A Tale of Two Eigenvalue Problems

Transcription

1 A Tale of Two Eigenvalue Problems Music of the Microspheres + Hearing the Shape of a Graph David Bindel Department of Computer Science Cornell University CAM Visit Talk, 28 Feb 214 Brown bag 1 / 22

2 The Computational Science & Engineering Picture Application Analysis Computation MEMS Smart grids Networks Linear algebra Approximation theory Symmetry + structure HPC / cloud Simulators Solvers Brown bag 2 / 22

3 Music of the Spheres! On the beats in the vibrations of a revolving cylinder or bell by G. H. Bryan, 189 Brown bag 3 / 22

4 Audio Track Audio Track The Beat Goes On Brown bag 4 / 22

5 Foucault in Solid State Brown bag 5 / 22

6 A General Picture =q 1 (t) +q 2 (t) q 2 q 2 q 1 q 1 Brown bag 6 / 22

7 A Small Application Northrup-Grumman HRG (developed c early 199s) Brown bag 7 / 22

8 A Smaller Application (Cornell) Brown bag 8 / 22

9 Uncritical FEA: Fail! Brown bag 9 / 22

10 The Perturbation Picture Perturbations split degenerate modes: Coriolis forces (good) Imperfect fab (bad, but physical) Discretization error (non-physical) Brown bag 1 / 22

11 Fun with Fourier Perfect geometry: Fourier expand modal displacements Imperfect geometry: Fourier expand geometric distortion, too! R 7! R + (R) B B Brown bag 11 / 22

12 Typical Fabrication Imperfections p= p=1 p=2 p=4 Imperfections are small and structured. Use for: Fast quantitative information via structured FEA Qualitative information for engineers in early design Fourier analysis, group theory, eigenvalue perturbations, FEA,... Brown bag 12 / 22

13 Analyzing Imperfect Rings Brown bag 13 / 22

14 Analyzing Imperfect Rings q 2 q 2 q 2 q 1 q 1 q 1 q 2 q 2 q 2 q 1 q 1 q 1 Brown bag 14 / 22

15 Read All About It! Yilmaz and Bindel Effects of Imperfections on Solid-Wave Gyroscope Dynamics Proceedings of IEEE Sensors 213, Nov 3 6. Thanks to DARPA MRIG + Sunil Bhave and Laura Fegely. Brown bag 15 / 22

16 Can One Hear the Shape of a Drum? r 2 u = u on u Brown bag 16 / 22

17 What Do You Hear? A B S Size of bottlenecks (Cheeger inequality) 2 Volume (Weyl law) asymptotic distribution of n Brown bag 17 / 22

18 Can One Hear the Shape of a Graph? From eigenvalues of adjacency, Laplacian, normalized Laplacian? Brown bag 18 / 22

19 What Do You Hear? Size of separators (Cheeger inequality) What about analogue of Weyl s law? Brown bag 19 / 22

20 What Do You Hear? What information hides in the eigenvalue distribution? Discretizations of Laplacian: something like Weyl s law Sparse random graphs: Wigner semicircular distribution Real networks: less well understood Computing all eigenvalues seems expensive! But estimating distributions is much less expensive Steal a method (KPM) from the physics literature Brown bag 2 / 22

21 Exploring Spectral Densities (with David Gleich) Consider spectrum of normalized Laplacian (random walk matrix) Approximate via KPM and compare to full eigencomputation Things we know Eigenvalues in [ 1, 1]; nonsymmetric in general Stability: change d edges, have j d apple ˆj apple j+d kth moment = probability of return after k-step random walk Eigenvalue cluster near 1 well-separated clusters Eigenvalue cluster near triangles connected by one node What else can we hear? Brown bag 21 / 22

22 x x What Do You Hear? Brown bag 22 / 22

23 Erdos Brown bag 22 / 22

24 Internet topology Brown bag 22 / 22

25 Marvel characters Brown bag 22 / 22

26 Marvel comics Brown bag 22 / 22

27 PGP Brown bag 22 / 22

28 Yeast Brown bag 22 / 22

29 Enron s (SNAP) Brown bag 22 / 22

30 Reuters911 (Pajek) Brown bag 22 / 22

31 US power grid (Pajek) Brown bag 22 / 22

32 DBLP 21 (LAW) 6 x N = , nnz = 16154, 8 s (1 moments, 1 probes) Brown bag 22 / 22

33 Hollywood 29 (LAW) 4 x N = , nnz = , 293 s (1 moments, 1 probes) Brown bag 22 / 22