QMC designs and covering of spheres by spherical caps

Transcription

1 QMC designs and covering of spheres by spherical caps Ian H. Sloan University of New South Wales, Sydney, Australia Joint work with Johann Brauchart, Josef Dick, Ed Saff (Vanderbilt), Rob Womersley and Yu Guang Wang

2 Given X N := {x 1,...,x N } S d R d+1, in this talk we relate two different quality measures::

3 Cubature the quality of equal weight (or Quasi-Monte Carlo) integration Q[X N ](f) := 1 N N k=1 f(x k ), as an approximation to the integral of f, I(f) := f(x)dσ d (x), S d where σ d denotes the normalised surface measure on S d.

4 Covering the covering radius or mesh norm, or fill distance ρ(x N ) := max x S d min arccos(x x k). 1 k N

5 Covering by spherical caps

6 Now move just one point

7 Covering by spherical caps

8 Covering by spherical caps

9 Trivial lower bound ρ(x N ) c d N 1/d We will say that the sequence X N ) has the optimal covering property if ρ(x N ) N 1/d

10 Worst case error Recall Q[X N ](f) := 1 N N j=1 f(x j ) S d f(x)dσ d (x) = I[X N ]. The worst case error in a Banach space B is defined by: wce(q[x N ];B)) } := sup{ Q[X N](f) I(f) : f B, f B 1, We are particularly interested in Sobolev spaces, B = Wp s(sd ), for 1 p.

11 Function spaces For f L 2 (S d ), f(x) = Z(d,l) f l,k Y l,k (x), where l=0 k=1 The Laplace-Fourier coefficients are defined by f l,k = (f,y l,k ) L2 (S d ) = f(x)y l,k (x)dσ d (x). S d Y l,k for k = 1,...,Z(d,l) is an L 2 orthonormal set of spherical harmonics of degree l: d Y l,k = λ l Y l,k. with λ l := l(l + d 1) the lth eigenvalue of d.

12 The Sobolev space W s p (Sd ) Then f W s p (Sd ) if (1 d )s/2 f L p (S d ), that is, if f W s p (S d ) := l=0 Z(d,l) k=0 (1 + λ l ) s/2 fl,k Y l,k Lp (S d ) <. The case s = 0 gives W s p (Sd ) = L p (S d ).

13 The main result The covering radius of a point set X N on S d is upper bounded by a power of the worst-case error in a Sobolev space : Theorem Let d 1,1 p,q such that 1/q + 1/p = 1 and s > d/p. For a positive integer N, let X N be an N-point set on S d. Then ρ(x N ) c s,d [wce(q[x N ];W s p (Sd ))] 1/(s+d/q),

14 Idea of the proof We need to show wce(q[x N ];W s p (Sd )) ρ(x N ) s+d/q, We construct a fooling function f W s p (Sd ) with support a circle of radius ρ(x N ) that contains no points of X N in its interior Then Q N f = 0, = wce(q[x N ];W s p (Sd )) If. f W s p (S d ) Finally, show If cρ d, and f W s p (S d ) ρ(x N ) s+d/p.

15 Rate of convergence of wce? 1/(s+d/q), ρ(x N ) c s,d [wce(q[x N ];W s p ))] (Sd This result raises the question: how fast can the wce decay> Definition. For s > 0, sequence of point sets (X N ) S d with N is a sequence of QMC designs for the Sobolev space Wp s(sd ) for some s > d/p, if there exists c(s,d) > 0, such that for all f Wp s(sd ) 1 f(x) N x X N f(x)dσ d (x) S c(s,d) d N f s/d H s. That is, it is a QMC design iff wce(q[x N ];W s p (Sd )) c(s,d) N s/d

16 QMC designs QMC designs were introduced previously (Brauchart, Saff, Sloan, Womersley 2014) in the Hilbert space setting (p = 2). Recall: (X N ) is a sequence of QMC designs iff wce(q[x N ];W s p (Sd )) c(s,d) N s/d This is the optimal rate of convergence in Wp s(sd ) for p = 2 proved by Hesse & IHS 2005 for d = 2, then Hesse 2006 for general d; for general p (0, ) by Brandolini et al 2013.

17 QMC designs lead to good covering Corollary of main theorem Let d 1,1 p,q with 1/q + 1/p = 1. For a fixed s > d/p, let (X N ) with N be any QMC design sequence for W s p (Sd ). Then there exist a constant c > 0 depending on d,s,p and the sequence (X N ) but not on N such that ρ(x N ) cn β/d for all N, where β := s/(s + d/q). In particular, if p = 1 (and thus β = 1 and s > d), then the sequence of covering radii ρ(x N ) has the optimal covering property, ρ(x N ) N 1/d.

18 Do QMC designs exist? We know they exist for all s because of what we know about spherical designs.

19 Spherical designs Definition: A spherical t-design on S d R d+1 is a set X N := {x 1,...,x N } S d such that 1 N N p(x j ) = j=1 S d p(x)dσ d (x) p P t. Here dσ d (x) is normalised measure on S d. So X N is a spherical t-design if the QMC (i.e. equal weight) cubature rule with these points integrates exactly all polynomials of degree t.

20 Spherical designs are good for integration Spherical designs are tools for numerical integration. The following theorem shows a good rate of convergence for sufficiently smooth functions f: Theorem. Let 1 p. Given s > d/p, there exists C(p,s,d) > 0 such that for every spherical t-design X N on S d there holds wce(q[x N ];W s p (Sd )) C(s,d) t s. For p = 2: Hesse & IHS, 2005, 2006 For general p: Brandolini et al.2013

21 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 order t d : N ( ) d+t/2 d + ( d+ t/2 d ( ) d+t/2 1 d, if t is even ), if t is odd Yudin (1997) established larger lower bounds, still of exact order t d.

22 Is N O(t d ) enough for a spher. t design? It has long been conjectured that c d t d points is enough, for some c d > 0, but until very recently there was no proof. Recently Bondarenko, Radchenko and Viazovska (Annals of Mathematics, 2013) proved this important existence result. We say that a sequence of spherical t designs on S d with n t d is an optimal order sequence of spherical designs. For such a sequence we have, for all p (0, ) and all s > d/p, wce(q[x N ];W s p (Sd )) N s/d.

23 But what is the constant? But what is the constant? The Bondarenko et al. constant is huge. For S 2 Chen, Frommer and Lang (2011) proved that (t + 1) 2 points is enough for all t up to 100. For S 2, 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).

24 Covering and spherical designs Reimer 2000, extending Yudin 1995, showed that a spherical t-design yields a covering radius ρ(x N ) c d t 1., (The result holds also for any positive weight cubature rule.) Thus an optimal order sequence (X N ) of spherical t-designs yields optimal covering: ρ(x N ) c d N 1/d. The difference in the present work is that we do not require any form of polynomial exactness.

25 Are there other QMC designs? There are many. Here are two different kinds: For 1 p and s d/p, minimisers of wce(q[x N ];W s p (Sd )) form a sequence of QMC designs for W s p (Sd ) Theorem. (Brauchart, Saff, IHS, Womersley, Math Comp, t2014) A sequence of N-point sets X N that maximize the sum of pairwise Euclidean distances is a sequence of QMC designs for W (d+1)/2 2 (S d ). Thus for S 2 the points that maximize the sum of Euclidean distances form a sequence of QMC designs for W 3/2 2.

26 The nested property of QMC designs Theorem. (Brauchart, Saff, IHS, Womersley, 2014, for p = 2) Given s > d/2, let (X N ) be a sequence of QMC designs for W s 2 (S d ). Then (X N ) is a sequence of QMC designs for all coarser s W 2 (S d ), i.e. for all s satisfying d/2 < s s. This result isn t trivial for larger s the set is smaller but we demand faster convergence. So there is some upper bound on the admissible values of s: s := sup{s : (X N ) is a sequence of QMC designs forh s }. We called s the QMC strength of the sequence (X N ).

27 The nested property for general p Our present proof yields a weaker result for general p unless we have quasi-uniformity. The mesh ratio of a sequence X N be defined as Definition γ(x N ) := ρ(x N) δ(x N ). A sequence (X N ) of N-point sets on S d is said to be quasi-uniform if the mesh ratios are uniformly bounded in N. Theorem Let 1 p < and s > d/p. A quasi-uniform QMC design sequence (X N ) for W s p (Sd ) is also a QMC design sequence for W s p (Sd ) for all s satisfying d/p < s s. We conjecture that the quasi-uniform condition can be removed.

28 Nested property So for each p [0, ] there is some upper bound on the admissible values of s: s p := sup{s : (X N) is a sequence of QMC designs forw s p (Sd )}. We call s p the (L p) strength of the sequence (X N ).

29 Idea of the proof of nested property Express the wce in terms of Bessel kernel, B (s) (x y) := (1 + λ l ) s/2 Z(d,l)P (d) l (x y). l=0 Specifically, wce(q[x N ];W s p (Sd )) = 1 N N j=1 B (s) (x j ) 1. Lq A Bernstein ineq. (using Mhaskar, Narcowich, Prestin, Ward 2010) gives 1 N N j=1 B (s ) (x j ) 1 c[γ(x N )] d/p N (s s )/d Lq 1 N N j=1 B (s) (x j ) 1. Lq

30 Candidates for sequences of QMC designs For the sphere S 2, some plausible candidates are: Random points, uniformly distributed on the sphere. NO at least for p = 2. 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 N log j=1 i=1 1 x j x i, N N j=1 i=1 1 x j x i.

31 (continued) Generalized spiral points (cf. Rakhmanov-S-Zhou; Bauer), with spherical coordinates (θ j,φ j ) given by θ j = cos 1 (1 2j 1 N ), φ j = 1.8 Nθ j mod 2π, j = 1,..., Distance points, which maximize N N j=1 i=1 x j x i. Sphericalt-designs with N = (t + 1) 2 /2 + 1 points.

32 WCE for W s 2 (S2 ) and s = 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 Number of points N

33 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

34 WCE for W s 2 (S2 ) and s = 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 Number of points N

35 WCE for W s 2 (S2 ) and s = 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 Number of points N

36 WCE for W s 2 (S2 ) and s = Random 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 N Number of points N

37 Integrating a smooth function Franke function in C (S 2 ) f ( x,y,z ) := 0.75exp( (9x 2) 2 /4 (9y 2) 2 /4 (9z 2) 2 /4) +0.75exp( (9x + 1) 2 /49 (9y + 1)/10 (9z + 1)/10) +0.5exp( (9x 7) 2 /4 (9y 3) 2 /4 (9z 5) 2 /4) 0.2exp( (9x 4) 2 (9y 7) 2 (9z 5) 2 ) And note: S 2 f(x)dσ 2 (x) = Q[X N ](f) I(f) wce(q[x N ];W s 2 (Sd )) f W s 2. Thus we should see in the error a rate of decay of order N s /2.

38 The Franke function

39 Integration errors for the Franke function 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 Number of points N

40 Estimates of s 2 s := sup{s : (X N ) is a sequence of QMC designs for W s 2 (S2 )}. Table 1: Estimates ofs 2 Point set s 2 Fekete 1.5 Equal area 2 Coulomb energy 2 Log energy 3 Generalized spiral 3 Distance 4 Spherical designs

41 Thus for S 2 : All the well known point set sequences we have looked at, except for random points, appear to be QMC designs for W2 s, 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 2 3/2. But the experimental s 2 in this case is s 2 = 4 there s a big gap in our knowledge! QMC designs can be valuable tools for numerical integration, especially if s p is large.

42 What does this tell us about covering? From our main theorem, IF we know that the strength of the QMC design X N in L 2 (S d ) is s 2 then we can conclude from the Main Theorem that ρ(x N ) N (β /d)+ǫ, where β = s 2 s 2 + d/q = s 2 s For example, for the case of minimum Coulomb energy points, IF s 2 = 2, then ρ(x N) cn 2/(3d)+ǫ.

43 Of course we have not PROVED that s 2 = 2 for the minimum Coulomb energy points. And perhaps we believe that minimal Coulomb energy points give optimal covering ρ(x N ) N 1/d? This would follow if we could prove for the minimum Coulomb energy points that X N is a QMC design for W s 1 for just one s > 2.

44 References J Brauchart, E Saff, I Sloan, R Womersley, Optimal order quasi-monte Carlo integration schemes on the sphere. Math Comp J Brauchart, J Dick, E Saff, I Sloan, Y Wang and R Womersley, Covering of spheres by spherical caps and worst-case error for equal weight cubature in Sobolev spaces, arxiv: v