A Tale of Two Eigenvalue ProblemsMusic of the Microspheres + Hearing the Shape of a Graph

David Bindel

Department of Computer ScienceCornell University

Brown Bag Talk, 26 Nov 2013

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The Computational Science & Engineering Picture

Application

Analysis Computation

MEMSSmart gridsNetworks

Linear algebraApproximation theorySymmetry + structure

HPC / cloudSimulatorsLittle languages

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The purpose of computing is insight, not numbers.Richard Hamming

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G. H. Bryan (1864–1928)

Fellow of the Royal Society (1895)Stability in Aviation (1911)Thermodynamics, hydrodynamics

Bryan was a friendly, kindly, veryeccentric individual...(Obituary Notices of the FRS)

... if he sometimes seemed a colossal buffoon, he himself didnot help matters by proclaiming that he did his best workunder the influence of alcohol.(Williams, J.G., The University College of North Wales, 1884–1927)

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Bryan’s Experiment

“On the beats in the vibrations of a revolving cylinder or bell”by G. H. Bryan, 1890

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Bryan’s Experiment Today

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The Beat Goes On

0.00 6.48 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90 5.00 5.10 5.20 5.30 5.40 5.50 5.60 5.70 5.80 5.90 6.00 6.10 6.20 6.30 6.40

Audio Track

0.00 8.36 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30 4.40 4.50 4.60 4.70 4.80 4.90 5.00 5.10 5.20 5.30 5.40 5.50 5.60 5.70 5.80 5.90 6.00 6.10 6.20 6.30 6.40 6.50 6.60 6.70 6.80 6.90 7.00 7.10 7.20 7.30 7.40 7.50 7.60 7.70 7.80 7.90 8.00 8.10 8.20

Audio Track

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The Beat Goes On

99.4

99.6

99.8

100

100.2

100.4

100.6

0 0.2 0.4 0.6 0.8 1

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 2 4 6 8 10 12 14

Free vibrations in a rotating frame (simplified):

¨

q+ 2�⌦J ˙

q+ !20q = 0, J ⌘

0 �1

1 0

�

Eigenvalue problem:��!2

I+ 2i!�⌦J+ !20

�q = 0.

Solutions: ! ⇡ ⌦0 ± �⌦. =) beating / ⌦!

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A General Picture

=q1(t) +q2(t)

q2

q1

q2

q1

q1(t)q2(t)

�⇡

cos(��⌦t) � sin(��⌦t)sin(��⌦t) cos(��⌦t)

� q01(t)q02(t)

�.

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Foucault in Solid State

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A Small Application

Northrup-Grummond HRG(developed c. 1965–early 1990s)

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A Smaller Application (Cornell)

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A Smaller Application (UMich, GA Tech, Irvine)

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Uncritical FEA: Fail!

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The Perturbation Picture

Perturbations split degenerate modes:Coriolis forces (good)Imperfect fab (bad, but physical)Discretization error (non-physical)

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Perfect Geometry

Free vibration problem in weak form

8w, b(w,a) + a(w,u) = 0.

wherea =

¨

u+ 2⌦⇥ ˙

u+⌦⇥ (⌦⇥ x) +

˙

⌦⇥ x

Expand u (and w) in Fourier series:

u(r, z, ✓) =

1X

m=0

�

c

m

(✓)uc

m

(r, z) +

1X

m=0

�

s

m

(✓)us

m

(r, z)

Discretize by finite elements:

M

¨

u

h

+C

˙

u

h

+K

¨

u

h

= 0

where C comes from Coriolis term (2b(w,⌦⇥ ˙

u)).Brown bag 16 / 46

Block Diagonal Structure

K =

2

666666666664

K

cc

0

K

ss

1

K

cc

1

K

ss

2

K

cc

2. . .

K

ss

M

K

cc

M

3

777777777775

Mass has same structure.

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Block Structure of Finite Element Matrix

Discretize 2b(w,⌦⇥ ˙

u):

C =

2

6664

C00 C01

C10 C11 C12

C21 C22 C23. . . . . . . . .

3

7775

Off-diagonal blocks come from cross-axis sensitivity:

⌦ = ⌦

z

e

z

+⌦

r✓

=

2

40

0

⌦

z

3

5+

2

4cos(✓)⌦

x

� sin(✓)⌦y

sin(✓)⌦x

+ cos(✓)⌦y

0

3

5 .

Neglect cross-axis effects (O(⌦

2/!20), like centrifugal effect).

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Analysis in Ideal Case

Only need to mesh a 2D cross-section!Compute an operating mode u

c

for the non-rotating geometry.Compute � = b(u

c

, ez

⇥ u

s

).Model: motion is approximately q1uc

+ q2us

, and

¨

q+ 2�⌦J ˙

q+ !20q = 0,

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Representing Imperfection

R 7! R+ ✏ (R)

B0 B

Write Fourier series for , too!

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Typical Fabrication Imperfections

p = 0 p = 1 p = 2 p = 4

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Block matrix structure

Ex: p = 2

K =

2

666666666664

K0 ✏ ✏2 ✏3

K1 ✏ ✏2

✏ K2 ✏ ✏2

✏ K3 ✏✏2 ✏ K4 ✏

✏2 ✏ K5

✏3 ✏2 ✏ K6. . .

3

777777777775

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Qualitative Information

Operating wave number m, perturbation number p:

p = 2m frequencies split by O(✏)

kp = 2m frequencies split at most O(✏2)

p 6 |2m frequencies change at O(✏2), no splitp = 1

p = 2m± 1 O(✏) cross-axis coupling.

Note:m = 2 affected at first order by p = 0 and p = 4

(and O(✏2) split from p = 1 and p = 2).m = 3 affected at first order by p = 0 and p = 6

(and O(✏2) split from p = 1 and p = 3).

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Analyzing Imperfect Rings

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Analyzing Imperfect Rings

q2

q1

q2

q1

q2

q1

q2

q1

q2

q1

q2

q1

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Read All About It!

Yilmaz and Bindel“Effects of Imperfections on Solid-Wave Gyroscope Dynamics”Proceedings of IEEE Sensors 2013, Nov 3–6.

Thanks to DARPA MRIG + Sunil Bhave and Laura Fegely.

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Can One Hear the Shape of a Drum?

�r2u = �u on ⌦

u = 0 on @⌦

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What Do You Hear?

S

A B

Size of bottlenecks (Cheeger inequality)

h 2

p�2

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What Do You Hear?

Volume (Weyl law)

lim

x!1

N(x)

xd/2= (2⇡)�d!

d

vol(⌦), N(x) = {# eigenvalues x}

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Can One Hear the Shape of a Graph?

From eigenvalues of adjacency, Laplacian, normalized Laplacian?

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What Do You Hear?

Size of separators (Cheeger inequality)

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What Do You Hear?

What information hides in the eigenvalue distribution?1 Discretizations of Laplacian: something like Weyl’s law2 Sparse random graphs: Wigner semicircular distribution3 “Real” networks: less well understood

But computing all eigenvalues seems expensive!

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Exploring Spectral Densities

Kernel polynomial method (see Weisse, Reviews of Modern Physics)Think of spectral distribution as a generalized function

Z 1

�1µ(x)f(x) dx =

1

N

NX

k=1

f(�k

)

Write f(x) =P1

j=1 cjTj

(x) and µ(x) =P1

j=1 dj�j

(x), whereR 1�1 �j

(x)Tk

(x) dx = �jk

Estimate dj

=

1N

tr(Tj

(A)) =

1N

E[zTTj

(A)z], z a random probevectorTruncate series for µ(x) and filter (avoid Gibbs)

Much cheaper than computing all eigenvalues!

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Exploring Spectral Densities (with David Gleich)

Consider spectrum of normalized Laplacian (random walk matrix)Approximate via KPM and compare to full eigencomputation

Things we knowEigenvalues in [�1, 1]; nonsymmetric in generalStability: change d edges, have

�j�d

ˆ�j

�j+d

kth moment = probability of return after k-step random walkEigenvalue cluster near 1 ⇠ well-separated clustersEigenvalue cluster near 0 ⇠ triangles connected by one node

What else can we “hear”?

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Erdos

−1 −0.5 0 0.5 1 1.50

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

20

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200

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Internet topology

−1 −0.5 0 0.5 1 1.50

2000

4000

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10000

12000

14000

16000

18000

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

50

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500

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Marvel characters

−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−200

0

200

400

600

800

1000

1200

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Marvel comics

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

1000

2000

3000

4000

5000

6000

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

50

100

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PGP

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500

3000

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

50

100

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500

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Yeast

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

100

200

300

400

500

600

700

800

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

60

70

80

90

100

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Enron emails (SNAP)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

2000

4000

6000

8000

10000

12000

14000

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Reuters911 (Pajek)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500

3000

3500

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US power grid (Pajek)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

100

200

300

400

500

600

700

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DBLP 2010 (LAW)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6x 10

4

N = 326186, nnz = 1615400, 80 s (1000 moments, 10 probes)

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Hollywood 2009 (LAW)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

4x 10

5

N = 1139905, nnz = 113891327, 2093 s (1000 moments, 10 probes)

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What Do You Hear?

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

120

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200

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

50

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450

500

−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−200

0

200

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800

1000

1200

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

50

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−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

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500

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

10

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90

100

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

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14000

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

500

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3500

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

100

200

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400

500

600

700

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6x 10

4

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

4x 10

5

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