Multivariate Chebyshev Polynomials - Hans Munthe-Kaashans.munthe-kaas.no/.../munthekaas_focm11.pdf · Munthe-Kaas and Ryland (Univ. of Bergen) Multivariate Chebyshev polynomials FoCM

Multivariate Chebyshev Polynomialsfrom group theory to PDE solvers

Hans Munthe-Kaas and Brett Ryland

1Department of MathematicsUniversity of Bergen

Norwayhttp://hans.munthe-kaas.nomailto:[email protected]

FoCM, Budapest 2011

Munthe-Kaas and Ryland (Univ. of Bergen) Multivariate Chebyshev polynomials FoCM ’11 1 / 34

Theme:Group (representation) theory in computational maths

Example:High order polynomial approximations

"Chebyshev polynomials are everywhere dense in Numerical Analysis"(G. Forsythe?)


Chebyshev 101 (classical theory)Definition

(1. kind) : Tk (x) = cos(kθ)

(2. kind) : Uk (x) = sin((k + 1)θ)/ sin(θ)

where x = cos(θ).

T0(x) = 1T1(x) = xT2(x) = 2x2 − 1T3(x) = 4x3 − 3x. . .

3-term recurrence

Tk+1(x) = 2xTk (x)− Tk−1(x)


Chebyshev 101 (classical theory)

Orthogonality -continuous and discrete

〈Tk ,T`〉C =

Z 1

−1

Tk (x)T`(x)√1− x2

dx

〈Tk ,T`〉Z =1N

Xxj∈{∗}

Tk (xj)T`(xj)

〈Tk ,T`〉E =1N

Xxj∈{o}

′Tk (xj)T`(xj)



Chebyshev interpolation

f (x) ≈N∑

k=0

a(k)Tk (x) , interpolating in extremal points {∗}

a(k) = 〈Tk , f 〉E/〈Tk ,Tk 〉E .

Note:{f (xj)}xj∈{∗} ↔ {ak}Nk=0 in O(N log N) flops via FCT.Lebesgue constant of Cheby-interpolation grows locarithmically

L{∗} = O(log N).

Exponentially fast convergence for analytic f (x).




f (x) ≈N∑

k=0


a(k) = 〈Tk , f 〉E/〈Tk ,Tk 〉E .

Clenshaw-Curtis quadrature∫ 1

−1f (x)dx ≈

N∑k=0

a(k)

∫ 1

−1Tk (x)dx .




f (x) ≈N∑

k=0


a(k) = 〈Tk , f 〉E/〈Tk ,Tk 〉E .

Pseudospectral derivation

f ′(x)dx ≈N∑

k=0

a(k)T ′k (x)dx =N∑

k=0

b(k)Tk (x)dx .

{a(k)} 7→ {b(k)} in O(N) flops by linear recursion.


Example: Curve length

L[f ] =

∫ b

a||f ′(t)||dt

Pseudospectral derivation + Clenshaw-Curtis integration.16 points on semicircle:

>> ChebyL(’semicircle’,16)

ans = 3.141592653589623

error = 1.7 · 10−13


Example: Curve length

L[f ] =

∫ b

a||f ′(t)||dt

Archimedes: Inscribed 2N-gon:

N = 16 points:error = 0.006

N = 2 · 106 points:error = 3.2 · 10−13


Spectral– and spectral element methods

Current practice ↑. ↑We want this.


Triangular ’lego bricks’ in spectral elements

⇒


MK 1989: "Symmetric FFTs, a general approach" (unpubl.)

Laplace–Dirichlet eigenfunctions by anti-symmetrization:

σ : mirror, ∆◦σ = σ◦∆∆u = λu

ua =12

(u − u◦σ)

⇒ ∆ua = λua, ua = 0 on mirror


Example: A2–Kaleidoscope

Rootsystem A2Weyl group A2

σi = I − 2αiα

Ti

αTi αi

, i = 1,2

W = 〈σ1, σ2〉

Laplacian eigenfunctions:

u(θ) = exp(ikθ)⇒ ∆u = λu

us =1|W |

∑g∈W

u◦g

∆us = λus (Neumann)



Rootsystem A2Weyl group A2

σi = I − 2αiα

Ti

αTi αi

, i = 1,2

W = 〈σ1, σ2〉

Laplacian eigenfunctions:

u(θ) = exp(ikθ)⇒ ∆u = λu

ua =1|W |

∑g∈W

det(g)u◦g

∆ua = λua (Dirichlet)



Rootsystem A2Properties of ua and us:

Continuous and discreteorthogonality.Gaussian quadrature.Symmetric FFTs forinterpolation, derivation,integration.Triangle based fastPoisson solvers.Sym. FFT havecomplicated data flow.Spectral convergence?


Weyl groupsHow to define ’caleidoscopes’ on general periodic domains?

DefinitionA root system is a subset of a euclidean space Φ = {αi} ⊂ 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.

3 If α ∈ Φ then the reflection σα = I − 2ααT

αT αleaves Φ invariant.

4 If α, β ∈ Φ then 2 αT βαT α∈ Z.

Rootsystem A2


Caleidoscopes (the Affine Weyl group)

DefinitionThe group generated by the reflections W = 〈σα|α ∈ Φ〉 is called the Weylgroup.

The set Λ = 〈θ 7→ θ + α | α ∈ Φ〉 is called the Root lattice.

The affine Weyl group W ′ = Λ o W is the group generated by all thesereflections and translations.

Rootsystem A2


Classification of Root systems

( Cartan–Weyl–Coxeter–Dynkin)

Dynkin diagram:

Nodes = generating mirrors.

Edges indicate mirror-angles

no edge : 90o

– one edge : 120o

= two edges : 135o

≡ three edges : 150o

Arrow separates long and shortroots.


Non-separable 2D cases: A2, B2 and G2

Rootsystem A2

Blue dots: Roots.

Blue arrows: Basis for rootsystem.

Red arrows: Fundamentaldominant weights.

Dotted lines: Mirrors in affineWeyl group.

Yellow triangle: Fundamentaldomain of affine Weyl group.

Red circles: Weights lattice.

Black dots: Downscaled rootlattice.


Root system B2:Rootsystem B2

Blue dots: Roots.








Root system G2:Rootsystem G2

Blue dots: Roots.








Multivariate Chebyshev polynomials

Let Φ be d-dimensional root system, W Weyl group and Λ root lattice.Let G = Rd/Λ be the ’root-periodic’ domain and G = Λ⊥ the reciprocal lattice(Fourier space).

DefinitionMultivariate Chebyshev polynomials Tk (x) are defined as follows for θ ∈ G,k ∈ G:

Tk (x) =1|W |

∑g∈W

ei(gk)T θ

xj (θ) =1|W |

∑g∈W

ei(gλj )T θ, λj = (0, . . . ,1, . . . ,0)T


Multivariate Chebyshev polynomialsDefinitionMultivariate Chebyshev polynomials Tk (x) are defined as follows for θ ∈ G,k ∈ G:

Tk (x) =1|W |

∑g∈W

ei(gk)T θ

xj (θ) =1|W |

∑g∈W

ei(gλj )T θ, λj = (0, . . . ,1, . . . ,0)T

Example: 1-D case

Φ = {−π, π}, W = {−1,1}

Tk (x) =12

(eikθ + e−ikθ) = cos(kθ)

x(θ) =12

(eiθ + e−iθ) = cos(θ).


Multivariate Chebyshev polynomialsDefinition

Tk (x) =1|W |

∑g∈W

ei(gk)T θ, xj (θ) =1|W |

∑g∈W

ei(gλj )T θ

Example: A2 case


Recurrence relations

T0 = 1, Tej = xj

T`Tk =1|W |

∑g∈W

Tk+gT `.

Classical (A1): xTk (x) =12

(Tk−1(x) + Tk+1(x))


Recurrence relations

T0 = 1, Tej = xj

T`Tk =1|W |

∑g∈W

Tk+gT `.

A2 recurrence:

z = x1, z = x2

T−1,0 = z, T0,0 = 1, T1,0 = zTn,0 = 3zTn−1,0 − 3zTn−2,0 + Tn−3,0

Tn,m = (3Tn,0Tm,0 − Tn−m,0)/2.


The example: A2

Fundamental domain of affine Weyl group mapped to Deltoid by θ 7→ x :

x1 =13

(cos θ1+cos θ2+cos(θ1−θ2)), x2 =13

(sin θ1−sin θ2−sin(θ1−θ2)).


Domains for B2 and G2 Chebyshev polynomials


A2 Chebyshev polynomials: 00r 10r 20r 30r | 10i 11r 21r 31r | 20i 21i 22r 32r


A2 Chebyshev polynomial T52, real part:


Problem: Deltoid 6= triangle

Possible solutions:1 Straighten the deltoid to triangle.2 Patch with overlap.3 Work with (overdetermined) frame based on triangular

trigonometric polynomials.


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 aregiven analytically.


Lebesgue constant in triangular interpolationLebesgue constant: L = ||I||∞, where I is the (multivariate) interpolation operator in the given nodes. Slow growth of theLebesgue constant is necessary for spectral convergence.Define Lebesgue function:

λ(x) =Xi∈I|`i (x)|,

where `i (x) is Lagrangian cardinal polynominal at node i , then L = ||λ(x)||∞ .

0 50 100 150 200 250 300

0

5

10

15

20

25

30

35

40

HesthavenUniform

Fekete

Chebyshev ∆ pts

No. of nodal points

Lebe

sgue

con

stan

t

Chebyshev on Deltoid

5 1510 20Polynomial degree

Bottom curve: Cheby-Lobattopoints on Deltoid. All othercurves: Interpolation points ontriangle:

Fekete points.

Image of C–L points bystraightening Deltoid totriangle.

Hesthaven electrostaticpoints.

Uniform meshpoints ontriangle.


A3 polynomials

The root system A3 (in 3D) is similar to the A2 case:The Voronoi region of the root lattice is the rhombicdodecahedron.The fundamental domain of the affine Weyl group is a tetrahedron.Inside this tetrahedron sits a regular octahedron.Under the coordinate change θ 7→ x , the tetrahedron maps to acusp-shaped domain, and the octahedron to a tetrahedron insidethis.Restriction to faces: A3 7→ A2.Restriction to lines: A3 7→ A1.


A3 fundamental domains


A3 domain after change of variables


Numerical algorithms

We have analytical formulas and algorithms forIntegrating Tk (x) over whole x-domain (deltoid etc.).Integrating Tk (x) over inscribed triangle (A2 case) and tetrahedron(A3 case).Computing ∇Tk (x) by recursion in Fourier domain.

All algorithms are FFT based, cost O(N log(N)). Optimizedsymmetrized transforms exist.


A spectral element method on a torus (periodic unit square) for thewave equation utt = uxx + uyy .Overlapping parts of the deltoids are projected onto the underlyingtriangles and boundary points are averaged to give a continuoussolution.

−0.2 0 0.2 0.4 0.6 0.8 1 1.2

0

0.2

0.4

0.6

0.8

1


(Animation of spectral element method on torus)


Dirichlet problem on L-shaped domain,decomposed into triangles

0

10

20

30

40

50

0

10

20

30

40

50

−20

0

20

40

60

80

100

solution N=48


Connections to Lie group representation theorySymmetrized exponentials⇒ Tk (x), first kind polynomials.

Anti-symmetrized exponentials⇒ Uk (x), second kind polynomials.

Uk (x) =1|W |

Pg∈W det(g)ei(g(k+ρ))T θ

det( ∂xi∂θj

), xj(θ) =

1|W |

Xg∈W

ei(gλj )T θ.

Equivalent to Weyl’s character formula:

ch(Vk ) =

Pg∈W det(g)eg(k+ρ)P

g∈W det(g)egρ .

Higher order irreducible representations of W lead to Chebyshev polynomials of’other kinds’.

Applications to the Fourier extension problem. (with D. Huybrechs).


Part 2: LGI for PDEs: Domain symmetries



Problem: Compute exp(tA) where A ≈ ∇2 on sphere.Exploit icosahedral symmetries.



{φj}←− 4 ⊂ R2

Spectral element discretizations:Triangular subdomains are now possible!


Icosahedral symmetries with reflections: G = C2 × A5

PROOF:

Cost of matrix exponential (Icoashedral symmetries):|G| = 120, direct computation 1203 = 1728000 operations.Block diagonalization by GFT:

13 + 13 + 33 + 33 + 33 + 33 + 43 + 43 + 53 + 53 = 488

⇒ factor 3500 reduction in cost!


Some referencesApproximation theory on triangles:M. Dubiner, ’Spectral methods on triangles and other domains’, J. of Sci. Computing, 1991.A.J. Sherwin, G.E. Karniadakis, ’A triangular spectral element method; applications to the incompressible Navier-Stokesequations’, Comput. meth. in appl. mech. and engineering.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 ontriangular 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 independentpartial 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.H. Weyl ’Character formula for Compact Lie groups’, late 1920’s.

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, B. Ryland, ’On Multivariate Cheb. Pol. and triangle based spectral elements’, ICOSAHOM proc. 2010.

S. Christiansen, H.Z. Munthe-Kaas, B. Owren, ’Topics in structure preserving discretization’, Acta Numerica 2011.


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Multivariate Chebyshev Polynomials - Hans Munthe-Kaashans.munthe-kaas.no/.../munthekaas_focm11.pdf · Munthe-Kaas and Ryland (Univ. of Bergen) Multivariate Chebyshev polynomials FoCM