Multiscale Methods. Introduction to Network Analysis 1

Transcription

1 Multiscale Methods In many complex systems, a big scale gap can be observed between micro- and macroscopic scales of problems because of the difference in physical (social, biological, mathematical, etc.) models and/or laws at different scales. Introduction to Network Analysis 1

2 Introduction to Network Analysis 2

3 In many problems, notwithstanding the fact that elementary parts of the system have a complicated (and even nondeterministic) behavior, their ensembles represent much more structured systems. High resolution Low resolution Two body scratch model Introduction to Network Analysis 3

4 Even when differences between models at different scales are not observed, an efficient approximation of the microscopic scale can be achieved by looking at the macroscopic scale with its substantially smaller number of elementary objects (such as variables and particles). Introduction to Network Analysis 4

5 The Multiscale Method Multiscale Multilevel Multigrid Multiresolution Introduction to Network Analysis 5

6 Computational Multiscale Methods Joseph Fourier Radiy Fedorenko Achi Brandt Functional analysis at multiple resolutions ( ) Smoothing, finite elements, two-level multigrid ( ) Popularization, first basic research (1977), algebraic multigrid (1980),... Introduction to Network Analysis 6

7 The Basic Observation Introduction to Network Analysis 7

8 Cycles Introduction to Network Analysis 8

9 Introduction to Network Analysis 9

10 Introduction to Network Analysis 10

11 Algebraic Multigrid Introduction to Network Analysis 11

12 Algebraic Multigrid Introduction to Network Analysis 12

13 Multilevel Algorithms for Combinatorial Optimization Problems Introduction to Network Analysis 13

14 Introduction to Network Analysis 14

15 Minimum Linear Arrangement, p=1 Introduction to Network Analysis 15

16 k-partitioning Introduction to Network Analysis 16

17 Simple Case: Coarsening by Contractions Introduction to Network Analysis 17

18 Introduction to Network Analysis 18

19 Weighted Aggregation (inspired by Algebraic Multigrid) Introduction to Network Analysis 19

20 Minimum Linear Arrangement Introduction to Network Analysis 20

21 Coarse nodes construction: C-points selection Introduction to Network Analysis 21

22 Interpolation weights Introduction to Network Analysis 22

23 Coarse Graph Introduction to Network Analysis 23

24 Coarse Graph Introduction to Network Analysis 24

25 Coarse Graph Introduction to Network Analysis 25

26 Introduction to Network Analysis 26

27 Uncoarsening: Interpolation for Minimum Linear Arrangement Introduction to Network Analysis 27

28 Relaxation Introduction to Network Analysis 28

29 Uncoarsening: Local Refinement, p=2 Introduction to Network Analysis 29

30 Uncoarsening: Local Refinement, p=2 Introduction to Network Analysis 30

31 Linear Arrangement: Spectral Approach Introduction to Network Analysis 31

32 Experimental Results: Linear Arrangement, p=2 [SRB] Multilevel algorithm for the minimum 2-sum problem, 2006 Ratios between multilevel and spectral algorithms Large-scale graphs Introduction to Network Analysis 32

33 Linear Arrangement, Larger powers Introduction to Network Analysis 33

34 Linear Arrangement, Larger powers Introduction to Network Analysis 34

35 Experimental Results: Linear Arrangement, p= [SRB] Multilevel algorithms for linear ordering problems, 2008 Ratios between multilevel and spectral algorithms Large-scale graphs Introduction to Network Analysis 35

36 Scalability. Dependence of running time on the graph size. Introduction to Network Analysis 36

37 The Minimum Workbound Problem Introduction to Network Analysis 37

38 Experimental Results: Minimum Workbound [SRB] Multilevel algorithms for linear ordering problems, 2008 Ratios between multilevel and spectral algorithms Large-scale graphs Introduction to Network Analysis 38

39 Refinement for k-partitioning solution inherited from coarse level refined solution Introduction to Network Analysis 39

40 More Examples: Dimensionality Reduction [FSS] Multilevel Nonlinear Dimensionality Reduction for Manifold Learning Introduction to Network Analysis 40

41 More Examples: Segmentation [SGSBB] Hierarchy and adaptivity in segmenting visual scenes, Nature, 2006 Introduction to Network Analysis 41

42 Segmentation: The pixel graph Introduction to Network Analysis 42

43 Segmentation: Multiscale Approach Introduction to Network Analysis 43

44 Two-dimensional layout problem Introduction to Network Analysis 44

45 Two-dimensional layout problem Introduction to Network Analysis 45

46 Material movement problem Introduction to Network Analysis 46

47 Introduction to Network Analysis 47

48 Two-dimensional layout problem: coarsening Variables Constraints Introduction to Network Analysis 48

49 Two-dimensional layout problem: example Mesh 64x64 + Random Edges Introduction to Network Analysis 49

50 Two-dimensional layout problem: VLSI Chip Original Multiscale Solver By Brandt, Ron Introduction to Network Analysis 50