Low-discrepancy point sets lifted to the unit sphere Johann S. - PowerPoint PPT Presentation

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)

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

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

Preliminaries U NIFORM D ISTRIBUTION ON THE UNIT SPHERE W ORST - CASE ERROR (WCE) C UI AND F REEDEN ’ S GENERALIZED D ISCREPANCY S TOLARSKY ’ S I NVARIANCE P RINCIPLE

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. � N � Q N [ f ]:= 1 I [ f ]:= S d f d σ d , f ( x k ) N k = 1 Equiv.: { X N } is asymptotically uniformly distributed over S d if a for every f ∈ C ( S d ) . N →∞ Q N [ f ] = I [ f ] lim a I.e., ν ( X ( s ) N ) → σ d as N → ∞ (in the weak- ∗ limit).

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

Cui and Freeden’s Generalized Discrepancy [SIAM J. SCI. COMPUT. 18:2 (1997)] . . . based on pseudodifferential operators yielding A Koksma-Hlawka like inequality √ � � � ≤ � Q N [ f ] − I [ f ] 6 D CF ( 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: N � 4 π [ D CF ( X N )] 2 = 1 − 1 log ( 1 + | x j − x k | / 2 ) 2 . N 2 j , k = 1 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 ) = 0. . . . an asymptotically uniformly distributed { X N } is ’equidistributed in C ( S 2 ) ’.

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 d n � E n ( f ) = 4 π f ( x k ) d n k = 1 applied to a function f ∈ B ( H ) , where B ( H ) is the unit ball in a certain Hilbert space H . ”

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 a ( s ) ℓ [ ℓ + ( d − 1 ) / 2 ] − 2 s = a ( d , s ) for some a ( d , s ) > 0, lim ℓ →∞ The worst-case error of Q N [ f ] �� � � � Q N [ f ] − I [ f ] � : f ∈ H s ( S d ) , � f � d , s ≤ 1 wce ( Q N ; H s a ( S d )) = sup . By Cauchy-Schwarz inequality N � � � 2 = 1 wce ( Q N ; H s a ( S d )) K ( a ; x j · x k ) N 2 j , k = 1 � � − S d K ( a ; x · y ) d σ d ( x ) d σ d ( y ) . S d Example A Example B

Stolarsky’s Invariance Principle for Euclidean Distance Spherical Cap centered at z with ’height’ t � � x ∈ S d : � x , z � ≤ t z ∈ S d , − 1 ≤ t ≤ 1 . C ( z ; t ):= , Spherical cap L 2 -discrepancy : �� 1 � 1 / 2 � � � 2 � � | X N ∩ C ( x ; t ) | D C � � L 2 ( X N ):= − σ d ( C ( x ; t )) d σ d ( x ) d t . � � N − 1 S d Theorem (Stolarsky [Proc. of the AMS 41 :2 (1973)]) � � � � 2 N � | x j − x k | + H d ( S d ) 1 D C L 2 ( X m ) = S d | x − y | d σ d ( x ) d σ d ( y ) . N 2 H d ( B d ) S d j , k = 1 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])

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 K d , s ( x , y ) = 2 V − 1 ( S d ) − | x − y | , x , y ∈ S d . Then N � � � 2 = V − 1 ( S d ) − 1 wce ( Q N ; H s ( S d )) | x i − x j | N 2 i , j = 1 � � 2 = H d ( S d ) D L 2 C ( X N ) . H d ( B d )

Point Constructions S PHERICAL D IGITAL N ETS S PHERICAL F IBONACCI L ATTICE A PPROXIMATIVE S PHERICAL D ESIGNS ( ersatz FOR SPHERICAL DESIGNS ? )

Area preserving map to S 2 & Consequences Map to S d Lambert cylindrical equal-area projection x := Φ s ( y ):= � Φ s ( φ 1 ( y 1 ) , φ 2 ( y 2 )) ∈ S 2 , where φ 1 ( y 1 ):= 2 π y 1 , φ 2 ( y 2 ):= 1 − 2 y 2 , and �� � � � 1 − t 2 cos φ, 1 − t 2 sin φ, t Φ s ( φ, t ):= . Lemma (B-Dick, Numerische Mathematik (2011)) D ( ∗ ) N ( S 2 , 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 . � 24 � � � 2 √ wce ( Q N ; H 3 / 2 ( S 2 )) D ∗ N ([ 0 , 1 ) 2 , R ( ∗ ) ; Z N ) . ≤ √ + 2 2 a 3

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 )) ≤ 11 J N ( Z N ) , Z N ⊆ [ 0 , 1 ) 2 .

D IGITAL N ETS AND S PHERICAL D IGITAL N ETS

Digital Nets Definition ( Star Discrepancy and local Discrepancy ) δ X N ( x ) = | X N ∩ [ 0 , x ) | D ∗ N ( X N ) = x ∈ [ 0 , 1 ] s | δ X N ( x ) | , sup − VOL ([ 0 , x )) N 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. 0 ≤ a i < b d i � a i s � b d i , a i + 1 � d 1 + · · · + d s = m − t (elementary interval) b d i i = 1 0 ≤ d 1 , . . . , d s ∈ Z Example Construction Theorem N ( X N ) ≤ C ( m − t ) s − 1 / b m − t . X N ( t , m , s ) -net in base b: D ∗

Low Discrepancy Point Sets and Sequences δ X N ( x ) = | X N ∩ [ 0 , x ) | D ∗ N ( X N ) = x ∈ [ 0 , 1 ] s | δ X N ( x ) | , sup − VOL ([ 0 , x )) N THM.: X N ( 0 , m , s ) -net in base b : D ∗ N ( X N ) ≤ C m s − 1 / b m − 1 . H AMMERSLEY [Ann. New York Acad. Sci. 86 1960] and H ALTON [Numer. Math. 2 1960] (i) for any N ≥ 2 there exists x 1 , . . . , x N ∈ [ 0 , 1 ) s s.t. N (( x 1 , . . . , x N )) = O (( log N ) s − 1 D ∗ ) N (ii) there exists a sequence ( x n ) n ≥ 1 in [ 0 , 1 ) s s.t. N (( x n ) n ≥ 1 ) = O (( log N ) s D ∗ ) N

Spherical Rectangle discrepancy Families of ’half-open’ axis-parallel rectangles ’lifted’ to S 2 Ω ∗ = { Φ s ( R 0 , y ) : 0 � y � 1 } . Ω = { Φ s ( R x , y ) : 0 � x ≺ y � 1 } , Definition ( Spherical Rectangle (Star) Discrepancy ) D ∗ N (Ω ∗ ; X N ):= sup D N (Ω; X N ):= sup | δ N ( R ; X N ) | , R∈ Ω ∗ | δ N ( R ; X N ) | . R∈ Ω Theorem (B-Dick, Numerische Mathematik) For Z N (a ( 0 , m , 2 ) -net in base b lifted to S 2 ) Recall � � � � b 2 b m + 1 m 9 + 1 1 2 b − 1 − 4 b + 3 D b m (Ω , Z N ) ≤ + , b + 1 b m b 2 m ( b + 1 ) 2 b � 9 � � b � b m (Ω ∗ , Z N ) ≤ b 2 / 4 b m + 1 m 4 + 1 1 2 − 1 4 b + 3 D ∗ + 4 − . b + 1 b m b b 2 m 4 ( b + 1 ) 2 By Roth [Mathematika (1954)] D N (Ω , X N ) ≥ D ∗ N (Ω ∗ , X N ) ≥ ( ⌊ log 2 N ⌋ + 3 ) / ( 2 8 N ) .

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 2 b √ . N • Z N . . . first N points of ( 0 , 2 ) -sequence √ 2 ( b 2 + b 3 / 2 ) J N ( Z N ) ≤ 4 √ . 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!