A continuous approach for distributing pointson the sphere

using fast Fourier transforms

13th International Conference Approximation Theory7 -10 March 2010, San Antonio

Manuel Graf in joint work with Daniel Potts 1

andRainer Backofen, Simon Praetorius, Axel Voigt, Thomas Witkowski 2

1Chemnitz University of Technology, Germany, supported by DFG2Dresden University of Technology, Germany, supported by DFG

Outline

1 Preliminaries

2 Fast Algorithms

3 Model of Crystal Growth

4 Numerical Approach

5 Numerical Examples

Preliminaries

sphere S2 := {x ∈ R3 : ‖x‖2 = 1}• spherical coordinates (θ, ϕ) ∈ [0, π]× [0, 2π)

x(θ, ϕ) := (sin θ cosϕ, sin θ sinϕ, cos θ)>

• metricdS2(x1, x2) := arccos x>1 x2

• measure∫S2

f (x)dµS2(x) =1

4π

∫ 2π

0

∫ π

0f (x(θ, ϕ)) sin θdθdϕ

Preliminaries

Hilbert space

L2(S2) :={

f : S2 → C :

∫S2

|f (x)|2 dµS2(x) <∞}

spherical harmonics Y kn of degree n and order k

Y kn (x(θ, ϕ)) := e−ikϕP

|k|n (θ)

polynomial space of degree N

ΠN(S2) := span{Y kn : n = 0, . . . ,N; k = −n, . . . , n}

dN := (N + 1)2

Preliminaries

scattered sampling set X = {xi : i = 0, . . . ,M − 1} ⊂ S2

• mesh norm

δ(X ) := 2 maxx∈S2

mini=0,...,M−1

dS2(x, xi )

• associated quadrature rule with weights wi ≥ 0

Q(X ) := {(xi ,wi ) : i = 0, . . . ,M − 1}

degree of exactness N if∫S2

f (x)dµS2(x) =M−1∑i=0

wi f (xi ) for all f ∈ ΠN(S2)

Preliminaries

Theorem (sufficient condition - Mhaskar, Narcowich, Ward 01;Filbir,Themistoclakis 08)

A sampling set X (S2) supports nonnegative quadrature weights fordegree of exactness N, if the mesh norm satisfies

δ(X (S2)) ≤ 1

306N.

Theorem (necessary condition - Yudin 95; Reimer 00)

If a sampling set X (S2) supports nonnegative quadrature weightsfor degree of exactness N = 2L− 1, then the mesh norm satisfies

δ(X (S2)) ≤ 2 arccos zL ≤4π

N + 2,

where zL is the greatest zero of the L-th Legendre polynomial.

Fast Algorithms

• nonequispaced spherical Fourier matrix

Y := (Y kn (xi )) ∈ CM×dN

for sampling set X = {xi : i = 0, . . . ,M − 1} ⊂ S2

• fast algorithms1 for evaluation of

f = Yf, fi := f (xi ) =N∑

n=0

n∑k=−n

f kn Y k

n (xi ) (nfsft)

and

f = Y∗diag(w)f, f kn =

M∑i=0

wi fiY kn (xi ) (adjoint nfsft)

in O(N2log2N + M) arithmetic operations

1NFFT Library (Keiner,Kunis,Potts): http//www.tu-chemnitz.de/∼potts/nfft

Fast Algorithms

HEALPix grid XHS of size M = 12S2 ∈ N

• used in cosmic microwavebackground data analysis

• lacks exact integration scheme

numerical computation of the quadrature weights wi

(G, Kunis, Potts (Appl. Comput. Harm. Anal. 2009))

S M N rel. residual time

175 367500 512 3.242945e-13 9h185 410700 512 1.918440e-14 35min375 1687500 1024 1.088695e-14 97min

Motivation: Grystal Growth on the Sphere

• particle ordering on the sphere(Thomson problem)

minimizeM−1∑i=0

M−1∑i 6=j=0

‖xi−xj‖−12

(Backofen, Voigt, Witkowski (Phys. Rev. E 2010))

• crystal defects, grain boundaryscars

• ...

Model of Crystal Growth

phase field crystal model

• density function on the sphere of radius r > 0

ρ : rS2 → R

• minimizing free energy functional

F(ρ) :=

∫rS2

−|∇rS2ρ|2 +1

2|∆rS2ρ|+

1

2(1− ε)ρ2 +

1

4ρ4dµrS2

• L2 gradient flow ∂tρ = − δFδρ (Swift, Hohenberg (Phys. Rev. A 1977))

• H−1 gradient flow ∂tρ = ∆ δFδρ (Elder et al. (Phys. Rev. Lett. 2002))

∂tρ(rx, t) = ∆rS2

[((∆rS2 + 1)2 − ε

)ρ(rx, t) + ρ(rx, t)3

]

Numerical Approach

spectral method

• approximation by a spherical polynomial of degree N

ρ(rx, t) :=N∑

n=0

n∑k=−n

ρkn(t)Y k

n (x), x ∈ S2, t ∈ [0,∞)

• in spectral domain the PDE becomes a system of ODEs

dρkn

dt= −n(n + 1)

r 2

[((1− n(n + 1)

r 2

)2

− ε

)ρkn + (ρ3)k

n

]

with n = 0, . . . ,N, k = −n, . . . , n

Numerical Approach

semi-implicit time-integration scheme

ρt+1 − ρt

4t= −n(n + 1)

r 2

[((1− n(n + 1)

r 2

)2

− ε

)ρt+1 + (ρ3

t )

]

computation of (ρ3t ), t ≥ 0

• given ρt ∈ CdN

• evaluate ρt on a given sampling set X ⊂ rS2 (nfsft)

• compute pointwise ρ3t

• get spherical Fourier coefficients (ρ3t )

(adjoint nfsft & quadrature)

Numerical Examples

numerical solution for an icosahedral symmetric initial density ρ0

on a sphere rS2 with radius r = 100 using a HEALPix grid of sizeS = 100 and a polynomial degree N = 114

t = 0 t = 150 t = 300

t = 900 t = 2400 t = 4500

Numerical Examples

the solution ρt after t = 4500 time steps in the Fourier domain

n

k

0 10 20 30 40 50 60 70 80 90 100 110

−110

−90

−70

−50

−30

−10

0

10

30

50

70

90

1100

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Numerical Examples

distributing many points on the sphere

• r = 600, ε = 0.4, ρ = −0.3︸ ︷︷ ︸≈ 100.000 particels

• Coulomb energy is only0.0004% above lower bound

1

2M2+cM

32 , c = −0.553051...

(Kuijlaars, Saff (Trans. Amer. Math. Soc. 1998))

Thank you for your attention!