Eigenvalue problems and NXP PowerPoint template (Title ... › casa › meetings › special ›...

NXP PowerPoint template (Title)Template for presentations (Subtitle)

Name

Subject

Project

MMMM dd, yyyy

Eigenvalue problems andmodel order reductionin the electronics industry

Joost Rommes [[email protected]]NXP Semiconductors/Corp. I&T/DTF/A&M/PDM/MathematicsO-MOORE-NICE!

COMSON Autumn School on MOR, TerschellingSeptember 21–25, 2009

Subject/Department, Author, MMMM dd, yyyy

CONFIDENTIAL 3

Introduction

Part I: Examples from the electronics industry

Part II: Large networks: MOR or fast solvers?

Part III: Eigenvalue problems in MOR

Concluding remarks

2/64NXP Semiconductors Corp. I&T/DTF, Joost Rommes, September 21–25, 2009


CONFIDENTIAL 3

Introduction

I More functionality on smaller chips

I Decreasing device sizes

I Increased complexity

I Need for fast design cycles

I Need for first-time-right

I Many transistors and . . .

I . . . many parasitic effects

I Need for advanced mathematicalmethods!


Outline

Part IExamples from the electronics industry

I Electro Static Discharge analysis

I Extracted parasitics

I Handle wafer simulations

I Layout optimization and place/route

I Pole-zero analysis and sensitivities



CONFIDENTIAL 3

Electro Static Discharge analysisDamaged interconnect that was too small to conduct current



CONFIDENTIAL 3

Extracted parasitics

Analogue design involves

1. Design circuit on schematic level

2. Then refine layout for parasitics

I Typical parasitics: R, L, C

I Up to millions of RLCs

I Heavy demand on analoguesimulator

Hence MOR and/or fast solvers are needed!



CONFIDENTIAL 3

Handle wafer simulations

I Noise propagates via handle wafer

I Where to place handle wafer contacts to limit noise?



CONFIDENTIAL 3

Handle wafer simulations

Two needs:

1. Given HW contacts, how big is noise? (AC analysis: sparsesolvers)

2. Equivalent circuit model for full system simulation (MOR)

Work with Maria Ugryumova, and Kapora/De Haas/Bakker (NXP)



CONFIDENTIAL 3

Layout optimization, place/route

Analogue design involves

1. Design circuit on schematic level

2. Optimize schematic circuit

3. Repeat on layout level, includingparasitics

I Can we optimize automatically?

I Can we place and routeautomatically?

I Yes, by parameterization andderivative-free optimization

Work with Beelen/Jeurissen/De Jong/Schilders (NXP)



CONFIDENTIAL 3

Pole-zero stability analysisFrequency response of circuit (1000 unknowns). Pole+994 · 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



CONFIDENTIAL 3

Sensitive eigenvalues

How to compute the eigenvalues that are most sensitive toparameter changes?

−35 −30 −25 −20 −15 −10 −5 00

2

4

6

8

10

12

14

16

18

real (1/s)

imag

(ra

d/s)

KPSS = 1

KPSS = 2,...,50

Work with Nelson Martins (CEPEL)


Outline

Part IILarge networks: MOR or fast solvers?



CONFIDENTIAL 3

Resistor network modelingKirchhoff’s Current Law and Ohm’s Law for resistors lead to[

R −PPT 0

] [ibv

]=

[0in

]or Gv = in

where

I N is number of resistors

I n is number of nodes

I R = diag(r1, . . . , rN) ∈ RN×N (resistances)

I P ∈ {−1, 0, 1}N×n (incidence matrix)

I ib ∈ RN (resistor currents)

I in ∈ Rn (injected node currents)

I v ∈ Rn (node voltages)

I G = PTR−1P ∈ Rn×n with G = GT ≥ 0



CONFIDENTIAL 3

Resistor network modeling

I Kirchhoff’s Current Law and Ohm’s Law for resistors lead to[R −P

PT 0

] [ibv

]=

[0in

]I Eliminating resistor currents ib (MNA):

Gv = in,

where G = PTR−1P ∈ Rn×n with G = GT ≥ 0

I Systems made independent by grounding v(gnd) = 0

I Systems are very sparse in general



CONFIDENTIAL 3

Results for computing resistor currents1. Reorder G (AMD)

2. Solve v from Gv = in (sparse: Cholesky)

3. Compute ib = R−1Pv

Table: CPU times for Spectre, Pstar, and fastR. * denotes error.

#nodes #Rs Spectre Pstar fastR Speedup

1 5k 6k 3s 2.15 0.2s 15 102 19k 46k 180s 149s 0.8s 225 1863 32k 94k 573s 400s 1.5s 382 2674 32k 99k 600s 400s 1.5s 400 2675 36k 188k 2900s* 481s 7s 414* 686 72k 476k 17700s 6600s 8s 2212 8257 560k 917k 64300s* * 21s 3060* ∞8 586k 1.5M * * 23s ∞ ∞9 2M 6M * * 330s ∞ ∞



CONFIDENTIAL 3

Reduction of large networks

I Speed-up design cycle and enable First-Time-Right

I Enlarge capacity of optimization software



CONFIDENTIAL 3

Reduction of resistor networks

I Given a very large resistor network described by I = GV , findan equivalent network with

(a) same terminals(b) same path resistances between terminals(c) n � n internal nodes(d) r � r resistors(e) easy (re)use in design flow (netlist)

Requirement (d) is essential in practice

I Use tools from graph theory to recover part of the structure



CONFIDENTIAL 3

Reduction of resistor networksEliminating all internal nodes not an option:

I satisfies conditions (a)–(c),

I but violates (d) and (e): r = (m2 −m)/2 resistors in ROM!

I example with 12738 nodes, 340 terminals, 21209 resistors

ROM has 0 internal nodes, same terminals, but 57630 resistors!



CONFIDENTIAL 3

Outline of reduction approach

1. Select components for reduction

2. Find important internal nodes

3. Eliminate non-important internal nodes

4. Synthesize reduced resistor network

I No approximations are made: resulting ROM is exact!

I Paper ”Efficient methods for large resistor networks” toappear in IEEE TCAD (with Wil Schilders)



CONFIDENTIAL 3

Step 1: Select components for reduction

1

23

4

5

6

78

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

81

76

7778

79

80

82

8384

85

86

91

87

88

89

90

92

93

9495

9697

98

99

100

101

102

103

104

105

10611

1

107

108

109

110

11211

4

11311

8

115

116

117

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

13413

9

135

136

137

138

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

15615

7

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

195

196

197

198

199

200

201

202

203

204

205

206

207

208

209

210

211

212

21321

4

215

216

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

246

247

248

249

250

251

252

253

254

255

256

257

258

259

260

261

262

263

264

265

266

267

268

269

270

271

272

273

274



CONFIDENTIAL 3

Step 2: Find important internal nodes

1

2 3

4

5 6

7

89

10

1112

13

1415

16

1718

1920

21

2223

24

2526

27

28

29

30

31

3233

34

3536

37

38

39

40

41

42

43 44

45

4647

48

49 50

5152

53

54

Graph and matrix reordering algorithms (AMD)



CONFIDENTIAL 3

Step 3: Eliminate non-important internal nodes

0 10 20 30 40 50 60

0

10

20

30

40

50

60

nz = 34810 10 20 30 40 50 60

0

10

20

30

40

50

60

nz = 1506

Figure: Left: 59 nodes, 1711 resistors. Right: 64 nodes, 721 resistors

I Left: eliminate all internal nodes

I Right: preserve five internal nodes

I Selective (AMD) elimination reduces fill-in

I Terminals are preserved



CONFIDENTIAL 3

Step 4: Synthesize reduced order resistor network

Reduced netlist can be (re)used in design flow



CONFIDENTIAL 3

Results for reducing networks

I Results for four resistor networks

I ESD analysis of realistic layouts (I–III)

I Parasitic interconnect reduction (IV)

Network I Network II Network III Network IVOrig ROM Orig ROM Orig ROM Orig ROM

#terminals 3260 1978 15299 8000#int nodes 99k 8k 101k 1888 1M 180k 46k 6k#resistors 161k 56k 164k 39k 1.5M 376k 67k 26k

#other devs 1874 1188 8250 29k#other nodes 0 0 0 11k

CPU red 130 s 140 s 1250 s 75 sCPU sim 67h 6h 20h 2h – 120h – 392sSpeed up 11x 10x ∞ ∞



CONFIDENTIAL 3

RC and RCLk networks

I Reduction not always needed (AC, pole-zero analysis)I If reduction is needed for many-terminal systems

I Use similar graph algorithmsI Elimination like in R networks not accurateI Use conventional MOR methods like

I Moment matching via Krylov [Celik, Grimme, . . . ]I Balanced truncation/ADI [Moore, Benner, Stykel, Freitas, . . . ]I Modal truncation via dominant poles [Martins, R., . . . ]I Spectral zero interpolation [Antoulas, Sorensen, Ionutiu, . . . ]

I Solves state reduction in general

I The big open question is:

How to reduce number of elements significantly?

Work with Roxana Ionutiu (linear) and Michael Striebel (nonlinear)


Outline

Part IIIEigenvalue problems in MOR



CONFIDENTIAL 3

Computation of system properties

Eigenvalues and eigenvectors are important forI Determining system stability

I Are there λ with Re(λ) > 0?

I Determining resonance frequenciesI Dominant poles and corresponding eigenvectors

I Determining parameter sensitivities

I PSS computations

I . . .



CONFIDENTIAL 3

The generalized eigenvalue problem (non-defective)

Given A,E ∈ Rn×n, find (λ, x, y) that satisfy

Ax = λEx, x 6= 0

y∗A = λy∗E , y 6= 0

An eigentriplet (λ, x, y) consists of

λ ∈ C eigenvalue

x ∈ Cn right eigenvector

y ∈ Cn left eigenvector

I (A,E ) has n eigenvalues (real / complex conjugated pairs)

I Corresponding eigenspaces need not be n-dimensional

I Bi-orthogonality: λi 6= λj ⇒ y∗j Exi = 0



CONFIDENTIAL 3

Eigenvalue decompositions

Complete eigenvalue decomposition (Λ,X ,Y ):

AX = EXΛ, Y ∗A = ΛY ∗E with Y ∗EX = I ,Y ∗AX = Λ

Λ = diag(λ1, λ2, . . . , λn) ∈ Cn×n

X = [x1, x2, . . . , xn] ∈ Cn×n

Y = [y1, y2, . . . , yn] ∈ Cn×n

In practice only interest in k � n eigentriplets: partial ED

AXk = EXkΛk , Y ∗k A = ΛkY ∗

k E with Y ∗k EXk = I ,Y ∗

k AXk = Λk

Λk = diag(λ1, λ2, . . . , λk) ∈ Ck×k

Xk = [x1, x2, . . . , xk ] ∈ Cn×k

Yk = [y1, y2, . . . , yk ] ∈ Cn×k



CONFIDENTIAL 3

Eigenvalue computations

Methods for complete eigendecompositions:

I QR method for AX = XΛ

I QZ method for AX = EXΛ

I Complexity O(n3), practical use up to n ≈ 3000

Methods for partial eigendecompositions:

I Krylov methods (Lanczos, Arnoldi)

I Newton based methods (Jacobi-Davidson [Sleijpen, Van derVorst (1995)])

I No dense matrix computations needed

I Careful selection strategies needed



CONFIDENTIAL 3

First-order dynamical systems

First-order SISO dynamical system{E x(t) = Ax(t) + bu(t)y(t) = c∗x(t) + du(t)

where

u(t), y(t), d ∈ R, input, output, direct i/o

x(t),b, c ∈ Rn, state, input-to-, -to-output

E ∈ Rn×n, system (descriptor) matrix

A ∈ Rn×n, system matrix



CONFIDENTIAL 3

Transfer functionFirst-order SISO dynamical system (d = 0):{

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 )



CONFIDENTIAL 3

Transfer function H(s) = c∗(sE − A)−1b (proper)

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



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)

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)

Hence pole λ with large∣∣∣ RRe(λ)

∣∣∣ causes peak (dominant)



CONFIDENTIAL 3

Dominant poles cause peaks in Bode-plot



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



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



CONFIDENTIAL 3

DPA: Complexity

Every iteration solves of

(skE − A)vk = b and (skE − A)∗wk = c

Can be done efficiently:

I Fill-in minimizing reordering: AMD [Davis, Duff]

I Reuse LU = skE − A as U∗L∗ = (skE − A)∗

I Scalable up to millions of unknowns for sparse systems

If solves cannot be done exactly (CPU/MEM):

I Jacobi-Davidson style methods [Sleijpen/Van der Vorst,Hochstenbach, R.]

I Recycling Krylov spaces [De Sturler]



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)]



CONFIDENTIAL 3

Convergence behavior: DPA vs. RQI



CONFIDENTIAL 3

Dominant Pole Algorithm (DPA) and extensionsDPA computes dominant poles of H(s) = c∗(sE − A)−1b

1. Newton scheme [M., Lima, Pinto (IEEE TPWRS 11(1) 1996]

2. Nice convergence behavior [R., Sleijpen (SIMAX 30(1) 2008)]

3. Subspace acceleration, selection, deflation: SADPA [R.,Martins (IEEE TPWRS 21(3) 2006)]

4. Poles of MIMO systems: SAMDP [R., Martins (IEEE TPWRS21(4) 2006)]

5. Dominant zeros [R., Martins, Pellanda (IEEE TPWRS 22(4)2007)]

6. Poles of second-order systems: QDPA [R., Martins (SISC30(4) 2008)]

7. Spectral zeros [Ionutiu, R., Antoulas (IEEE TCAD 27(12)2008]

8. Sensitive poles: SPA [R., Martins (IEEE TPWRS 23(2) 2008)]



CONFIDENTIAL 3

Computation of most sensitive eigenvalues

I Suppose system matrix A depends on parameter p

I Sensitivity of eigenvalue is given by

∂λ

∂p= y∗

∂A

∂px

where y and x are left and right eigenvectors

I In stabilizer design one is interested in eigenvalues mostsensitive to parameter changes

I See [R., Martins, IEEE TPWRS 23(3) 2008] for more details



CONFIDENTIAL 3

Computation of most sensitive eigenvalues

−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: Root locus of most sensitive eigenvalues for 13k state system.



CONFIDENTIAL 3

Residues and sensitivitiesI Recall

H(s) = c∗(sI − A)−1b =n∑

i=1

Ri

s − λi,

where residues Ri are

Ri = (c∗xi )(y∗i b),

and (λi , xi , yi ) are eigentriplets

I If ∂A∂p has rank 1:

∂λ

∂p= y∗

∂A

∂px = y∗(bc∗)x = (y∗b)(c∗x) = (c∗x)(y∗b)

I Sensitive eigenvalues of A are dominant poles of (A,b, c)

I Apply DPA to (A,b, c) to compute sensitive eigenvalues!



CONFIDENTIAL 3

The case rank(∂A∂p ) > 1

I Derivative of A to p can be written as ∂A∂p =

∑ki=1 bic

∗i

I Sensitivity of eigenvalue is given by

∂λ

∂p= y∗

∂A

∂px =

k∑i=1

(y∗bi )(c∗i x) = trace(C ∗x)(y∗B).

where y and x are left and right eigenvectors

I SAMDP can be used to compute sensitive eigenvalues

I More effective alternative is to update right-hand sides

b =∂A

∂pvk and c =

∂A

∂p

∗wk



CONFIDENTIAL 3

Sensitive Pole Algorithm (SPA)

1: Set v0,w0 ∈ Rn s.t. Apv0 6= 0 and A∗pw0 6= 02: for k = 0, 1, 2, . . . do3: b = Apvk/‖Apvk‖2

4: c = A∗pwk/‖A∗pwk‖2

5: Solve vk+1 ∈ Cn from (skE − A)vk+1 = b6: Solve wk+1 ∈ Cn from (skE − A)∗wk+1 = c7: Compute the new pole estimate

sk+1 = sk −c∗vk+1

w∗k+1Evk+1

8: The pole λ = sk+1 with x = vk+1/‖vk+1‖2 andy = wk+1/‖wk+1‖2 has converged if

‖Ax− sk+1Ex‖2 < ε

9: end for46/64

NXP Semiconductors Corp. I&T/DTF, Joost Rommes, September 21–25, 2009


CONFIDENTIAL 3

Multiple parameters

I A ≡ A(p1, p2, . . . , pk) depends on multiple parameters pi

I Derivative (gradient) of λ is given by:

∇λ = (y∗∂A

∂p1x, . . . , y∗

∂A

∂pkx)

I Sensitivity in unit direction di ∈ Rk for pi

sensitivity(λ,d) ≡ |∇λ · d|

Note directional derivative can be written as

∇λ · d =k∑

i=1

diy∗ ∂A

∂pix = y∗Apx

I Use SPA (steps 3 and 4) with Ap where Ap =∑k

i=1 di∂A∂pi

I Entries of d typically equal to parameter increments



CONFIDENTIAL 3

41-State power system

I Descriptor realization has dimension n = 136

I Power system stabilizer gain (Kpss) controlled by a116,115

aj116,115 = j , (j = 2, . . . , 50)

I Most sensitive poles with respect to Kpss:

pole y∗ ∂A∂p x symbol hits SPA hits RQI

−22.1 + 4.64i −1.24 + 2.00i x 48% 35%−5.37 + 5.64i +0.10 + 1.23i circle 23% 4%+0.53 + 5.41i −0.13 + 0.02i square 7% 3%−1.28 + 4.37i −0.04− 0.11i * 1% 3%



CONFIDENTIAL 3

−35 −30 −25 −20 −15 −10 −5 00

2

4

6

8

10

12

14

16

18

real (1/s)

imag

(ra

d/s)

KPSS = 1

KPSS = 2,...,50

Figure: Full spectra of the 41-state power system matrices (Aj ,E ), where

aj116,115 = j (j = 1, . . . , 50), computed by the QZ method.



CONFIDENTIAL 3

−35 −30 −25 −20 −15 −10 −5 00

2

4

6

8

10

12

14

16

18

real (1/s)

imag

(ra

d/s)

Λ(A1)

SASPA(A1)

SASPA(Ai)

Figure: Sensitive pole traces for (Aj ,E ), where aj116,115 = j

(j = 1, . . . , 50), computed by SASPA with s0 = 1i (six poles every run).



CONFIDENTIAL 3

BIPS/97: 1664-state system

I Descriptor realization has dimension n = 13250

I Single and multiple parameter root loci

Table: Generators, PSS gain ranges, and increments for BIPS(cf. Figure 5 and 6).

generator element gain (min, max) increment

Xingo a4971,4970 (0, 15) 0.5Paulo Afonso IV a4901,4900 (0, 15) 0.5

Itaipu Kp a4973,4972 (0, 2.2) 0.0733Itaipu Kw a4978,4977 (0, 10.35) 0.345



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.



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.



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



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?



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).



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).



CONFIDENTIAL 3

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).



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?



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?



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



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



CONFIDENTIAL 3

Thank you!

[email protected]/site/rommes

MOOREN I C E!



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!


Eigenvalue problems and NXP PowerPoint template (Title ... › casa › meetings › special ›...

Documents

Transcript of Eigenvalue problems and NXP PowerPoint template (Title ... › casa › meetings › special ›...