Low-discrepancy point sets lifted to the unit sphere

Transcription

1 Low-discrepancy point sets lifted to the unit sphere Johann S. Brauchart School of Mathematics and Statistics, UNSW MCQMC 2012 (UNSW) February 17 based on joint work with Ed Saff (Vanderbilt) and Josef Dick, Ian Sloan, Rob Womersley (UNSW) and Christoph Aistleitner (TU-Graz)

2 Rob Womersley ( Mathematics & Statistics in all Spheres of Life

3 How do you uniformly arrange N (repulsive) particles on a sphere?

4 How do you quasi-uniformly arrange N (repulsive) particles on a sphere? pre-scribed density, well-separated, small holes

5 Preliminaries UNIFORM DISTRIBUTION ON THE UNIT SPHERE WORST-CASE ERROR (WCE) CUI AND FREEDEN S GENERALIZED DISCREPANCY STOLARSKY S INVARIANCE PRINCIPLE

6 Uniform Distribution on the Unit Sphere S d in R d+1 Definition A sequence {X N } is asymptotically uniformly distributed over S d if # {k : x k,n B} lim = σ d (B) N N for every σ d -measurable clopen set B in S d. Informally: A reasonable set gets a fair share of points as N becomes large. I[f ]:= f d σ d, S d Q N [f ]:= 1 N N f (x k ) Equiv.: {X N } is asymptotically uniformly distributed over S d if a k=1 lim Q N[f ] = I[f ] for every f C(S d ). N a I.e., ν(x (s) N ) σ d as N (in the weak- limit).

7 Sobolev Space over S d Sequence of positive real numbers a = (a (s) l ) l 0 satisfying lim l a (s) l = a(d, s) for some a(d, s) > 0, 2s [l + (d 1)/2] Define an inner product for functions f and g in C (S d ) by means of (f, g) H s a := l=0 Z (d,l) k=1 1 a (s) l ˆf (d) l,k ĝ(d) l,k, where the Laplace-Fourier coefficients are given by (d) (d) ˆf l,k = (f, Y l,k ) L 2 (S d ) = f (x)y (d) l,k (x) d σ d(x). S d The Sobolov space H s a(s d ) then is the completion of the space { } Z (d,l) f C (S d 1 ] 2 (d) ) : [ˆf l,k < l=0 k=1 a (s) l with respect to the Sobolev norm H s a := (, ) H s a.

8 Cui and Freeden s Generalized Discrepancy [SIAM J. SCI. COMPUT. 18:2 (1997)]... based on pseudodifferential operators yielding A Koksma-Hlawka like inequality QN [f ] I[f ] 6 DCF (X N ) f H 3/2 (S 2 ), where f is from the Sobolev space H 3/2 (S 2 ). D CF (X N ) in terms of elementary function expressible: 4π [D CF (X N )] 2 = 1 1 N 2 N j,k=1 log (1 + x j x k /2) 2. Definition (Equidistribution in H 3/2 (S 2 )) A sequence {X N } is equidistributed in H 3/2 (S 2 ) if lim N D CF(X N ) = an asymptotically uniformly distributed {X N } is equidistributed in C(S 2 ).

9 Interpretation as Worst Case Error (WCE) Sloan and Womersley [Adv. Comput. Math. 21 (2004)]... show that D CF (X N ) has a natural interpretation, as the worst-case error (apart from a constant factor) for the equal-weight cubature rule E n (f ) = 4π d n d n k=1 f (x k ) applied to a function f B(H), where B(H) is the unit ball in a certain Hilbert space H.

10 Worst Case Error (WCE) in a Sobolev Space H s a(s d ) Sequence of positive real numbers a = (a (s) l ) l 0 satisfying lim l a (s) l = a(d, s) for some a(d, s) > 0, 2s [l + (d 1)/2] The worst-case error of Q N [f ] wce(q N ; H s a(s d )) = sup { QN [f ] I[f ] : f H s (S d ), f d,s 1 }. By Cauchy-Schwarz inequality [ wce(qn ; H s a(s d )) ] 2 = 1 N 2 S d N K (a; x j x k ) j,k=1 S d K (a; x y) d σ d (x) d σ d (y). Example A Example B

11 Stolarsky s Invariance Principle for Euclidean Distance Spherical Cap centered at z with height t C(z; t):= { x S d : x, z t }, z S d, 1 t 1. Spherical cap L 2 -discrepancy: { 1 2 1/2 DL C 2 (X N ):= X N C(x; t) σ d (C(x; t)) N d σ d (x) d t}. 1 S d Theorem (Stolarsky [Proc. of the AMS 41:2 (1973)]) 1 N 2 N j,k=1 x j x k + H d(s d ) [ ] 2 H d (B d DL C ) 2 (X m ) = x y d σ d (x) d σ d (y). S d S d Stolarsky s proof makes use of Haar integrals over SO(d + 1). Reproved using reproducing kernel Hilbert space techniques (B-Dick [Proc. of the AMS, in press])

12 Sum of distance integral: V 1 (S d ) = S d x y d σ d (x) d σ d (y). S d Theorem (B. Womersley) Let d 2 and H s (S d ), s = (d + 1)/2, with reproducing kernel Then K d,s (x, y) = 2V 1 (S d ) x y, x, y S d. [ wce(qn ; H s (S d )) ] 2 = V 1 (S d ) 1 N N 2 x i x j i,j=1 = H d(s d ) [ 2 H d (B d D L 2 C N)] ) (X.

13 Point Constructions SPHERICAL DIGITAL NETS SPHERICAL FIBONACCI LATTICE APPROXIMATIVE SPHERICAL DESIGNS (ersatz FOR SPHERICAL DESIGNS?)

14 Area preserving map to S 2 & Consequences Lambert cylindrical equal-area projection Map to S d x:=φ s (y):= Φ s (φ 1 (y 1 ), φ 2 (y 2 )) S 2, where φ 1 (y 1 ):=2πy 1, φ 2 (y 2 ):=1 2y 2, and ( ) Φ s (φ, t):= 1 t 2 cos φ, 1 t 2 sin φ, t. Lemma (B-Dick, Numerische Mathematik (2011)) D ( ) N (S2, Ω ( ) ; Z N ) = D ( ) N ([0, 1)2, R ( ) ; Z N ). Theorem (B-Dick, Numerische Mathematik (2011)) Sequence (Z N ) N 2 s.t. Z N = Φ 2 (Z N ), Z N [0, 1) 2. [ ] ( 2 wce(q N ; H 3/2 24 a (S 2 )) + 2 ) 2 DN([0, 1) 2, R ( ) ; Z N ). 3

15 Definition (Isotropic Discrepancy) J N (Z N ):=D N ([0, 1] 2, A; Z N ), where A is the family of convex subsets of [0, 1] 2 Theorem (Aistleitner-B-Dick, submitted) D C (Φ(Z N )) 11J N (Z N ), Z N [0, 1) 2.

16 DIGITAL NETS AND SPHERICAL DIGITAL NETS

17 Digital Nets Definition (Star Discrepancy and local Discrepancy) D N(X N ) = sup δ XN (x), x [0,1] s δ XN (x) = X N [0, x) N VOL([0, x)) Digital Net (Characterization) A (t, m, s)-net in base b with N = b m points has the property that each elementary interval contains exactly b t points. s i=1 [ ai b d i, a ) 0 a i < b d i i + 1 d b d d s = m t i 0 d 1,..., d s Z (elementary interval) Example Construction Theorem X N (t, m, s)-net in base b: D N(X N ) C (m t) s 1 /b m t.

18 Low Discrepancy Point Sets and Sequences D N(X N ) = sup δ XN (x), x [0,1] s δ XN (x) = X N [0, x) N VOL([0, x)) THM.: X N (0, m, s)-net in base b: D N(X N ) C m s 1 /b m 1. HAMMERSLEY [Ann. New York Acad. Sci ] and HALTON [Numer. Math ] (i) for any N 2 there exists x 1,..., x N [0, 1) s s.t. D N((x 1,..., x N )) = O( (log N)s 1 ) N (ii) there exists a sequence (x n ) n 1 in [0, 1) s s.t. D N((x n ) n 1 ) = O( (log N)s ) N

19 Spherical Rectangle discrepancy Families of half-open axis-parallel rectangles lifted to S 2 Ω = {Φ s (R x,y ) : 0 x y 1}, Ω = {Φ s (R 0,y ) : 0 y 1}. Definition (Spherical Rectangle (Star) Discrepancy) D N (Ω; X N ):= sup δ N (R; X N ), D N(Ω ; X N ):= sup δ N (R; X N ). R Ω R Ω Theorem (B-Dick, Numerische Mathematik) For Z N (a (0, m, 2)-net in base b lifted to S 2 ) D b m(ω, Z N ) b2 m b + 1 b m + 1 b m Db m(ω, Z N ) b2 /4 m b + 1 b m + 1 b m By Roth [Mathematika (1954)] ( ) + 1 b b 2m ( ) + 1 b b 2m Recall ( 2b 1 4b + 3 (b + 1) 2 ), ( b b + 3 4(b + 1) 2 D N (Ω, X N ) D N(Ω, X N ) ( log 2 N + 3)/(2 8 N). ).

20 Isotropic & Spherical Cap Discrepancy Theorem (Aistleitner-B-Dick, submitted) Z N... (0, m, 2)-net in base b (N = b m ) J N (Z N ) 4 2b N. Z N... first N points of (0, 2)-sequence J N (Z N ) 4 2(b 2 + b 3/2 ) N. Remark Optimal rate of convergence for J N (cf. Beck and Chen, 1987 & 2008): N 2/3 (log N) c for some 0 c 4!

21 WCE of Numerical Integration Rules based on Digital Nets (Sobol points) on Spheres m swce N 3/2 swce e e e e e e e e e e e e e e e e e e e e Conjecture (B-Dick, Numerische Mathematik) d = 2 d = 200 A sequence of Sobol points in [0, 1) s (s 2) lifted to the s-sphere achieves optimal convergence rate of the sum of distances.

22 FIBONACCI LATTICE AND SPHERICAL FIBONACCI LATTICES

23 Fibonacci lattice points in [0, 1] 2 Fibonacci sequence: F 1 = 1, F 2 = 1, and F n+1 = F n + F n 1, n 1. Fibonacci lattice in [0, 1] 2 ( k f k :=, F n { kfn 1 F n }), 0 k < F n, where {x} = x x for nonnegative real numbers x. The set F n := {f 0,..., f Fn 1} is called a Fibonacci lattice point set.

24 Fibonacci lattice points on the unit sphere S 2 Lambert transformation Φ : [0, 1] 2 S 2 Φ(x, y) = ( ) 2 cos(2πy) x x 2, 2 sin(2πy) x x 2, 1 2x. Note: Area preserving map. Fibonacci lattice on S 2 The spherical Fibonacci lattice points are then given by and the point set z k = Φ(f k ), 0 k < F n, Z n = {z 0,..., z Fn 1} is the spherical Fibonacci lattice point set.

25 Separation results (B Dick, work in progress) Minimum distance: d min (P) = min 0 k<l<n y k y l, P = {y 0,..., y N 1 } Rd+1. Theorem (Fibonacci points in [0, 1] 2 ) d min (F 2m+1 ) = 1 F 2m+1, m 1, d min (F 2m ) = 2Fm F 2m, m 2. Theorem (Fibonacci points in S 2 ) d min (Z 2m+1 ) 1 F 2m+1, m 1. Conjecture (Fibonacci points in S 2 ) d min (Z n) = 2 Fn, n 2. n Fn Zndmin (Zn) ± ± ± ± ± ± ± ± ± ±

26 Isotropic & Spherical Cap Discrepancy Corollary (Aistleitner-B-Dick, submitted) { D C 44 2/F m if m is odd, (Z Fm ) 44 8/F m if m is even. Table: D(Z Fm )... maximum of the absolute values of the local discrepancies of Z Fm at the spherical caps centered at the spherical Fibonacci points Z Fm. m Fm D(Z Fm ) Fm 3/4 log Fm D(Z Fm ) Fm 3/4 log Fm m Fm D(Z Fm ) Fm 3/4 log Fm D(Z Fm ) F 3/4 m log Fm

27 Spherical Cap L 2 -discrepancy (B Dick, work in progress) [ 4 D L C ] 2 4 (Zn) = Fn 2 Fn 1 z j z k j,k=0 [ [ ] 2 n Fn 4 D L C ] 2 (Zn) 2 Fn 3/2 4Fn 3/2 D L C (Zn) e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e cf. B [Uniform Distribution Theory 6:2 (2011)]

28 SEQUENCES OF APPROXIMATE SPHERICAL DESIGNS

29 Motivation Lower bound in terms of number of nodes: Theorem (He-Sl 2005 (d = 2); He-Sl 2006 (d 2)) wce(q N ; H s (S d )) c (d, s) N s/d. Upper bound in terms of exactness degree: Theorem (He-Sl 2005 (H 3/2 (S 2 )); He-Sl 2006 (H s (S 2 ), s > 1); B.-He 2007 (H s (S d ), s > d/2)) (i) Q N(t) is exact for all polynomials of degree t, and (ii) (Q N(t) ) t N satisfies a local regularity condition (Property (R) ). a wce(q N(t) ; H s (S d )) C (d, s) t s. a Automatically fulfilled for positive weight numerical integration rules (Reimer 2000). Optimality: There are always Q N(t) with N(t) C t d.

30 Spherical Designs Definition (Delsarte, Goethals and Seidel 1977 ) Spherical t-designs x 1,..., x N(t) with N(t) points have the property S d P(x) d σ d (x) = 1 N(t) for all polynomials P with deg P t. N(t) P(x j ) j=1

31 Theorem (Upper Bound) A sequence of N(t)-point spherical designs (X N(t) ) t 1 on S d has the property that there exists C(s, d) > 0, independent of t and N(t), s. t. 1 C(s, d) sup f (x) f (x) d σ d (x) N(t) x X S t s as t, d N(t) f H s (S d ), f H s 1 where H s denotes the norm in the Sobolev space H s. H 3/2 (S 2 ): K. Hesse & I. H. Sloan [Bull. Austral. Math. Soc. 71 (2005)] H s (S 2 ), s > 1: K. Hesse & I. H. Sloan [J. Approx. Theory 141 (2006)] H s (S d ), s > d/2: B. & K. Hesse [Constr. Approx. 25 (2007)]

32 Approximate Spherical Designs Definition (B-Saff-Sloan-Womersley (work in progress)) A sequence (X N(t) ) t 1 on S d, with t and N(t), is said to be a sequence of approximate spherical t-designs for H s (S d ) for some s > d/2 if there exists c(s, d) > 0, independent of t and N(t), s. t. sup f H s (S d ), f H s 1 1 f (x) f (x) d σ d (x) N(t) x X S d N(t) c(s, d) t s as t. Proposition Spherical designs are approximative spherical designs for all s > d/2.

33 Approximate Spherical Designs are better than average Let Set Definition A(x y) = l=1 A N [X N ]:= 1 N 2 a l }{{} 0 N Z (d, l)p (d) l (x y). j=1 k=1 N A(x j x k ). Spherical average of A N [X N ] over all N-point config. on S d EA N := S d A N [{x 1,..., x N }] d σ d (x 1 ) d σ d (x N ). S d Proposition EA N = A(1) N.

34 Average WCE E [ wce(q N ; H s a(s d )) ] 2 K (a; 1) a (s) = 0. N Average Approximate Integration E 1 N N N 2 K t (a; x j x k ) a (s) 0 = K t(a; 1) a (s) 0. N j=1 k=1 Compare with optimal rate N 2s/d if N = O(t d ).

35 Characterization Theorem (B-Saff-Sloan-Womersley (work in progress)) Let s > 1. a = (a (s) l ) l 0 monotonically decreasing positive numbers satisfying lim l a (s) l = a(d, s) for some a(d, s) > 0, 2s [l + (d 1)/2] a (s) l (l + 1/2) 2s a(2; s)= O(l 2 ) a (s) l+1 (l + 3/2) 2s a (s) l (l + 1/2) 2s = O(l 3 ). Then a sequence (X N(t) ) t 0 of N(t)-point configurations is a sequence of approximate spherical t-designs for H s a(s 2 ) iff 1 N 2 N N j=1 k=1 K t (a; x j x k ) a (s) 0 = O(t 2s ) as t.

36 Variational characterization of sph. designs; Gr-Ti 1993, Sl-Wo 2009 An N(t)-point config. which minimizes the non-negative functional 1 N 2 N N j=1 k=1 K t (a; x j x k ) a (s) 0 and, in addition, evaluates it to zero is already a spherical t-design.

37 Families of Approximate Spherical Designs Theorem (B-Saff-Sloan-Womersley (work in progress)) Let d 2, d/2 < s < d/ The maximizer of the generalized distance functional N x j x k 2s d. j,k=1 form a sequence of approximate spherical t-designs for H s (S d ), provided N(t) t d.

38 The End Thank You!

39 Numerical Integration in the Unit Cube Koksma-Hlawka Inequality N 1 1 f (x n ) f (x) dx N [0,1] D N(X N )V HK (f ). s n=0 Return

40 Example (Digital (0, 3, 2)-net in base 2 with 8 points)

41 Example (Digital (0, 3, 2)-net in base 2 with 8 points)

42 Example (Digital (0, 3, 2)-net in base 2 with 8 points)

43 Example (Digital (0, 3, 2)-net in base 2 with 8 points)

44 Example (Digital (0, 3, 2)-net in base 2 with 8 points)

45 Example (Digital (0, 3, 2)-net in base 2 with 8 points) Return

46 Construction of (0, m, s)-nets by means of Sobol pts Let l = b 1 + b b n 2 n 2 n. The j-th component of the l-th point in the Sobol sequence x l,j = 2 n [b 1 v 1,j b n v n,j ], v k,j = m k,j 2 n k. The direction numbers {m 1,j, m 2,j, } are recursively defined by s j 1 m k,j = 2 ν a ν,j m k ν,j 2 s j m k sj,j m k sj,j, k s j + 1, ν=1 in terms of the coefficients of the primitive polynomial x s j + a 1,j x s j a sj 1,jx + 1 over Z 2 and the initial direction numbers m 1,j,..., m sj,j. (For the first component one may chose m k,1 = 1 for all k.) decoded as the integer a 1,j 2 s j a sj 1,j 2 is Return

47 88 x x x x2 86 Example Figure: 2-dim. projections of Sobol points in [0, 1)200 with N = 256, 512. Return

48 Proof. Substitution of d σ d (x d ) = ω d 1 ω d ( 1 τ 2 d ) d/2 1 d τd d σ d 1 (x d 1 ) d = 2, 3,..., yields σ 1 (Φ 1 (R y )) = 2πy 1 0 d φ/(2π) = y 1 and σ d (Φ d (R y )) = ω d 1 ω d 1 t d ( 1 τ 2 d ) d/2 1 d τd = T d (t d ) σ d 1 (Φ d 1 (R y )), where (since 2 d 1 ω d 1 /ω d = 1/B(d/2, d/2)) T d (t) = ω d 1 ω d 1 t = I (1 t)/2 (d/2, d/2). ( 1 τ 2 ) d/2 1 d τ = 2 d 1 ω d 1 ω d Φ d 1 (R y) (1 t)/2 Note that T 2 (t) = (1 t)/2 and t d = 1 2y d (2 d s). 0 d σ d 1 (x d 1 ) u d/2 1 d u Return

49 A(J; X N )... number of points of X N in J Definition (Local Discrepancy) δ N (J; X N ):= A(J; X N) N VOL(J). Return

50 Beck (1984) To any N-point cfg. {x 1,..., x N } on S d there exists a spherical cap C s.t. c 1 N 1/2 1/(2d) < # {k : x k C} N σ d (C). There exists an N-point cfg. {x 1,..., x N } on S d s.t. for any spherical cap C # {k : x k C} N σ d (C) < c 2 N 1/2 1/(2d) log N. Return

51 d = 2; N = 2,..., Return

52 d = 200; N = 2,..., Comparison, pseudo-random points Return 10 5

53 N Return

54 Spherical Designs Much progress has been made since their introduction by Delsarte, Goethals and Seidel [Geometriae Dedicata 6 (1977)]: existence of spherical designs in general and existence of so-called tight spherical designs in particular, their construction by algebraic and by variational means culminating in a recently proposed proof of the N 2 -conjecture (or N d -conjecture) of Korevaar and Meyers [Integral Transform. Spec. Funct. 1 (1993)] that N(t) = O(t d ) points are not only necessary but also sufficient to form a spherical t-design by Andriy Bondarenko, Danylo Radchenko and Maryna Viazovska (arxiv: v3 [math.mg]). Return

55 Property (R) Return Definition A sequence (Q N(t) ) t 1 of weighted numerical integration rules Q N(t) on S d is said to have the property (R) (or to be quadrature regular ), if there exist positive numbers c 0 and c 1 independent of n with c 1 π/2, such that for all t 1 the weights w 1,..., w N(t) associated with the nodes x 1,..., x N(t) satisfy N(t) j=1, x j C(x;φ) w j c 0 σ d (C(x; c 1 /t)), where C(x; φ):={y S d : x y cos φ} denotes the spherical cap.

56 Lifting Digital Nets to the Unit Sphere in R d+1, d 2 Cylindrical Coordinates on the d-sphere x = x s = ( 1 t 2 s x s 1, t s ), x 2 = ( 1 t 2 2 x 1, t 2 ), x 1 = ( cos φ, sin φ ) 1 t d 1, x d S d (d = 1,..., s) 0 φ 2π. Arbitrary x S s represented through angle φ and heights t 2,..., t s x = x(φ, t 2,..., t s ), 0 φ 2π, 1 t 2,..., t s 1. A point y = (y 1,..., y s ) [0, 1) s mapped to x S s using ϕ 1 (y 1 ) = 2πy 1, ϕ d (y d ) = 1 2y d (d = 2, 3,..., s) and cylinder coordinates x = Φ s (y) = x ( ϕ 1 (y 1 ), ϕ 2 (y 2 ),..., ϕ s (y s ) ).

57 Area preserving map Normalized surface area measure σ d on S d d σ 1 (x 1 ) = d φ/(2π), d σ d (x d ) = ω d 1 ω d ( 1 t 2 d ) d/2 1 d td d σ d 1 (x d 1 ), d = 2, 3,.... Let R y = [0, y 1 ) [0, y s ) [0, 1) s. Proposition s σ s (Φ s (R y )) = y 1 I yd (d/2, d/2). d=2 Proof Regularized Beta function I z (a, b) = B z (a, b)/ B(a, b), B z (a, b) = z 0 u a 1 (1 u) b 1 d u.

58 Let VOL(R z ) = z 1 z 2 z s for z [0, 1) s. Chose y 1 = z 1, y 2 = z 2 and y d such that I yd (d/2, d/2) = z d. Then again σ s (Φ s (R y )) = z 1 z 2 z s. Area preserving map Ψ s : ( z 1,..., z s ) Φs ( ( h 1 d (z 1), h 1 d (z 2),..., h 1 d (z s) ) ). h 1 d (y) is the inverse of h 1(y) = y and h d (y) = I y (d/2, d/2), d 2. I z 1 d d, d = 2,..., 10 and d = z Return

59 Example A Return From the well-known identity l=1 1 ( l (l + 1) P l(t) = 1 2 log 1 + one derives the sequence a = (a (s) l ) l 0 with 1 t 2 ), t [ 1, 1], a (3/2) 0 = 1, a (3/2) 1 l = l (l + 1) (2l + 1), l 1. The Sobolev space H (3/2) a (S 2 ) is a reproducing kernel Hilbert space with reproducing kernel K (a; x y) = K (3/2) H (x, y) = a l=0 ( a (3/2) x y ) l (2l + 1) P l (x y) = 2 2 log Considered by Cui and Freeden (1997); generalized discrepancy. a (3/2) l (l + 1/2)2 l + 1/2 = (l + 1/2) 3 l (l + 1) 2l + 1 = /8 + O(l 3 ) as l. l 2

60 Example B Return Let d/2 < s < d/ The weights a (s) 0 = V d 2s (S d ) = 2 2s 1 Γ((d + 1)/2)Γ(s) πγ(d/2 + s), a (s) l = V d 2s (S d ) (d/2 s) l (d/2 + s) l, in the sequence a = (a (s) l ) l 0 define the reproducing kernel K (a; x y) = l=0 a (s) l Z (d, l) P (d) l (x y) = 2V d 2s (S d ) x y 2s d, for the Sobolev space H (s) a (S d ), d/2 < s < d/ a (s) l Γ((d + 1)/2)Γ(s) = + a (d; s) + O(l 3 ) as l. (l + 1/2) 2s π[ Γ(d/2 s)] l 2