mit, spring 2017groups.csail.mit.edu/.../8_laplacian_operator.pdf · “no, but…” has to be a...
TRANSCRIPT
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Justin SolomonMIT, Spring 2017
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SIGN
MISTAKES
LIKELY
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Lots of (sloppy) math!
http://www.wgnflag.com/xcart/images/P/G-50_StopSeatBeltsStockSign.jpg
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http://www.takamine.com/templates/default/images/gclassical.pnghttp://firsttimeprogrammer.blogspot.com/2015/05/spectrogram-representation-of-b-note.html
Length of string
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“No, but…”
Has to be a weird drum Spectrum tells you a lot!
https://en.wikipedia.org/wiki/Hearing_the_shape_of_a_drum
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http://pngimg.com/upload/hammer_PNG3886.png
You can learn a lot about a shape by hitting it (lightly) with a hammer!
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What can you learn about its shape from
vibration frequencies and oscillation patterns?
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Make “vibration modes” more precise
Progressively more complicated domains Line segments
Regions in ℝ𝟐
Graphs
Surfaces/manifolds
Next time: Discretization, applications
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Theorem. Suppose 𝑨 ∈ ℂ𝒏×𝒏 is Hermitian. Then, 𝑨 has an orthogonal basis of 𝒏eigenvectors. If 𝑨 is positive definite, the corresponding eigenvalues are nonnegative.
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Line segments
Regions in ℝ𝟐
Graphs
Surfaces/manifolds
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Eigenfunctions:
“Dirichlet boundary conditions”
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Minus second derivative operator!
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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.
https://www.math.ku.edu/~matjohn/Teaching/S16/Math951/HilbertSchmidt.pdf
Hilbert space: Space with inner productSeparable: Admits countable, dense subsetCompact operator: Bounded sets to relatively compact setsSelf-adjoint: 𝐾𝑣, 𝑤 = ⟨𝑣, 𝐾𝑤⟩
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Image by E. Vouga
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Line segments
Regions in ℝ𝟐
Graphs
Surfaces/manifolds
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Wave equation:
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Gradient operator:
divergencegradient
http://www.gamasutra.com/db_area/images/feature/4164/figy.png, https://en.wikipedia.org/wiki/Gradient
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On board:1. Positive:
2. Self-adjoint:
“Dirichlet boundary conditions”
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On board:
“Laplace equation”“Harmonic function”
Images made by E. Vouga
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Images made by E. Vouga
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Coordinate-independent (important!)Images made by E. Vouga
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Point constraints are ill-advisedhttp://wp.production.patheos.com/blogs/johnbeckett/files/2015/01/shutterstock_81695467.jpg
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Small eigenvalue: Small Dirichlet Energyhttp://www.math.udel.edu/~driscoll/research/gww1-4.gif
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Line segments
Regions in ℝ𝟐
Graphs
Surfaces/manifolds
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Function:One value per vertex
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What is the Dirichlet energy of a function on a graph?
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Orient edges arbitrarily
-1 +2
-1 0
0 0
1
-1
00
2
0
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v
w
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https://en.wikipedia.org/wiki/Laplacian_matrix
Symmetric Positive definite
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What is the smallest eigenvalue
of the graph Laplacian?
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Fiedler vector (“algebraic connectivity”)
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Value at v is average of neighboring values
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Graph Laplacian encodes lots of information!
Example: Kirchoff’s TheoremNumber of spanning trees equals
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Line segments
Regions in ℝ𝟐
Graphs
Surfaces/manifolds
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http://www.ieeta.pt/polymeco/Screenshots/PolyMeCo_OneView.jpg
Map points to real numbers
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Suppose 𝒇: 𝑺 → ℝ and take 𝒑 ∈ 𝑺. For 𝒗 ∈ 𝑻𝒑𝑺, choose a curve 𝜶: −𝜺, 𝜺 → 𝑺 with
𝜶 𝟎 = 𝒑 and 𝜶′ 𝟎 = 𝒗. Then the differential of 𝒇 is 𝒅𝒇: 𝑻𝒑𝑺 → ℝ with
http://blog.evolute.at/
On the board (time-permitting):• Does not depend on choice of 𝜶• Linear map
Following Curves and Surfaces, Montiel & Ros
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Following Curves and Surfaces, Montiel & Ros
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Images made by E. Vouga
Decreasing E
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On the board:“Motivation” from finite-dimensional linear algebra.
Laplace-Beltrami operator
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Things we should check (but probably won’t):• Independent of choice of basis
• 𝚫 = 𝛁 ⋅ 𝛁
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Laplace-Beltrami coincides with Laplacian on ℝ𝟐
when 𝝈 takes 𝒙, 𝒚 axes to orthonormal vectors.
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http://alice.loria.fr/publications/papers/2008/ManifoldHarmonics//photo/dragon_mhb.png
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https://www.youtube.com/watch?v=CGiiSlMFFlI
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https://www.youtube.com/watch?v=Fyzqd2_T09Q
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https://www.youtube.com/watch?v=3uMZzVvnSiU
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Theorem (Courant). The n-th eigenfunctionof the Dirichlet boundary value problem has at most n nodal domains.
https://en.wikipedia.org/wiki/Ernst_Chladni
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Heat equationhttp://graphics.stanford.edu/courses/cs468-10-fall/LectureSlides/11_shape_matching.pdf
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https://en.wikipedia.org/wiki/Spherical_harmonics
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http://trecs.se/pix/pseudosphereNormalVectorField4096.png
Intuition:Laplacian measures difference with neighbors.
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Justin SolomonMIT, Spring 2017