Magdeburg, Germany feng@mpi-magdeburg.mpg.de
https://www.mpi-magdeburg.mpg.de/3668354/mor_ss19
2
Overview
Linearization-based MOR
Quadratic MOR
Bilinearization-based MOR
Variational analysis-based MOR
Trajectory piece-wise linear MOR
Proper orthogonal decomposition (POD)
References
3
Linearization-based MOR
Original large ODE
)()()()(/
tCXtytBuXfdtEdX
=+=
)(Xf
AXXf =)(~
Linearization: approximate by a linear function )(Xf
)()(~1
)(])(,[/ 00
tCXty
tuXDXfBAXdtEdX f
=
−+=
)()(
))(()(21)()()(
00
00000
XXDXf
XXXHXXXXDXfXf
f
fT
f
−+≈
+−−+−+=
Taylor series expansion:
)()(~)()()(/ 00
tCXtytBuXXDXfdtEdX f
=
+−+=
.,1,0,])[(,~},,,{izationorthogonal
10
1
21
=−==
=−− irEAEsMBAr
rMrMrMrVi
i
j
,...1,0,)( 1 == − irEAM ii
B~
4
Example
1 2 3 N-3 N-2 N-1 N
C C C C C C Ci(t)
Y. Chen, MIT thesis 1999.
)()(
),(
0
01
)(
)()(
)()()()(
1
11
3221
211
tLXty
tu
xxg
xxgxxg
xxgxxgxxgxg
dtdX
nn
kkkk
=
+
−
−−−
−−−−−−
=
−
+−
1)( 40 −+= xexg x
xxgg
xgxggxg
41)0(')0(!2
)0('')0(')0()( 2
=+≈
+++=
)()(
),(
0
01
4141
418241
4182414282
1
11
321
21
tLXty
tu
xx
xxx
xxxxx
dtdX
nn
kkk
=
+
−
−−
+−+−
=
−
+−
5
Example
0 2 4 6 8 100
0.005
0.01
0.015
0.02
0.025
time, u(t)=0, t<3, u(t)=1, t>=3;
volta
ge a
t nod
e 1
+ q=10 __ full simulation
6
Example
0 2 4 6 8 100
0.005
0.01
0.015
0.02
0.025
time, u(t)=0, t<3, u(t)=1, t>=3;
volta
ge a
t nod
e 1
+ q=10 -- q=20 __full simulation
7
Example
0 2 4 6 8 100
0.005
0.01
0.015
0.02
0.025
time, u(t)=0, t<3, u(t)=1, t>=3;
volta
ge a
t nod
e 1
+ q=10 -- q=20 ...linear, no reduction __ full simulation
8
Quadratic MOR )(Xf )(Xg
)(Xf
)(Xg
nqRZVZX q <<∈≈ ,,
)()(ˆ)(~/
tCVZtytuBVWVZVZVAVZVdtEVdZV TTTTTT
=++=
Approximate by a quadratic polynomial
)()()()(/
tCXtytBuXfdtEdX
=+=
Taylor series expansion:
)()(~)(~/
tCXtytuBWXXAXdtEdX T
=++=
))(()(21)()(
))(()(21)()()(
00000
00000
XXXHXXXXDXf
XXXHXXXXDXfXf
fT
f
fT
f
−−+−+≈
+−−+−+=
.,1,0,])[(,~)(
},,,{izationorthogonal1
01
0
21
=−=−=
=−− irEAEsMBAEsr
rMrMrMrVi
i
j
9
Example
1 2 3 N-3 N-2 N-1 N
C C C C C C Ci(t)
Y. Chen, MIT thesis 1999.
)()(
),(
0
01
)(
)()(
)()()()(
1
11
3221
211
tLXty
tu
xxg
xxgxxg
xxgxxgxxgxg
dtdX
nn
kkkk
=
+
−
−−−
−−−−−−
=
−
+−
1)( 40 −+= xexg x
22'
2'
80041!2
)0('')0()0(
!2)0('')0()0()(
xxxgxgg
xgxggxg
+=++≈
+++=
)()(
),(
0
01
)(800
)(800)(800
)(800)(800)(800800
4141
418241
4182414182
21
21
21
232
221
221
21
1
11
321
21
tLXty
tu
xx
xxxx
xxxxxxx
xx
xxx
xxxxx
dtdX
nn
kkkk
nn
kkk
=
+
−
−−−
−−−−−−
+
−
−−
+−+−
=
−
+−
−
+−
10
Example
1 2 3 N-3 N-2 N-1 N
C C C C C C Ci(t)
Y. Chen, MIT thesis 1999.
)()(
),(
0
01
)(800
)(800)(800
)(800)(800)(800800
4141
418241
4182414282
21
21
21
232
221
221
21
1
11
321
21
tLXty
tu
xx
xxxx
xxxxxxx
xx
xxx
xxxxx
dtdX
nn
kkkk
nn
kkk
=
+
−
−−−
−−−−−−
+
−
−−
+−+−
=
−
+−
−
+−
)()(~)(~/
tLXtytuBWXXAXdtdX T
=++=
W is a tensor, it has n matrices, the ith matrix corresponds to the ith element of the nonlinear vector.
11
Example
1 2 3 N-3 N-2 N-1 N
C C C C C C Ci(t)
Y. Chen, MIT thesis 1999.
−
−
=∈ ×
0000
000000800800008001600
1
nnRW
W
1
1
000000
00000800800008000800
000800800
0000
+
−
−−
−
=∈ × ii
i
RW nni
1,,1 +− iii
−−
=∈ ×
800800008008000000
000000
nnn RW
=
XWX
XWX
XWX
WXX
nT
iT
T
T
1
=
VZWVZ
VZWVZ
VZWVZ
WVZVZ
nTT
iTT
TT
TT
1
12
Example
0 2 4 6 8 100
0.005
0.01
0.015
0.02
0.025
time, u(t)=0, t<3, u(t)=1, t>=3;
volta
ge a
t nod
e 1
-- quadratic MOR, q=10 …linear, no reduction __ full simulation
13
Example
0 2 4 6 8 100
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
time, u(t)=0, t<3, u(t)=1, t>=3;
volta
ge a
t nod
e 1
-- quadratic MOR, q=10 …quadratic MOR, q=20 __ full simulation
14
Example
0 2 4 6 8 100
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
time, u(t)=0, t<3, u(t)=1, t>=3;
volta
ge a
t nod
e 1
-- quadratic MOR, q=10 …quadratic, no reduction __ full simulation
15
Bilinearization-bsed MOR
)(Xf )(Xg
)(Xf
)(XgnqRZVZX q <<∈≈⊗ ,,
)()(ˆ)()(/
tLVZtytuBVtVZuNVVZAVdtdZ TTT
=++= ⊗⊗⊗
Approximate by quadratic polynomial , but written into Kronecker product
)()()()(/
tLXtytBuXfdtdX
=+=
Taylor series expansion:
XXAXAXf
XXXHXXXXDXf
XXXHXXXXDXfXf
fT
f
fT
f
⊗++=
−−+−+≈
+−−+−+=
210
00000
00000
)(
))(()(21)()(
))(()(21)()()(
)()()()(/
tXLtytuBtuXNXAdtdX
⊗⊗
⊗⊗⊗⊗⊗⊗
=
++=
2, nNRX N ≈∈⊗
16
⊗
=⊗
XXX
X
=⊗
0B
B [ ]0LL =⊗
⊗+⊗
=⊕
11
21
0 AIIAAA
A
⊗+⊗
=⊕
000
BIIBN
)()()()(/
tXLtytuBtuXNXAdtdX
⊗⊗
⊗⊗⊗⊗⊗⊗
=
++=)()(
)()(/tLXty
tBuXfdtdX=
+=
Carleman bilinearization
Carleman bilinearization: [1] W.J. Rugh, Nolinear System Theory, The John Hopkins University Press, Boltimore, 1981. [2] S. Sastry, Nonlinear Systems: Analysis, Stability and Control, Springer, New York, 1999.
=⊗
BaBa
BaBaBA
mnm
n
1
111
Kronecker product
Bilinearization-bsed MOR Exercise
17
)()(
)()(/tXLty
tuBtuXNXAdtdX⊗⊗
⊗⊗⊗⊗⊗⊗
=
++=How to compute V?
∑∞
=
=1
)()(n
n tyty
nnn
t
nreg
n
t
n dtdtttuttttuttthty 1210 21)(
0)()(),,,()( −−−−−= ∫ ∫
⊗⊗⊗⊗ ⊗−
⊗⊗
= BeNeNeLttth tAtAtATn
regn
nn 11),,,( 21)(
Volterra series expression of bilinear system According to the theory in [Rugh 1981], the output response of the bilinear system can be expressed into Volterra series,
BAAsINAAsINAAsINAAsILBAIsNAIsNAIsNAIsLsssh
nnTn
nnT
nreg
n111
1111
2111
1111
11
12
11
121
)(
)()()()()1(
)()()()(),,,(−−−−−−−−−
−−−−
−−−−
−
−−−−−=
−−−−=
Laplace transform (drop for simplicity): ⊗
Bilinearization-bsed MOR
18
)()(
)()(/tXLty
tuBtuXNXAdtdX⊗⊗
⊗⊗⊗⊗⊗⊗
=
++=How to compute V?
BAAsINAAsINAAsINAAsILBAIsNAIsNAIsNAIsLsssh
nnTn
nnnreg
n111
1111
2111
1111
11
12
11
121
)(
)()()()()1(
)()()()(),,,(−−−−−−−−−
−−−−
−−−−
−
−−−−−=
−−−−=
Laplace transform:
++++=− −−−− in
inn sAsAIAsI 111)(
∑ ∑∞
=
∞
=
−−−−− −−=1 1
11
121
)(
1
111)1(),,,(n
nnn
l l
llllln
nn
regn BANNALAsssssh
BANNALAllm lllnn
nn 11)1(),,( 1−−− −−=
Multimoments:
Bilinearization-bsed MOR
19
)()(
)()(/tXLty
tuBtuXNXAdtdX⊗⊗
⊗⊗⊗⊗⊗⊗
=
++=How to compute V?
∑ ∑∞
=
∞
=
−−−−− −−=1 1
11
121
)(
1
111)1(),,,(n
nnn
l l
llllln
nn
regn BANNALAsssssh
BANNALAllm lllnn
nn 11)1(),,( 1−−− −−=
Multimoments:
},,{span},{}{range 1
1
1111 BABABAAKV q
q−−−− ==
},,{span},{}{range 112
,11
111
−−
−−
−−
−−− == j
qjjjqj NVANVANVANVAAKV j
j
},,{colspan}{range 1 JVVV =
)()(ˆ)()(/
tLVZtytuBVtVZuNVVZAVdtdZ TTT
=++= ⊗⊗⊗
Reduced model:
Bilinearization-bsed MOR
20
Example
0 2 4 6 8 100
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
time, u(t)=0, t<3, u(t)=1, t>=3;
volta
ge a
t nod
e 1
original systemquadratic MOR, q=20quadratic, no reductionbilinearization, q=20
V for bilinearization MOR: },{colspan}{range 21 VVV =
})(,,){(span}{range 1911 BABAV −⊗−⊗=
)}1(:,){(span}{range 11
2 NVAV −⊗=
V for quadratic MOR:
},,{span}{range 201 BABAV −−=
21
Variational analysis-based MOR
+⊗⊗+⊗++≈
+−−+−+=
XXXAXXAXAXf
XXXHXXXXDXfXf fT
f
3210
0000
)(
))(()(21)()()(
Taylor series expansion:
)()()()(/
tLXtytBuXfdtdX
=+=
)()()(~~/ 21
tLXtytuBXXAXAdtdX
=+⊗+=
)()()(~~/ 321
tLXtytuBXXXAXXAXAdtdX
=+⊗⊗+⊗+=
++++== )()()(),0(),( 33
22
1 tXtXtXtXtX ααααα
Variational analysis:
)()()(~~/ 21
tLXtytuBXXAXAdtdX
=+⊗+= α
)()()(~~/ 321
tLXtytuBXXXAXXAXAdtdX
=+⊗⊗+⊗+= α
or
or
Original system:
.0),0( that so ,0)( if ,0)( :Assume ==== tXtutX α
22
+++= )()()()( 33
22
1 tXtXatXtX αα
Variational analysis [11]:
)()()(~~/ 321
tLXtytuBXXXAXXAXAdtdX
=+⊗⊗+⊗+= α
)()()(~~)]()()[(
)]()[(
)(/)(
33
22
133
22
133
22
13
33
22
133
22
12
33
22
1133
22
1
tLXtytuBXXXXXXXXXA
XXXXXXAXXXAdtXXXd
=++++⊗+++⊗++++
+++⊗++++
+++=+++
αααααααααα
αααααα
αααααα
)(~~)(/)(: 111 tuBtXAdttdX +=α
)()(/)(: 1122122 XXAtXAdttdX ⊗+=α
)()()(/)(: 1113122123133 XXXAXXXXAtXAdttdX ⊗⊗+⊗+⊗+=α
Variational analysis-based MOR
23
+++= )()()()( 33
22
1 tXtXatXtX αα
Variational analysis:
)()()(~~/ 321
tLXtytuBXXXAXXAXAdtdX
=+⊗⊗+⊗+= α
)(~~)(/)(: 111 tuBtXAdttdX +=α
)()(/)(: 1122122 XXAtXAdttdX ⊗+=α
)()()(/)(: 1113122123133 XXXAXXXXAtXAdttdX ⊗⊗+⊗+⊗+=α
}~,,~{span 11
111111 BABAVZVX q−−=≈
},,{span 2121
122222 AAAAVZVX q−−=≈
]},[,],,[{span 321321
133332 AAAAAAVZVX q−−=≈
333
222
11
33
22
1
33
22
1 )()()()(
ZVZVZVXXX
tXtXatXtX
ααα
ααα
αα
++≈
++≈
+++=
},,{span)( 321 VVVtX ≈∈
Variational analysis-based MOR
24
Original system:
)()()()(/
tLXtytBuXfdtdX
=+=
)()()(~~/ 321
tLXtytuBXXXAXXAXAdtdX
=+⊗⊗+⊗+=
≈
},,{span)( 321 VVVtX ≈∈
Compute V: },,{orth)(range 321 VVVV =
Reduced model: )()(ˆ
)(~~/ 321
tLVZtytuBVVZVZVZAVVZVZAVVZAVdtdZ TTTT
=+⊗⊗+⊗+=
VZtX ≈)(
Variational analysis-based MOR
25
Example
V for bilinearization MOR:
},{colspan}{range 21 VVV =
)}1(:,{span}{range 12 NVAV −=
V for Variational analysis MOR:
},{span}{range)}6:1(:,{span}{range
}{orth
},,{span}{range
21
02
112
20
2
4111
VVVVAV
AVBABAV
==
=
=
−
−−
0 2 4 6 8 100
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
time, u(t)=0, t<3; u(t)=1, t>=3
volta
ge a
t nod
e 1
original exact output responseBilinearization MOR, q=20, relative error=0.0078Variational analysis MOR, q=9, relative error=0.0073
})(,,){(span}{range 1911 BABAV −⊗−⊗=
26
Example
V for [Phillips 2000] method:
V for our method [Feng 2014]:
},{span}{range)]}6:1(:,[{span}{range
}{orth
},,{span}{range
21
02
112
20
2
4111
VVVVAV
AVBABAV
==
=
=
−
−−
},{span}{range)}6:1(:,{span}{range
)(
},,{span}{range
21
022
11210
2
4111
VVVVV
VVAAVBABAV
==
⊗=
=−
−−
0 2 4 6 8 100
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
time, u(t)=0, t<0.3; u(t)=1, t>=0.3
volta
ge a
t nod
e 1
originalour method, q=9, relative error=0.0073Phillips 2003 method, q=9, relative error=0.0076
27
Trajectory piece-wise linear MOR
)(Xf)(1 Xg
)(2 Xg
)(4 Xg
)(3 Xg
2X 3X
4X
)()()()(/
tLXtytBuXfdtdX
=+=
Original system:
)()(
,)(/1
0
tLXty
BuXgwdtdXs
iii
=
+=∑−
=
1,,1,0),()()( −=−+= siXXAXfXg iiii
iXk
jjkjki x
XfaaA
∂
∂==
)(),(
1X
28
Trajectory piece-wise linear MOR
)()()()(/
tLXtytBuXfdtdX
=+=
Original system:
)()(
,)(/1
0
tLXty
BuXgwdtdXs
iii
=
+=∑−
=
))((
1,,1,0),()()(
iiii
iiii
XAXfXA
siXXAXfXg
−+=
−=−+=
)()(
),()(/1
0
tLXty
tBuwBXAwdtdX ii
s
iii
=
++=∑−
=
iiiiii XAXfBBBB −== )(],,[~
How to compute V?
},,{span}{range1,,1,0}~,,~{span}{range
11
1
−
−−
=−==
s
iq
iiii
VVVsiBABAV i
29
Trajectory piece-wise linear MOR
)()()()(/
tLXtytBuXfdtdX
=+=
Original system:
)()(ˆ
),((/1
0
tLVZty
tBuVwBVVZAVwdtdZ Tii
Ts
ii
Ti
=
++=∑−
=
Trajectory piece-wise linear system:
Reduced model:
)()(
),()(/1
0
tLXty
tBuwBXAwdtdX ii
s
iii
=
++=∑−
=
30
Proper orthogonal decomposition (POD)
POD and SVD SVD:
nmTT RD
YVUVUY ×∈Σ=
=Σ= :
000
or
s.t. ,),,( and ),,(exist there,matrix any For 11nn
nmm
mnm RvvVRuuURY ××× ∈=∈=∈
ly.respective and of columns first theincluding matrices thebe and Let ).,,(diag Here, 1
VUdVUD dd
dσσ =
It is obvious,
∑ ∑∑∑∑
∑∑
= ====
===
==
==
==Σ=⇒
==
d
i
d
iiRji
d
iiRiji
m
kkj
dki
d
iiij
Td
d
iiij
TddTdd
iiij
Tdij
Tdi
d
ij
Tddn
uyuuuyuYUuYU
uVDUUuVDVDuy
VDUyyY
mm
1 1111
111
1
.,,))((
))()(())(())((
)(),,(
Y can be represented in terms of d linearly independent columns of . dU
31
Proper orthogonal decomposition (POD)
Definition
.,1 ,~,~ s.t. minarg}{1
~,,~11
ljiuuu ijRji
n
jj
Ruu
lii mm
l
≤≤== ∑=∈
= δε
.rank of basis POD called are }{ vectors the},,,1{For 1 ludl lii =∈
: of columns theingapproximatin ions,approximat rank all among optimal, is }{ basis POD The 1
Ylu l
ii =
2
1||~~,|| Here, mm RiRij
l
ijj uuyy=Σ−=ε
32
Model Order Reduction using POD
)())(()(
ROM get the to Use.3
)~,,~( ,~~
of from rank of vectorsPOD Get the 2.
),( snapshots get the tosystemnonlinear original theSolve 1.
1
1
tBuVtVzfVdt
tdzEVV
VuuVVUX
XSVDq
xxX
TTT
qT
tt N
+=
=Σ=
=
Algorithm MOR using POD
How to deal with ? An effective way is to approximate the nonlinear function by projecting it onto a subspace with dimension , that approximates the subspace spanned by the snapshots of the nonlinear function.
))(( tVzf
nl <<
),,( ),())(( 1f
lfff uuUtcUttxf =≈
33
DEIM
).()( s.t. ,],,[ matrix a find : toequivalent is This
1tcUPtfPReeP fTTmn
l=∈= ×
℘℘
?,,1, indices especify th tohow and compute toHow liU i =℘
Compute :
U
).,,( 3.)( : toApply 2.
)).(,),((matrix a into ))(( of snapshots eCollect th 1.
1
1
Fl
Ff
TFF
Nt
uuUVUFFSVD
xfxfFtxf
=
Σ=
=
:points at )( esinterpolat )( that require we,)( determine To nltftcUtc f <<
thenr,nonsingula is Suppose UPT
).()()()( that,so
)()()()()(
1
1
tfPUPUtcUtf
tfPUPtctcUPtfP
TfTff
TfTfTT
−
−
=≈
=⇒=
34
DEIM
Using DEIM to decide the indices: Algorithm Discrete Empirical Interpolation Method (DEIM) { }
{ }
{ }
for end .8
], [ ], [ .7
||max]|,[| .6 .5
),,( where,for )( Solve 4.do to2for .3
][ ],[ ],[ .2
|| max]|,.[|1
],[ :Output
Ffor basis POD :Input
T11
11
11
1
1
1
℘℘
←℘←←
=℘−=
==
=
℘=℘==
=℘
∈℘℘=℘
℘
−
℘
=
i
Fi
ff
i
fFi
iFi
TfT
Ff
F
lTl
li
Fi
iePPuUU
rUur
uPUPli
ePuUu
Ru
ρα
ααααα
ρ
35
DEIM
Come back to :
))(( tVzfV T
)).(()())(( 1 tVzfPUPUtVzf TT −≈
. oft independen is ROM solving during ))(( ofn Computatio).(~ where
)))((,)),((())((
))(()())(( Finally,
1
1
ntVzfVtVzx
tVzftVzftVzfP
tVzfPUPUVtVzfV
T
TT
TTTT
l
=
=
≈
℘℘
−
can be precomputed before solving the ROM
36
References
Quadratic MOR: [1] Yong Chen, “Model order reduction for nonlinear systems”, MIT master thesis, 1999. Bilinearization MOR: [2] J. R. Phillips, “Projection Frameworks for Model Reduction of Weakly Nonlinear Systems”, Proc. IEEE/ACM DAC, 184-189, 2000. [3] Z. Bai, and D. Skoogh, “A projection method for model reduction of bilinear dynamical systems”, Linear Algebra Appl., 415: 406-425, 2006. [4] L. Feng and P. Benner, A Note on Projection Techniques for Model Order Reduction of Bilinear Systems, AIP Conf. Proc. 936: 208-211, 2007.
Model Order Reduction of Parametric Systems 37 21.06.2019
Variational analysis MOR: [5] J. R. Phillips, “Automated Extraction of Nonlinear Circuit Macromodels ”, Proc. of IEEE Custom Itegrated Circuits Conference, 451-454, 2000. [6] L. Feng, X. Zeng, Ch. Chiang, D. Zhou, Q, Fang “Direct nonlinear order reduction with variational analysis”, Proc. of Design, Automation and Test in Europe Conference and Exhibition, 2:1316-1321, 2004. Trajectory piece-wise linear (polynomial) MOR: [7] M. Rewienski and J. White, “ A Trajectory Piecewise-Linear Approach to Model Order Reduction and Fast Simulation of Nonlinear Circuits and Micromachined Devices”, IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., 22(2):155-170, 2003. [8] M. Rewienski and J. White, “ Model order reduction for nonlinear dynamical systems based on trajectory piecewise-linear approximations ”, Linear Algebra Appl., 415(2-3):426-454, 2006.
References
Model Order Reduction of Parametric Systems 38 21.06.2019
POD: [9] S. Volkwein, “Model reduction using proper orthogonal decomposition ”, Lecture notes, June 7, 2010. [10] S. Schaturantabut and D. C. Sorensen, “Nonlinear model reduction via discrete empirical interpolation”, SIAM J. Sci. Comput. 32(5): 2737-2764, 2010. A very good book for nonlinear system: [11] W. J. Rugh, “Nonlinear system theory, the Volterra/Wiener Approach”, The Johns Hopkins University Press, 1981.
References
Recommended