cs 468, spring 2012 - research | mit...
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http://alice.loria.fr/publications/papers/2008/ManifoldHarmonics//photo/dragon_mhb.png
Justin SolomonMIT, Spring 2019
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The Laplacian
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Wave equation:
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Laplacian is a differentialoperator!
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Finite element method (FEM)
http://www.stressebook.com/wp-content/uploads/2014/08/Airbus_A320_k.jpg
First-order Galerkin
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https://xkcd.com/1201/
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Laplacian (second derivative)
Gradient(first derivative)
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One derivative, one integral
Gradient(first derivative)
Kinda-sorta cancels out?
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Which basis?
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Not the only wayto approximate the Laplacian operator.
Divided differences Higher-order elements Boundary element methods Discrete exterior calculus …
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“Test function”
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Can recover function from dual
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Use Laplacian without evaluating it!
Space of test functions (no boundary!):
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Choose one of each:
Function space
Test functionsOften the same!
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Image courtesy K. Crane, CMU
One “hat function” per vertex
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Linear combination of hats(piecewise linear)
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One vector per face
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One scalar per face
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Sum scalars per facemultiplied by face areas
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Linear along edges
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Linear along edges
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Length of e23 cancels“base” in A
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Case 1: Same vertex
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Case 2: Different vertices
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THE COTANGENT LAPLACIAN
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http://nylander.wordpress.com/2006/05/24/finite-element-method-fem-solution-to-poisson%E2%80%99s-equation-on-triangular-mesh/
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Multiply by Laplacian matrix!
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Product of hats is quadratic
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Just do the integral“Consistent” approach
Approximate some more
Redefine g
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Just do the integral“Consistent” approach
Approximate some more
Redefine g
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Diagonal elements:Norm of 𝒉𝒊
Off-diagonal elements:Overlap between 𝒉𝒊 and 𝒉𝒋
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Rows sum to one ring area / 3
Involves only vertex and its neighbors
Partitions surface area
Issue: Not diagonal!
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Approximate with diagonal matrixhttp://users.led-inc.eu/~phk/mesh-dualmesh.html
Won’t make big difference for smooth functions
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Area/3 to each vertexhttp://www.alecjacobson.com/weblog/?p=1146
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Cotangent Laplacian LPer-vertex function to integral of its Laplacian against each hat
Area weights AIntegrals of pairwise products of hats(or approximation thereof)
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Must integrate to zero
Determined up to constant
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Strong form
Weak form
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2 3 4 5 6 7 8 9 10
1005025
On board: FEM approach?
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https://www.femtable.org/
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“Laplacian Eigenmaps for Dimensionality Reduction and Data Representation”Belkin & Niyogi 2003
Tricky parameter to choose
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http://alice.loria.fr/publications/papers/2008/ManifoldHarmonics//photo/dragon_mhb.png
Justin SolomonMIT, Spring 2019