QMC Designs on the Sphere Ian H. Sloan University of New South - - PowerPoint PPT Presentation

QMC Designs on the Sphere

Ian H. Sloan

University of New South Wales, Sydney, Australia Uniform distribution and QMC Methods, RICAM 2013 Joint with J Brauchart, EB Saff and R Womersley

QMC designs

• Definition. A sequence of point sets (XN) ⊂ Sd with N → ∞ is a

sequence of QMC designs for the Sobolev space Hs(Sd), for some s > d/2, if there exists c(s, d) > 0, such that for all f ∈ Hs(Sd)

• 1

N

• x∈XN

f(x) −

• Sd f(x) d σd(x)
• ≤ c(s, d)

N s/d fHs.

Here d σd(x) is normalised measure on Sd.

QMC designs

• Definition. A sequence of point sets (XN) ⊂ Sd with N → ∞ is a

sequence of QMC designs for the Sobolev space Hs(Sd), for some s > d/2, if there exists c(s, d) > 0, such that for all f ∈ Hs(Sd)

• 1

N

• x∈XN

f(x) −

• Sd f(x) d σd(x)
• ≤ c(s, d)

N s/d fHs.

Here d σd(x) is normalised measure on Sd.

This is the optimal rate of convergence in Hs(Sd)

K Hesse and IHS 2005 for d = 2, Hesse 2006 for general d.

QMC designs

• Definition. A sequence of point sets (XN) ⊂ Sd with N → ∞ is a

sequence of QMC designs for the Sobolev space Hs(Sd), for some s > d/2, if there exists c(s, d) > 0, such that for all f ∈ Hs(Sd)

• 1

N

• x∈XN

f(x) −

• Sd f(x) d σd(x)
• ≤ c(s, d)

N s/d fHs.

Here d σd(x) is normalised measure on Sd.

This is the optimal rate of convergence in Hs(Sd)

K Hesse and IHS 2005 for d = 2, Hesse 2006 for general d.

The idea grew from properties of spherical designs.

Spherical designs

Definition: A spherical t-design on Sd ⊂ Rd+1 is a set XN := {x1, . . . , xN} ⊂ Sd such that 1 N

N

• j=1

p(xj) =

• Sd

p(x) d σd(x) ∀p ∈ Pt.

Spherical designs

Definition: A spherical t-design on Sd ⊂ Rd+1 is a set XN := {x1, . . . , xN} ⊂ Sd such that 1 N

N

• j=1

p(xj) =

• Sd

p(x) d σd(x) ∀p ∈ Pt. So XN is a spherical t-design if the equal weight cubature rule with these points integrates exactly all polynomials of degree ≤ t.

A spherical 50-design

This is a Womersley spherical 50-design with 1302 ≈ 1

2512 points.

Spherical designs are good for integration

Spherical designs are tools for numerical integration. The following theorem (Hesse & IHS, 2005, 2006) shows a good rate of convergence for sufficiently smooth functions f: Theorem. Given s > d/2, there exists C(s, d) > 0 such that for every spherical t-design XN on Sd there holds

• 1

N

• x∈XN

f(x) −

• Sd f(x) d σd(x)
• ≤ C(s, d)

ts fHs.

How many points for a spherical t-design?

It is known (Seymour & Zaslavsky, 1984) that for every t ≥ 1 (and for every

dimension of the sphere) there always exists a spherical design.

But how many points does a spherical t-design need?

There is no possible upper bound because ...

How many points for a spherical t-design?

It is known (Seymour & Zaslavsky, 1984) that for every t ≥ 1 (and for every

dimension of the sphere) there always exists a spherical design.

But how many points does a spherical t-design need?

There is no possible upper bound because ...

Delsarte, Goethals, Seidel (1977) established lower bounds of exact

• rder td:

N ≥       

• d+t/2

d

• +
• d+t/2 − 1

d

• ,

if t is even

• d+⌊t/2⌋

d

• ,

if t is odd

How many points for a spherical t-design?

It is known (Seymour & Zaslavsky, 1984) that for every t ≥ 1 (and for every

dimension of the sphere) there always exists a spherical design.

But how many points does a spherical t-design need?

There is no possible upper bound because ...

Delsarte, Goethals, Seidel (1977) established lower bounds of exact

• rder td:

N ≥       

• d+t/2

d

• +
• d+t/2 − 1

d

• ,

if t is even

• d+⌊t/2⌋

d

• ,

if t is odd Yudin (1997) established larger lower bounds, still of exact order td.

Is (constant ×td) enough points?

It has long been conjectured that cdtd points is enough, for some cd > 0, but until very recently there was no proof. Recently Bondarenko, Radchenko and Viazovska (Annals of Mathematics, 2013) proved this important existence result.

Is (constant ×td) enough points?

It has long been conjectured that cdtd points is enough, for some cd > 0, but until very recently there was no proof. Recently Bondarenko, Radchenko and Viazovska (Annals of Mathematics, 2013) proved this important existence result. But what is the constant? The Bondarenko et al. constant is huge.

Is (constant ×td) enough points?

It has long been conjectured that cdtd points is enough, for some cd > 0, but until very recently there was no proof. Recently Bondarenko, Radchenko and Viazovska (Annals of Mathematics, 2013) proved this important existence result. But what is the constant? The Bondarenko et al. constant is huge. For S2 Chen, Frommer and Lang (2011) proved that (t + 1)2 points is enough for all t up to 100. For S2, we believe that (t + 1)2 points is enough for all t. Even N ≈ 1

2(t + 1)2 seems to be enough (R. Womersley, private

communication).

Efficient spher. designs are QMC designs

Clearly, every sequence of spherical t-designs is a sequence of QMC designs for Hs(Sd), for all s > d/2, iff N ≍ td as t → ∞, since Theorem. Given s > d/2, there exists C(s, d) > 0 such that for every spherical t-design XN on Sd there holds

• 1

N

• x∈XN

f(x) −

• Sd f(x) d σd(x)
• ≤ C(s, d)

ts fHs. If N ≍ td this gives

• 1

N

• x∈XN

f(x) −

• Sd f(x) d σd(x)
• ≤ c(s, d)

N s/d fHs.

Are there other QMC designs?

We think there are many. Here’s one that is certain:

• Theorem. (J Brauchart, EB Saff, IH Sloan, R Womersley, Math Comp, to appear)

A sequence of N-point sets X∗

N that maximize the sum of pairwise

Euclidean distances is a sequence of QMC designs for H(d+1)/2(Sd).

Are there other QMC designs?

We think there are many. Here’s one that is certain:

• Theorem. (J Brauchart, EB Saff, IH Sloan, R Womersley, Math Comp, to appear)

A sequence of N-point sets X∗

N that maximize the sum of pairwise

Euclidean distances is a sequence of QMC designs for H(d+1)/2(Sd). Thus for S2 the points that maximize the sum of Euclidean distances form a sequence of QMC designs for H3/2.

Are there other QMC designs?

We think there are many. Here’s one that is certain:

• Theorem. (J Brauchart, EB Saff, IH Sloan, R Womersley, Math Comp, to appear)

A sequence of N-point sets X∗

N that maximize the sum of pairwise

Euclidean distances is a sequence of QMC designs for H(d+1)/2(Sd). Thus for S2 the points that maximize the sum of Euclidean distances form a sequence of QMC designs for H3/2. To prove this and other things we need some machinery. But first:

The nested property of QMC designs

• Theorem. (Brauchart, Saff, IHS, Womersley, op. cit.)

Given s > d/2, let (XN) be a sequence of QMC designs for Hs(Sd). Then (XN) is a sequence of QMC designs for all coarser Hs′(Sd), i.e. for all s′ satisfying d/2 < s′ ≤ s.

This result isn’t trivial – for the smaller set Hs(Sd) we demand faster convergence.

The nested property of QMC designs

• Theorem. (Brauchart, Saff, IHS, Womersley, op. cit.)

Given s > d/2, let (XN) be a sequence of QMC designs for Hs(Sd). Then (XN) is a sequence of QMC designs for all coarser Hs′(Sd), i.e. for all s′ satisfying d/2 < s′ ≤ s.

This result isn’t trivial – for the smaller set Hs(Sd) we demand faster convergence.

So there is some upper bound on the admissible values of s: s∗ := sup{s : (XN) is a sequence of QMC designs for Hs}. We call s∗ the QMC index of the sequence (XN).

Generic QMC designs

If s∗ = +∞, we say the sequence of QMC designs is “generic”. Every sequence of spherical t-designs with N ≍ td as t → ∞ is a generic sequence of QMC designs.

We don’t know if there are other interesting examples.

The Sobolev space Hs(Sd)

With λℓ := ℓ (ℓ + d − 1) (λℓ is the ℓth eigenvalue of −∆∗

d),

Hs(Sd) =

• f ∈ L2(Sd) :

∞

• ℓ=0

Z(d,ℓ)

• k=1

(1 + λℓ)s

• fℓ,k
• 2

< ∞

• .

Thus H0(Sd) = L2(Sd). Here Laplace-Fourier coefficients

• fℓ,k = (f, Yℓ,k)L2(Sd) =
• Sd f(x)Yℓ,k(x) d σd(x).

Yℓ,k for k = 1, . . . , Z(d, ℓ) is an orthonormal set of spherical harmonics of degree ℓ: ∆∗

d Yℓ,k = −λℓYℓ,k.

Norms for Hs(Sd)

It is useful to allow also other equivalent norms for Hs(Sd): Let (a(s)

ℓ

)ℓ≥0 satisfy a(s)

ℓ

≍ (1 + λℓ)−s ≍ (1 + ℓ)−2s. Inner product and norm for f, g ∈ Hs(Sd) (f, g)Hs:=

∞

• ℓ=0

Z(d,ℓ)

• k=1

1 a(s)

ℓ

• fℓ,k

gℓ,k, fHs:=

• (f, f)Hs.

It is easily seen that Hs(Sd) is embedded in C(Sd) iff s > d/2.

Worst case error

Let Q[XN](f) := 1 N

N

• j=1

f(xj) ≈

• Sd f(x) d σd(x).

Then the worst case error in Hs(Sd) is defined by: wce(Q[XN]; Hs(Sd)) := sup

• Q[XN](f) − I(f)
• : f ∈ Hs(Sd), fHs ≤ 1
• ,

Worst case error

Let Q[XN](f) := 1 N

N

• j=1

f(xj) ≈

• Sd f(x) d σd(x).

Then the worst case error in Hs(Sd) is defined by: wce(Q[XN]; Hs(Sd)) := sup

• Q[XN](f) − I(f)
• : f ∈ Hs(Sd), fHs ≤ 1
• ,

QMC designs defined in terms of wce: A sequence (XN) of N point configurations on Sd is a sequence of QMC designs for Hs iff there exists c(s, d) > 0 such that wce(Q[XN]; Hs(Sd)) ≤ c(s, d) N s/d .

Reproducing kernel for Hs(Sd), s > d/2

K(s)(x, y)=

∞

• ℓ=0

Z(d,ℓ)

• k=1

a(s)

ℓ

Yℓ,k(x)Yℓ,k(y) =

∞

• ℓ=0

a(s)

ℓ

Z(d, ℓ)P (d)

ℓ

(x · y), where P (d)

ℓ

Legendre polynomial associated with Sd. It is a zonal kernel: i.e. K(s)(x, y) = K(s)(x · y).

Reproducing kernel for Hs(Sd), s > d/2

K(s)(x, y)=

∞

• ℓ=0

Z(d,ℓ)

• k=1

a(s)

ℓ

Yℓ,k(x)Yℓ,k(y) =

∞

• ℓ=0

a(s)

ℓ

Z(d, ℓ)P (d)

ℓ

(x · y), where P (d)

ℓ

Legendre polynomial associated with Sd. It is a zonal kernel: i.e. K(s)(x, y) = K(s)(x · y). The reproducing kernel properties are easily verified: K(s)(·, x)∈ Hs(Sd), x ∈ Sd, (f, K(s)(·, x))Hs= f(x), x ∈ Sd, f ∈ Hs(Sd).

WCE in terms of the reproducing kernel

Recall: For a(s)

ℓ

≍ (1 + ℓ)−2s, K(s)(x · y) :=

∞

• ℓ=0

a(s)

ℓ Z(d,ℓ)

• k=1

Yℓ,k(x)Yℓ,k(y) =

∞

• ℓ=0

a(s)

ℓ

Z(d, ℓ)P (d)

ℓ

(x · y) is a reproducing kernel for Hs(Sd).

WCE in terms of the reproducing kernel

Recall: For a(s)

ℓ

≍ (1 + ℓ)−2s, K(s)(x · y) :=

∞

• ℓ=0

a(s)

ℓ Z(d,ℓ)

• k=1

Yℓ,k(x)Yℓ,k(y) =

∞

• ℓ=0

a(s)

ℓ

Z(d, ℓ)P (d)

ℓ

(x · y) is a reproducing kernel for Hs(Sd). Theorem

• wce(Q[XN]; Hs(Sd))

2 = 1 N 2

N

• j=1

N

• i=1

K(s)(xj · xi) . Here K(s)(x · y) :=

∞

• ℓ=1

a(s)

ℓ

Z(d, ℓ)P (d)

ℓ

(x · y) = K(s)(x · y) − a(s)

0 .

Optimal QMC designs

Recall: K(s)(x · y) := K(s)(x · y) − a(s)

0 .

Let X∗

N = {x∗ 1, · · · , x∗ N}, for N = 2, 3, 4, . . . be a sequence of

minimizers of

N

• j=1

N

• i=1

K(s)(xj · xi),

Optimal QMC designs

Recall: K(s)(x · y) := K(s)(x · y) − a(s)

0 .

Let X∗

N = {x∗ 1, · · · , x∗ N}, for N = 2, 3, 4, . . . be a sequence of

minimizers of

N

• j=1

N

• i=1

K(s)(xj · xi),

• Theorem. (op cit) There exists c(s, d) > 0 such that for all N ≥ 2

wce(Q[X∗

N]; Hs(Sd)) ≤ c(s, d)

N s/d . Consequently, (X∗

N) is a sequence of QMC designs for Hs(Sd).

Optimal QMC designs

Recall: K(s)(x · y) := K(s)(x · y) − a(s)

0 .

Let X∗

N = {x∗ 1, · · · , x∗ N}, for N = 2, 3, 4, . . . be a sequence of

minimizers of

N

• j=1

N

• i=1

K(s)(xj · xi),

• Theorem. (op cit) There exists c(s, d) > 0 such that for all N ≥ 2

wce(Q[X∗

N]; Hs(Sd)) ≤ c(s, d)

N s/d . Consequently, (X∗

N) is a sequence of QMC designs for Hs(Sd).

Proof: Use the fact that spherical designs with N ≍ td exist, and satisfy the bounds in the theorem for all s > d/2. The minimizer for a particular s > d/2 must be at least as good.

QMC designs from distance kernels

Let V (Sd) :=

• Sd
• Sd |x − y| d σd(x) d σd(y). Then it happens that

2V (Sd) − |x − y| is a reproducing kernel for H(d+1)/2(Sd)

QMC designs from distance kernels

Let V (Sd) :=

• Sd
• Sd |x − y| d σd(x) d σd(y). Then it happens that

2V (Sd) − |x − y| is a reproducing kernel for H(d+1)/2(Sd), and that correspondingly K(d+1)/2(x, y) := K(d+1)/2 − a(d+1)/2 = V (Sd) − |x − y| .

QMC designs from distance kernels

Let V (Sd) :=

• Sd
• Sd |x − y| d σd(x) d σd(y). Then it happens that

2V (Sd) − |x − y| is a reproducing kernel for H(d+1)/2(Sd), and that correspondingly K(d+1)/2(x, y) := K(d+1)/2 − a(d+1)/2 = V (Sd) − |x − y| . Thus wce(Q[XN]; H(d+1)/2(Sd)) =  V (Sd) − 1 N 2

N

• j=1

N

• i=1

|xj − xi|  

1/2

is the corresponding WCE in H(d+1)/2(Sd).

QMC designs from distance kernels

Let V (Sd) :=

• Sd
• Sd |x − y| d σd(x) d σd(y). Then it happens that

2V (Sd) − |x − y| is a reproducing kernel for H(d+1)/2(Sd), and that correspondingly K(d+1)/2(x, y) := K(d+1)/2 − a(d+1)/2 = V (Sd) − |x − y| . Thus wce(Q[XN]; H(d+1)/2(Sd)) =  V (Sd) − 1 N 2

N

• j=1

N

• i=1

|xj − xi|  

1/2

is the corresponding WCE in H(d+1)/2(Sd).

• Corollary. A sequence of N-point sets X∗

N that maximize the sum of

Euclidean distances is a sequence of QMC designs for H(d+1)/2(Sd).

Generalized distance kernels

In the same way, a kernel for Sobolev spaces with s ∈ (d/2, d/2 + 1) is given by (Hubbert & Baxter 2012, Brauchart & Womersley 201?) K(s)

gd (x, y) := 2V2s−d(Sd) − |x − y|2s−d

where V2s−d := V2s−d(Sd) :=

• Sd
• Sd |x − y|2s−d d σd(x) d σd(y).

A reproducing kernel for s > d/2 + 1

For d/2 + L < s < d/2 + L + 1, with L = 1, 2, · · · , a version of the reproducing kernel for Hs(Sd) is (Brauchart & Womersley 201?) K(s)

gd (x, y) :=

• 1 − (−1)L+1

Vd−2s(Sd) +QL(x · y) + (−1)L+1 |x − y|2s−d, where QL(x · y) :=

L

• ℓ=1

βℓ P (d)

ℓ

(x · y), βℓ = · · · .

A reproducing kernel for s > d/2 + 1

For d/2 + L < s < d/2 + L + 1, with L = 1, 2, · · · , a version of the reproducing kernel for Hs(Sd) is (Brauchart & Womersley 201?) K(s)

gd (x, y) :=

• 1 − (−1)L+1

Vd−2s(Sd) +QL(x · y) + (−1)L+1 |x − y|2s−d, where QL(x · y) :=

L

• ℓ=1

βℓ P (d)

ℓ

(x · y), βℓ = · · · . This is important because it gives us a practical tool for computing worst-case errors for (almost) all values of s.

QMC designs are better than average

We might wonder: perhaps ALL sequences of point sets are QMC designs? No, because:

QMC designs are better than average

We might wonder: perhaps ALL sequences of point sets are QMC designs? No, because: Theorem Given s > d/2,

• E
• wce(Q[XN]; Hs(Sd))

2

• = b(s, d)

N 1/2 for some constant b(s, d) > 0, where the points x1, . . . , xN are independently and uniformly distributed on Sd. Hence for s > d/2 optimal order quadrature error O(N −s/d) is not attained by random points.

QMC designs are better than average

We might wonder: perhaps ALL sequences of point sets are QMC designs? No, because: Theorem Given s > d/2,

• E
• wce(Q[XN]; Hs(Sd))

2

• = b(s, d)

N 1/2 for some constant b(s, d) > 0, where the points x1, . . . , xN are independently and uniformly distributed on Sd. Hence for s > d/2 optimal order quadrature error O(N −s/d) is not attained by random points. On the other hand ...

Randomized equal area points

. Suppose we have a sequence (DN = {Dj,N, . . . , DN,N}) of equal area partitions of Sd with small diameter: ∪N

j=1Dj,N= Sd,

Dj,N ∩ Dk,N = ∅, j = k, σd(Dj,N) = 1/N ∀j, diamDj,N ≤ c/N 1/d ∀j. This time we distribute our N random points in a stratified way, with

• ne point to each Dj,N. The result is very different:

Theorem (op cit.) Let XN = {x1,N, . . . , xN,N}, where xj,N is chosen randomly from Dj,N with respect to uniform measure on Dj,N. Then, for d/2 < s < d/2 + 1, β′ N s/d ≤

• E
• {wce(Q[XN]; Hs(Sd))}2

≤ β N s/d . N ≍ td. Thus on average, randomized equal area points will form a QMC design sequence for d/2 < s < d/2 + 1.

(But not for s > d/2 + 1.)

Candidates for sequences of QMC designs

For the sphere S2, some plausible candidates are:

Random points, uniformly distributed on the sphere. NO Equal area points (cf. Rakhmanov-S-Zhou). Fekete points which maximize the determinant (cf.

Sloan-Womersley).

• Log. energy points and Coulomb energy points, which minimize

N

• j=1

N

• i=1

log 1 |xj − xi|,

N

• j=1

N

• i=1

1 |xj − xi|.

(continued)

Generalized spiral points (cf. Rakhmanov-S-Zhou; Bauer), with

spherical coordinates (θj, φj) given by θj = cos−1(1 − 2j − 1 N ), φj = 1.8 √ Nθj mod 2π, j = 1, . . . , N

Distance points, which maximize

N

• j=1

N

• i=1

|xj − xi|.

Spherical t-designs with N = ⌈(t + 1)2/2⌉ + 1 points.

WCE for Hs(S2) and s = 1.5

10

1

10

2

10

3

10

4

10

−4

10

−3

10

−2

10

−1

10 Number of points N Random 1.18 N−0.52 Fekete 0.90 N−0.75 Equal area 0.93 N−0.75 Coulomb energy 0.90 N−0.75 Log energy 0.90 N−0.75 Generalized spiral 0.91 N−0.75 Distance 0.90 N−0.75 Spherical design 0.91 N−0.75

There are many overlapping curves

Random 1.18 N −0.52 Fekete 0.90 N −0.75 Equal area 0.93 N −0.75 Coulomb energy 0.90 N −0.75 Log energy 0.90 N −0.75 Generalized spiral 0.91 N −0.75 Distance 0.90 N −0.75 Spherical design 0.91 N −0.75

WCE for Hs(S2) and s = 2.5

10

1

10

2

10

3

10

4

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10 Number of points N Random 2.25 N−0.53 Fekete 0.42 N−1.04 Equal area 0.82 N−0.98 Coulomb energy 1.03 N−1.23 Log energy 1.18 N−1.26 Generalized spiral 1.24 N−1.26 Distance 1.17 N−1.26 Spherical design 1.22 N−1.26

WCE for Hs(S2) and s = 3.5

10

1

10

2

10

3

10

4

10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10 Number of points N Random 4.92 N−0.52 Fekete 0.13 N−0.82 Equal area 1.80 N−0.95 Coulomb energy 0.17 N−1.06 Log energy 1.50 N−1.58 Generalized spiral 2.51 N−1.55 Distance 3.50 N−1.77 Spherical design 3.59 N−1.77

WCE for Hs(S2) and s = 4.5

10

1

10

2

10

3

10

4

10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10 10

1

Number of points N Random 10.22 N −0.52 Fekete 0.35 N −0.79 Equal area 4.09 N −0.95 Coulomb energy 0.29 N −1.01 Log energy 1.29 N −1.48 Generalized spiral 4.82 N −1.53 Distance 7.00 N −2.00 Spherical design 18.49 N −2.31

Integrating a smooth function

Franke function in C∞(S2) f

• x, y, z
• := 0.75 exp(−(9x − 2)2/4 − (9y − 2)2/4 − (9z − 2)2/4)

+0.75 exp(−(9x + 1)2/49 − (9y + 1)/10 − (9z + 1)/10) +0.5 exp(−(9x − 7)2/4 − (9y − 3)2/4 − (9z − 5)2/4) −0.2 exp(−(9x − 4)2 − (9y − 7)2 − (9z − 5)2)

• S2 f(x)dσ2(x) = 0.532865250084389...

Integrating a smooth function

Franke function in C∞(S2) f

• x, y, z
• := 0.75 exp(−(9x − 2)2/4 − (9y − 2)2/4 − (9z − 2)2/4)

+0.75 exp(−(9x + 1)2/49 − (9y + 1)/10 − (9z + 1)/10) +0.5 exp(−(9x − 7)2/4 − (9y − 3)2/4 − (9z − 5)2/4) −0.2 exp(−(9x − 4)2 − (9y − 7)2 − (9z − 5)2)

• S2 f(x)dσ2(x) = 0.532865250084389...

And note: |Q[XN](f) − I(f)| ≤ wce(Q[XN]; Hs(Sd)) fHs. Thus we should see in the error a rate of decay of order N −s∗/2.

The Franke function

Integration errors for the Franke function

10

1

10

2

10

3

10

4

10

−13

10

−12

10

−11

10

−10

10

−9

10

−8

10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10 Number of points N Random 0.28 N−0.51 Fekete 0.03 N−0.90 Equal area 0.24 N−0.96 Coulomb energy 0.04 N−1.17 Log energy 0.47 N−1.68 Generalized spiral 0.68 N−1.71 Distance 3.23 N−2.14 Spherical design

Estimates of s∗ for d = 2

s∗ := sup{s : (XN) is a sequence of QMC designs for Hs(S2)}.

Table 1: Estimates of s∗ for d = 2

Point set s∗ Fekete 1.5 Equal area 2 Coulomb energy 2 Log energy 3 Generalized spiral 3 Distance 4 Spherical designs ∞

Thus for S2:

All the well known point set sequences we have looked at, except for random points, appear to be approx. spherical designs for Hs, for a significant range of values of s.

Thus for S2:

All the well known point set sequences we have looked at, except for random points, appear to be approx. spherical designs for Hs, for a significant range of values of s. But so far this has been proved only for point sets that maximize the sum of distances, and only for s up to 3/2. (And we have proved similar results for generalized sums of distances for 1 < s < 2 ).

Thus for S2:

All the well known point set sequences we have looked at, except for random points, appear to be approx. spherical designs for Hs, for a significant range of values of s. But so far this has been proved only for point sets that maximize the sum of distances, and only for s up to 3/2. (And we have proved similar results for generalized sums of distances for 1 < s < 2 ). For the sum of distances points we have therefore proved that s∗ ≥ 3/2. But the experimental s∗ in this case is s∗ = 4 – there’s a big gap in our knowledge!

Thus for S2:

All the well known point set sequences we have looked at, except for random points, appear to be approx. spherical designs for Hs, for a significant range of values of s. But so far this has been proved only for point sets that maximize the sum of distances, and only for s up to 3/2. (And we have proved similar results for generalized sums of distances for 1 < s < 2 ). For the sum of distances points we have therefore proved that s∗ ≥ 3/2. But the experimental s∗ in this case is s∗ = 4 – there’s a big gap in our knowledge! QMC designs can be valuable tools for numerical integration, especially if s∗ is large.

WCE for Hs(S2) and s = 1.5

10

1

10

2

10

3

10

4

10

−4

10

−3

10

−2

10

−1

10 Number of points N Random 1.18 N−0.52 Fekete 0.90 N−0.75 Equal area 0.93 N−0.75 Coulomb energy 0.90 N−0.75 Log energy 0.90 N−0.75 Generalized spiral 0.91 N−0.75 Distance 0.90 N−0.75 Spherical design 0.91 N−0.75

A testable conjecture

That all of our point sets (except the random ones) are essentially

• ptimal in Hs(S2).