Properties of Spherical Fibonacci Points. Johann S. Brauchart - PowerPoint PPT Presentation

Properties of Spherical Fibonacci Points. Johann S. Brauchart j.brauchart@tugraz.at 22. April 2017 OPTIMAL POINT CONFIGURATIONS AND ORTHOGONAL POLYNOMIALS 2017, CIEM CASTRO URDIALES

E XPLICIT P OINT S ET C ONSTRUCTION

F IBONACCI L ATTICE P OINTS IN THE S QUARE [ 0 , 1 ] 2

Fibonacci sequence (OEIS: A000045): F 0 := 0 , F 1 := 1 , F 2 := 1 , F n + 1 := F n + F n − 1 , n ≥ 1 . Fibonacci lattice in [ 0 , 1 ] 2 � k � �� k F n − 1 F n : , , 0 ≤ k < F n . F n F n { x } is fractional part of real x .

1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 n = 14: F n = 377.

1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 n = 15: F n = 610.

Fibonacci Quart. 8 1970 no. 2, 185–198.

L ∞ Discrepancy of Fibonacci Lattice Point Sets √ Golden Ratio : φ = 1 + 5 . 2 F n − 1 / F n is n th convergent of 1 φ − 1 = 0 + . 1 1 + 1 + ... So � D ( F n ; · ) � ∞ ≍ n ≍ log N .

L 2 Discrepancy of sym. Fibonacci Lattice Point Sets

Theorem (Bilyk, Temlyakov, & Yu, 2012) 2 = 1 8 S n + 17 36 + c n ; · ) � 2 � D ( F ′ F 2 n ≍ n 1 / 2 ≍ ( log F n ) 1 / 2 , where c depends on parity of F n and F n − 1 � S n := 1 1 � 2 . � � 2 � F 2 sin π kF n − 1 sin π k n k = 1 F n F n . . . requires O ( F n ) steps for computation (vs. O ( F n log F n ) via Warnock’s formula).

S PHERICAL F IBONACCI L ATTICE P OINTS

Area preserving Lambert transformation Φ : [ 0 , 1 ] 2 → S 2  �   2 cos ( 2 π y ) x − x 2 �  Φ ( x , y ) = x − x 2 2 sin ( 2 π y ) 1 − 2 x

Illumination Integrals

P ROPERTIES

N UMERICAL I NTEGRATION VIA Quasi Monte Carlo (QMC) methods � N � S d f d σ d ≈ 1 f ( x j ) . N j = 1

Rationale Good properties of f are preserved by not making a change of variables. Costs: finding of “good” node sets.

Uniform distribution on S d Definition ( X N ) is asymptotically uniformly distributed on S d if # { k : x k , N ∈ B } lim = σ d ( B ) N N →∞ for every Riemann-measurable set B in S d . Quantification Informally: A reasonable set gets a fair share of points as N becomes large. Equivalent definition ( X N ) is asymptotically uniformly distributed on S d if � N � 1 lim f ( x k ) = S d f d σ d N N →∞ k = 1 for every f ∈ C ( S d ) . Quantification

S PHERICAL D ESIGNS

Definition (Delsarte, Goethals and Seidel, 1977) Spherical t -designs { x 1 , . . . , x N } ⊂ S d satisfy � N � S d P ( x ) d σ d ( x ) = 1 P ( x j ) N j = 1 for all polynomials P with deg P ≤ t . Theorem ( Bondarenko, Radchenko and Viazovska, 2013 ) There exists c d such that: for every N ≥ c d t d there is a spherical t-design with N points.

A sequence ( Z ∗ N t ) of spherical t -designs with N t points of exactly the optimal order ( N t ≍ t d ) of points has the remarkable property that � � � � � ≤ c s N − s / d � Q [ Z N ∗ t ]( f ) − I ( f ) � f � H s t for all f ∈ H s ( S d ) and all s > d 2 . The order of N t cannot be improved.

O PTIMAL NODE SETS

The reproducing kernel Hilbert space approach provides an elegant and powerful method to compute the worst-case error of a QMC rule for functions from the unit ball in a Sobolev space H s ( S d ) , s > d 2 ; i.e., for f ∈ H s , Q [ X N ]( f ) − I ( f ) � � N � f , 1 = K ( x , x j ) − I ( K ( x , · )) H s , N j = 1 � �� � R [ X N ]( x ) where K is a reproducing kernel for H s and R [ X N ] the “representer” of the error.

In particular, the distance kernel K ( x , y ) := 1 − c d | x − y | yields an invariance principle for the WCE, N � � � 2 1 | x j − x k | + 1 wce ( Q [ X N ] , K ) N 2 c d j , k = 1 � � = S d | x − y | d σ d ( x ) σ d ( y ) , S d named after Stolarsky (JSB-Dick, 2013 ). � 1 � Proof exploits K ( x , y ) = S d 1 C ( x , t ) ( z ) 1 C ( y , t ) ( z ) d σ d ( z ) d t . − 1

A sequence ( X ∗ N ) of maximal sum-of-distance N -point sets define QMC rules that satisfy | Q [ X N ∗ ]( f ) − I ( f ) | ≤ c s ′ N − s ′ / d � f � H s ′ 2 < s ′ ≤ d + 1 for all f ∈ H s ′ ( S d ) and all d 2 . ∗ The order of N cannot be improved. ∗ Open: Determine strength of ( X ∗ N ) .

DPPs and Worst-case Errors Masatake Hiraro (Aichi Prefectural University) (MCQMC 2016 at Standford): N -point spherical ensembles on S 2 yield average WCE of order N − s / 2 , 1 < s < 2; also results for harmonic ensembles on S d ;

Spherical Fibonacci Points — see Discrepancy results

L OW - DISCREPANCY SEQUENCES ON THE SPHERE

� Motivated by classical (up to log N optimal) results of J. Beck ( 1984), a sequence ( X N ) is of low-discrepancy if � log N � D ( X N , · ) � ∞ ≤ c 1 N 1 / 2 + 1 / ( 2 d ) . Unresolved Question: Unlike in the unit cube case, there are no known explicit low-discrepancy constructions on the sphere.

Spherical cap L ∞ Discrepancy Spherical cap L ∞ -discrepancy � � � | Z N ∩ C | � � D C L ∞ ( Z N ) := sup − σ d ( C ) � N C Corollary (Aistleitner-JSB-Dick, 2012 ) �� √ D C L ∞ ( Z F m ) ≤ 44 8 F m and numerical evidence that for some 1 2 ≤ c ≤ 1 , L ∞ ( Z F m ) = O (( log F m ) c F − 3 / 4 D C ) as F m → ∞ . m RMK: A. Lubotzky, R. Phillips and P . Sarnak (1985, 1987) ) ≪ ( log N ) 2 / 3 N − 1 / 3 with numerical have D C L ∞ ( X LPS N evidence indicating O ( N − 1 / 2 ) .

ln-ln plot of spherical cap L ∞ -discrepancy of point set families.

Should be compared with . . . Theorem (Aistleitner-JSB-Dick, 2012) � � c C D C N 1 / 2 ≤ E L ∞ ( X N ) ≤ N 1 / 2 . Coulomb Surprisingly: Random Theorem (Götz, 2000) c N ) ≤ C log N N 1 / 2 ≤ D C L ∞ ( X ∗ N 1 / 2 , X ∗ N minimizing the Coulomb potential energy N N � � 1 | x j − x k | . j = 1 k = 1 j � = k

Spherical Cap L 2 Discrepancy Let D ( X N , C ) := | X N ∩ C | − σ d ( C ) be the local N discrepancy function w.r.t. spherical caps C . The L 2 -discrepancy � D ( X N , · ) � 2 satisfies N � 1 | x j − x k | + 1 � D ( X N , · ) � 2 2 N 2 c d j , k = 1 � � = S d | x − y | d σ d ( x ) σ d ( y ) , S d an invariance principle first shown by Stolarsky ( 1973; JSB-Dick, 2013;); i.e., maximizers of the sum of distances have optimal � D ( X N , · ) � 2 . † † The precise large N behavior is closely related to minimal Riesz energy asymptotics (JSB, 2011).

Optimal Spherical Cap L 2 Discrepancy Conjecture (B, 2011 ) L 2 ( X N ) ∼ A 2 N − 3 / 4 + · · · D C as N → ∞ , where � � 8 π � 1 / 2 3 √ A 2 = [ − ζ ( − 1 / 2 )] L − 3 ( − 1 / 2 ) 2 3 = 0 . 44679 . . . .

Spherical Cap L 2 -discrepancy (B–Dick, work in progress) Fn − 1 � 2 4 1 � DC � � � 4 L 2 ( Z n ) = − � z j − z k � � F 2 � 3 n j , k = 0 � 2 � 2 � � F − 3 / 2 4 F 3 / 2 DC DC n Fn 4 ( Z n ) ( Z n ) n n L 2 L 2 3 2 6.2622e-01 3.5355e-01 1.7712 4 3 3.2188e-01 1.9245e-01 1.6725 5 5 1.2865e-01 8.9442e-02 1.4384 6 8 5.7129e-02 4.4194e-02 1.2926 7 13 2.4622e-02 2.1334e-02 1.1540 8 21 1.1107e-02 1.0391e-02 1.0688 9 34 5.0965e-03 5.0440e-03 1.0103 10 55 2.3683e-03 2.4516e-03 0.9660 11 89 1.1064e-03 1.1910e-03 0.9289 12 144 5.2192e-04 5.7870e-04 0.9018 13 233 2.4792e-04 2.8116e-04 0.8817 14 377 1.1837e-04 1.3661e-04 0.8665 15 610 5.6680e-05 6.6375e-05 0.8539 16 987 2.7240e-05 3.2249e-05 0.8446 17 1597 1.3119e-05 1.5669e-05 0.8372 18 2584 6.3331e-06 7.6130e-06 0.8318 19 4181 3.0598e-06 3.6989e-06 0.8272 20 6765 1.4808e-06 1.7972e-06 0.8239 21 10946 7.1699e-07 8.7320e-07 0.8211 22 17711 3.4756e-07 4.2426e-07 0.8192 23 28657 1.6848e-07 2.0613e-07 0.8173 24 46368 8.1756e-08 1.0015e-07 0.8162 25 75025 3.9663e-08 4.8662e-08 0.8150 26 121393 1.9257e-08 2.3643e-08 0.8145 27 196418 9.3470e-09 1.1487e-08 0.8136 28 317811 4.5399e-09 5.5814e-09 0.8133 29 514229 2.2041e-09 2.7118e-09 0.8128 30 832040 1.0708e-09 1.3176e-09 0.8127 31 1346269 5.1999e-10 6.4018e-10 0.8122 0.7985 cf. B [Uniform Distribution Theory 6 :2 (2011)]