IFT CONTINUOUS LAPLACIAN Mikhail Bessmeltsev

Transcription

1 IFT CONTINUOUS LAPLACIAN Mikhail Bessmeltsev

2 Famous Motivation

3 An Experiment

4 Unreasonable to Ask? Length of string

5 1D: length of a string u x, t - string height Boundary conditions: u 0, t = 0 = u l, t PDE: 2 u = t 2 c2 2 u x 2

6 Standing waves? (on the board)

7 Can you hear the length of an interval?

8 Spoiler Alert No, but Has to be a weird drum Spectrum tells you a lot!

9 Rough Intuition You can learn a lot about a shape by hitting it (lightly) with a hammer!

10 Spectral Geometry What can you learn about its shape from vibration frequencies and oscillation patterns?

11 Objectives Make vibration modes more precise Progressively more complicated domains Line segments Regions in R 2 Graphs Surfaces/manifolds Next time: Discretization, applications

12 Vector Spaces and Linear Operators

13 In Finite Dimensions

14 Recall: Spectral Theorems in C n Theorem. Suppose A C n n is Hermitian. Then, A has an orthogonal basis of n eigenvectors. If A is positive semidefinite, the corresponding eigenvalues are nonnegative.

15 Our Progression Line segments Regions in R 2 Graphs Surfaces/manifolds

16 Minus Second Derivative Operator Dirichlet boundary conditions Eigenfunctions:

17 Physical Intuition: Wave Equation Minus second derivative operator!

18 Observation

19 Hilbert-Schmidt Theorem Theorem. Let H 0 be an infinite-dimensional, separable Hilbert space and let K L(H) be compact and self-adjoint. Then, there exists a countable orthonormal basis of H consisting of eigenvectors of K. Hilbert space: Space with inner product Separable: Admits countable, dense subset Compact operator: Bounded sets to relatively compact sets Self-adjoint: Kv, w = v, Kw

20 Our Progression Line segments Regions in R 2 Graphs Surfaces/manifolds

21 Planar Region Wave equation:

22 Typical Notation divergence gradient Gradient operator:

23 Positivity, Self-Adjointness Dirichlet boundary conditions On board: 1. Positive: 2. Self-adjoint:

24 Dirichlet Energy On board: Images made by E. Vouga Laplace equation Harmonic function

25 Images made by E. Vouga Harmonic Functions

26 Intrinsic Operator Images made by E. Vouga Coordinate-independent

27 Another Interpretation of Eigenfunctions Small eigenvalue: smooth function

28 Our Progression Line segments Regions in R 2 Graphs Surfaces/manifolds

29 Basic Setup Function: One value per vertex

30 Dirichlet energy of a function on a graph?

31 Differencing Operator Orient edges arbitrarily

32 Dirichlet Energy on a Graph v w

33 (Unweighted) Graph Laplacian Symmetric Positive definite

34 Second-Smallest Eigenvector Fiedler vector ( algebraic connectivity )

35 Mean Value Property Value at v is average of neighboring values

36 For More Information Graph Laplacian encodes lots of information! Example: Kirchoff s Theorem Number of spanning trees equals

37 Hear the Shape of a Graph?

38 Our Progression Line segments Regions in R 2 Graphs Surfaces/manifolds

39 Scalar Functions Map points to real numbers

40 Differential of a Map Suppose f: S R and take p S. For v T p S, choose a curve α: ε, ε S with α 0 = p and α 0 = v. Then the differential of f is df: T p S R with On the board (time-permitting): Does not depend on choice of α Linear map Following Curves and Surfaces, Montiel &

41 Gradient Vector Field Following Curves and Surfaces, Montiel &

42 Images made by E. Vouga Dirichlet Energy Decreasing E

43 From Inner Product to Operator On the board: Motivation from finite-dimensional linear algebra. Laplace-Beltrami operator

44 What is Divergence? Things we should check (but probably won t): Independent of choice of basis Δ =

45 Eigenfunctions

46 Chladni Plates

47 Performance Art?

48 Practical Application

49 Additional Connection to Physics Heat equation fall/lectureslides/11_shape_matching.pdf

50 Spherical Harmonics

51 Weyl s Law

52 Laplacian of xyz function Intuition: Laplacian measures difference with neighbors.