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Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
On Multivariate Chebyshev Polynomials
From Group Theory to Numerical Analysis
Hans Munthe-Kaas1
1 Department
of Mathematics
University of Bergen
http://hans.munthe-kaas.no
mailto:hans@math.uib.no
MAIA, Ålesund 2007
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
Outline
1 Motivation: PDEs with discrete domain symmetries
2 Multivar. Chebyshev pol. and approximation on triangles
3 Computing with Multivar. Chebyshev pol.
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
Motivation: PDEs and domain symmetries
2
Heat equation ut = u on sphere.
2
commutes with rotations and reflections.
Discretize respecting icosahedral symmetry group.
Matrix exp(A), where AP = PA for domain symmetries P, is fast
(by non-commutative Fourier analysis).
We would like: Spectral Elements in space, Lie group integrators
in time.
HOWTO: Spectral elements based on triangular subdivisions?
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
icosahedral symmetries...
G = C2 × A5
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
Fast matrix exponential on the sphere
We seek exp(A) where AP = PA for all permutation matrices
induced by icosahedral symmetry group G = C2 × A5 .
Irreducible representations of G = C2 × A5 ⇒
Matrix block diagonalization ⇒
3500 times faster matrix exponential.
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
Approximation on triangles; some references
Approximation theory on triangles:
J. Hesthaven, ’From Electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex’, SIAM J.
Num. Anal., 1998.
M. A. Taylor, B. A. Wingate and R. E. Vincent, ’An algorithm for computing Fekete points in the Triangle’, SIAM J.
Num. Anal., 2000.
F. X. Giraldo and T. Warburton, ’A nodal triangle-based spectral element method for shallow water equations on
the sphere’, J. Comp. Phys., 2005.
T. Lyche , K. Scherer, ’On the p-norm condition number of the multivariate triangular Bernstein basis’, J. of Comp.
and Appl. Math., 2000.
R. T. Farouki, T. N. T. Goodman, and T. Sauer, ’Construction of an orthogonal bases for polynomials in Bernstein
form on triangular and simplex domains’, Comp. Aided Geom. Design, 2003.
Multivariate Chebyshev polynomials:
T. H. Koornwinder (1974), ’Orthogonal polynomials in two variables which are eigenfunctions of two algebraically
independent partial differential operators I–IV’, Indiag. Math. 36.
R. Eier and R. Lidl (1982), ’A class of orthogonal polynomials in k variables, Math. Ann. 260.
M. E. Hoffman and W. D. Withers (1988), ’Generalized Chebyshev Polynomials Associated with Affine Weyl
Groups’, Transactions of the AMS 308.
Fast transforms for Multivariate Chebyshev pol.:
H. Munthe-Kaas, ’Symmetric FFTs, a general approach’, in PhD thesis NTNU, 1989.
M. Püschel and M. Rötteler, ’Cooley-Tukey FFT like algorithm for the discrete triangle transform, tech. rep. (2005).
This talk:
H. Z. Munthe-Kaas, ‘On group Fourier analysis and symmetry preserving discretizations of PDEs’, J. Phys. A 39
(2006), no. 19, 5563–5584.
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
Classical Chebyshev pol. obtained by folding ...
Consider periodic domain G = R/2πZ with Fourier basis
(k , θ) = exp(ik θ), for k ∈ G = Z.
We say that (·, ·) : G × G → C is a pairing between dual abelian
groups, satisfying
(k , θ + θ ) = (k , θ) · (k , θ ), (k + k , θ) = (k , θ) · (k , θ).
Let a ’mirror group’ W = {1, −1} act on G, and define
symmetrized Fourier basis
1 1
(k , θ)s = (k , gθ) = (g ∗ k , θ)
|W | |W |
g∈W g∈W
1
= (exp(ik θ) + exp(−ik θ)) = cos(k θ).
2
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
Classical Chebyshev pol. obtained by folding ...
Consider periodic domain G = R/2πZ with Fourier basis
(k , θ) = exp(ik θ), for k ∈ G = Z.
We say that (·, ·) : G × G → C is a pairing between dual abelian
groups, satisfying
(k , θ + θ ) = (k , θ) · (k , θ ), (k + k , θ) = (k , θ) · (k , θ).
Let a ’mirror group’ W = {1, −1} act on G, and define
symmetrized Fourier basis
1 1
(k , θ)s = (k , gθ) = (g ∗ k , θ)
|W | |W |
g∈W g∈W
1
= (exp(ik θ) + exp(−ik θ)) = cos(k θ).
2
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
... Classical Chebyshev pol. and folding
Define the change of variables
x(θ) = (1, θ)s = cos(θ).
This defines Chebyshev polynomials
Tk (x) = (k , θ)s for k = {0, 1, . . .}.
Note: Tk are polynomials because of recursion:
T0 (x) = 1
T1 (x) = x
2T1 (x)·Tk (x) = Tk +1 (x) + Tk −1 (x),
which follows from the properties of the dual pairing!
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
... Classical Chebyshev pol. and folding
Define the change of variables
x(θ) = (1, θ)s = cos(θ).
This defines Chebyshev polynomials
Tk (x) = (k , θ)s for k = {0, 1, . . .}.
Note: Tk are polynomials because of recursion:
T0 (x) = 1
T1 (x) = x
2T1 (x)·Tk (x) = Tk +1 (x) + Tk −1 (x),
which follows from the properties of the dual pairing!
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
Chebyshev recursion
1
x = T1 = (exp(iθ) + exp(−iθ))
2
1
Tk = (exp(ik θ) + exp(−ik θ))
2
1
xTk = (Tk +1 + Tk −1 )
2
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
Multivariate Chebyshev polynomials
Rootsystem A2
General construction:
1 Pick a reflection group W
acting on Rn , generating a
discrete translation group
Λ ⊂ Rn . Let G = Rn /Λ.
(G is parallelogram in
example).
2 Exponentials on G fold to
’cosine’ functions Tk (θ) on
fundamental domain (triangle
in example).
3 Variable change xi = xi (θ),
where xi are the
W -symmetrized generators
of G = Zn , yields Multivar.
b
Chebyshev polynomials
Tk (x).
Problem: Classify all possible such
constructions!
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
Root systems
How to define ’mirror groups’ on general periodic domains?
Definition
A root system is a subset Φ of a euclidean space E such that
1 Φ is finite, spans E and does not contain 0.
2 If α ∈ Φ then the only multiples of α in Φ are ±α.
T
αα
3 If α ∈ Φ then the reflection σα = I − 2 αT α leaves Φ invariant.
T
α β
4 If α, β ∈ Φ then 2 αT α ∈ Z.
(Theory originates from classification of semisimple Lie algebras).
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
Kaleidoscopes (the Affine Weyl group)
Definition
The group generated by the reflections W = σα |α ∈ Φ is called the
Weyl group.
The set Λ = θ → θ + α | α ∈ Φ is called the Root lattice.
The affine Weyl group W = Λ W is the group generated by all
these reflections and translations.
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
Classification of Root systems
( Cartan–Weyl–Coxeter–Dynkin)
Dynkin diagram:
Nodes = generating mirrors.
Edges indicate mirror-angles
no edge : 90o
– one edge : 60o
= two edges : 45o
≡ three edges : 30o
Arrow separates long and short
roots.
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
Non-separable 2D cases: A2 , B2 and G2
Rootsystem A2
Blue dots: Roots.
Blue arrows: Basis for root
system.
Red arrows: Fundamental
dominant weights.
Dotted lines: Mirrors in affine
Weyl group.
Yellow triangle: Fundamental
domain of affine Weyl group.
Red circles: Weights lattice.
Black dots: Downscaled root
lattice.
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
Root system B2 :
Rootsystem B2
Blue dots: Roots.
Blue arrows: Basis for root
system.
Red arrows: Fundamental
dominant weights.
Dotted lines: Mirrors in affine
Weyl group.
Yellow triangle: Fundamental
domain of affine Weyl group.
Red circles: Weights lattice.
Black dots: Downscaled root
lattice.
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
Root system G2 :
Rootsystem G2
Blue dots: Roots.
Blue arrows: Basis for root
system.
Red arrows: Fundamental
dominant weights.
Dotted lines: Mirrors in affine
Weyl group.
Yellow triangle: Fundamental
domain of affine Weyl group.
Red circles: Weights lattice.
Black dots: Downscaled root
lattice.
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
Multivariate Chebyshev polynomials
Let Φ be a d-dimensional root system, W be the Weyl group and Λ
the root lattice.
Let G = Rd /Λ be the ’root-periodic’ domain, G = Zd the dual group
and (k , θ) = exp(ik T θ) be the Fourier basis (dual pairing).
Definition
Multivariate Chebyshev polynomials Tk (x) are defined as follows:
1 1
(k , θ)s = (k , gθ) = (g T k , θ)
|W | |W |
g∈W g∈W
xk (θ) = (ek , θ)s ek = (0, . . . , 1, . . . , 0)T
Tk (x) = (k , θ)s .
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
Recurrence relations
The ’Mother of all’ recurrence relations between Tk (x) is the
following:
T0 (x) = 1 (1)
Tej (x) = xj (2)
Tk (x)T (x) = αk , (m)|Wm|Tm (x), (3)
m∈W \G
b
where α is given by a convolution
1
αk , (m) = δgk +h ,m . (4)
|W |2
g,h∈W
c
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
Recurrence example:
A2 recurrence:
z = x1 , z = x2
T−1,0 = z, T0,0 = 1, T1,0 = z
Tn,0 = 3zTn−1,0 − 3zTn−2,0 + Tn−3,0
Tn,m = (3Tn,0 Tm,0 − Tn−m,0 )/2.
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
The example: A2
Fundamental domain of affine Weyl group mapped to Deltoid by
variable change θ → x.
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
The Deltoid
The Deltoid, or 3-cusp Steiner hypocycloide, was first introduced by
L. Euler in 1745 in discussion of caustic patterns in optics like this
reflection in a bathroom mirror:
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
Deltoid drawn by Spirograph
The deltoid can be constructed as a ’Spirograph’ curve obtained by
drawing the perimeter of a circle of radius 2/3 rolling inside a circle of
radius 1. If the rolling is stopped at regular angles 2π/k , we get lines
that cross each other at 1-D Chebyshev–Gauss and
Chebyshev–Gauss–Lobatto points.
The deltoid circumscribes the diameter of a rolling circle
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
-3 -2 -1 0 1 2 3
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
Straightening the deltoid
We have constructed a coordinate map which straightens the deltoid to a triangle. The map has analytically
computable jacobian. It is well behaved away from the corners, but has corner singularities due to the cusps of the
deltoid. Interpolation points are given analytically.
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
Lebesgue constant in triangular interpolation
Lebesgue constant: L = ||I||∞ , where I is the (multivariate) interpolation operator in the given nodes. Slow growth
of the Lebesgue constant is necessary for spectral convergence.
Define Lebesgue function: X
λ(x) = | i (x)|,
i∈I
where i (x) is Lagrangian cardinal polynominal at node i, then L = ||λ(x)||∞ .
No. of nodal points
0 50 100 150 200 250 300
40
35 Uniform Hesthaven Bottom curve: Cheby-Lobatto
points on Deltoid. All other
30 curves: Interpolation points on
triangle:
Chebyshev ∆ pts
25 Fekete points.
Lebesgue constant
Image of C–L points by
20 straightening Deltoid to
triangle.
15 Fekete Hesthaven electrostatic
points.
10
Uniform meshpoints on
Chebyshev on Deltoid triangle.
5
0
5 10 15 20Polynomial degree
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
Lebesgue function on Deltoid
0.8
0.6
0.4
0.2
0
5
-0.2
4
-0.4 3
2
-0.6
1
-0.8
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2
The Lebesgue function is nice and nearly uniform over the domain.
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
Lebesgue function on Deltoid straightened to Triangle
The Lebesgue function is still nice and quite uniform.
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
Unified treatment of all Chebyshev families
Dynkin diagram
⇓
Cartan matrix
⇓
Matrix representation of W
⇓
Symmetric FFTs
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
Symmetric FFTs for multivar. Chebyshev
Cooley–Tukey is based on duality theory of abelian groups:
If H < G then H = G/H ⊥ and G/H = H ⊥ ,
thus Cooley-Tukey can be based on any subgroup decomposition of
the lattice.
Algorithm:
Pick G by downscaling root lattice with factor Mk = c · 2k .
Pick sequence of subgrids H1 < H2 < · · · < Hk −1 < G, where Hj
is root lattice downscaled by factor Mj .
Divide-and-conquer in C–T style. Take care of symmetries in
primal space, and dual symmetries in dual space. Compute only
in fundamental domains. Cosets of Hj + q contain either all the
original symmetries, or are equivalent to other cosets Hj + r .
5
Cost: N log2 (N), (N: gridpoints in fundamental domain).
2
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
Computing with Multivar. Chebyshev pol.
Beautiful properties of multivar. Chebyshev:
Fast algorithms for interpolation, differentiation and integration,
based on group theory (crystallographic groups and duality).
Continuous and discrete orthogonalities, both for
Chebyshev–Gauss and Chebyshev–Gauss–Lobatto points, thus
high order integration rules.
Lebesque constant for interpolation problem grows
logartihmically, just like 1-D case. Thus the discrete nodes are
good interpolation points.
Optimal ∞-norm characterization of Tk generalize from 1D to 2D
(and higher?)
Recurrence relations.
The root lattice of A1 , A2 , A3 , D4 , D5 , E6 , E7 , E8 are densest
lattice packings for these dimensions. Thus optimal in terms of
sampling theory. (Densest in primal space implies loosest in
dual space.)
H. Z. Munthe-Kaas MAIA ’07 Multivariate Chebyshev Polynomials
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