Efficient Spherical Designs with Good Geometric Properties Rob Womersley, R.Womersley@unsw.edu.au School of Mathematics and Statistics, University of New South Wales 45 -design with N = 1059 and symmetrtic 45 -deisgn with N = 1038 (Shanghai Jiao Tong University) Efficient spherical designs April, 2015 1 / 34
Outline Examples Spherical designs 1 Evaluating A t,N,ψ ( X N ) Spheres and point sets Degrees of freedom for S 2 Aims Numerical results Spherical polynomials Geometric properties 3 Number of points Mesh norm Characterizations Separation 2 Nonlinear equations Mesh ratio Variational characterizations Conclusions 4 (Shanghai Jiao Tong University) Efficient spherical designs April, 2015 2 / 34
Spherical designs Spheres and point sets Unit sphere Unit sphere � � S d = x ∈ R d +1 : | x | = 1 Sets of points X N = { x 1 , . . . , x N } ⊂ S d d +1 � | x | 2 = x · x x · y = x i y i , i =1 Distance | x − y | 2 = 2(1 − x · y ) Euclidean distance: x , y ∈ S d , Geodesic distance: x , y ∈ S d , dist ( x , y ) = arccos( x · y ) Can choose points or given points (scattered data) Want sequences of point sets X N , often as part of integration/approximation problem (Shanghai Jiao Tong University) Efficient spherical designs April, 2015 3 / 34
Spherical designs Spheres and point sets Geometric quality of point set Spherical cap centre z ∈ S d , radius α � � x ∈ S d : dist ( x , z ) ≤ α C ( z ; α ) = Separation (twice packing radius): δ X N = min i � = j dist ( x i , x j ) Mesh norm (covering radius): h X N = max j =1 ,...,N dist ( x , x j ) min x ∈ S d 2 h X N Mesh ratio: ρ X N = δ X N ≥ 1 Desire: ρ X N ≤ c (Shanghai Jiao Tong University) Efficient spherical designs April, 2015 4 / 34
Spherical designs Aims Aims Numerical integration (cubature) � N � Q N ( f ) := w j f ( x j ) ≈ I ( f ) := S d f ( x ) dω ( x ) j =1 Equal weights w j = | S d | /N, j = 1 , . . . , N (Quasi Monte-Carlo rules) Degree of precision t if exact for all polynomials of degree ≤ t Spherical t -design is a set X N of N points such that � N � 1 1 ∀ p ∈ P t ( S d ) , p ( x j ) = S d p ( x ) dω ( x ) | S d | N j =1 N point, equal weight w j = | S d | cubature rule, degree of precision t N Efficient: Low number of points N Good geometric properties: Quasi-uniform: Mesh ratio ρ X ≤ c (Shanghai Jiao Tong University) Efficient spherical designs April, 2015 5 / 34
Spherical designs Spherical polynomials Spherical Polynomials � S d � Space P t ≡ P t of spherical polynomials of degree at most t Dimension of space of homogeneous harmonic polynomials of degree ℓ Z ( d, ℓ ) = (2 ℓ + d − 1)Γ( ℓ + d − 1) Z ( d, 0) = 1; , Γ( d )Γ( ℓ + 1) Orthonormal basis Y ℓ,k , ℓ = 0 , 1 , 2 , . . . , k = 1 , . . . , Z ( d, ℓ ) � S d � is D ( d, t ) = Z ( d + 1 , t ) ≍ t d Dimension P t Addition Theorem Z ( d,ℓ ) � Y ℓ,k ( x ) Y ℓ,k ( y ) = Z ( d, ℓ ) | S d | P ( d +1) ( x · y ) , ℓ k =1 ( z ) = P ( d − 2 , d − 2 2 ) 2 ( z ) Normalized Gegenbauer polynomial P ( d +1) ℓ ℓ P ( d − 2 , d − 2 2 ) 2 (1) ℓ Jacobi polynomial P ( α,β ) ( z ) for z ∈ [ − 1 , 1] ℓ (Shanghai Jiao Tong University) Efficient spherical designs April, 2015 6 / 34
Spherical designs Number of points Spherical t -designs – Number of points N Delsarte, Goethals and Seidel (1977) N point t -design on S d � d + m � 2 if t = 2 m + 1 , d N ≥ N ∗ ( d, t ) := � d + m � � d + m − 1 � + if t = 2 m. d d ⇒ N ≥ dim P ⌊ t/ 2 ⌋ ( S d ) Positive weight cubature, degree of precision t = On S 2 : N ∗ (2 , t ) = ( t + 1)( t + 3) / 4 for t odd; ( t + 2) 2 / 4 for t even Improved by Yudin (1997) by exponential factor ( e/ 4) d +1 as t → ∞ . Bannai and Damerell (1979, 1980) Tight spherical t -designs if achieve lower bounds Cannot exist on S 2 except for t = 1 , 2 , 3 , 5 Seymour and Zaslavsky (1984) t -designs exist for N sufficiently large Bondarenko, Radchenko and Viazovska (2011, 2013, 2015) On S d spherical t -designs exist for N ≥ c d t d well-separated spherical t -designs exist for N ≥ c ′ d t d (Shanghai Jiao Tong University) Efficient spherical designs April, 2015 7 / 34
Spherical designs Number of points Existence Results for S 2 Bajnok (1991)construction with N = O ( t 3 ) n points z 1 , . . . , z n , t -design on [ − 1 , 1] Regular m -gon at latitudes z j N = mn point t -design if m ≥ t + 1 Korevaar and Meyers (1993) N = O ( t 3 ) Both depend on t -designs for interval [ − 1 , 1] Set of n points z j ∈ [ − 1 , 1] : � 1 n � 2 p ( z j ) = p ( z ) dz ∀ p ∈ P t ([ − 1 , 1]) n − 1 j =1 ⇒ n = O ( t 2 ) points Equal weights = Survey Gautschi (2004) Tensor product constructions based on 1-D existence result (Shanghai Jiao Tong University) Efficient spherical designs April, 2015 8 / 34
Spherical designs Number of points Evidence for S 2 Hardin and Sloane (1996) Summary of known results for S 2 Conjecture N = t 2 2 (1 + o (1)) � � N = ( t + 1) 2 = dim P t ( S 2 ) Start from extremal (maximum determinant) points Sloan, W. (2004) Under-determined system of equations Use interval methods to verify a nearby solution Chen and W. (2006) Chen, Frommer, Lang (2009) An, Chen, Sloan, W. (2010) (Shanghai Jiao Tong University) Efficient spherical designs April, 2015 9 / 34
Spherical designs Number of points Number of points, dimension of space D ( d, t ) = dimension of space of polynomials of degree ≤ t on S d DGS lower bound N ∗ ( d, t ) Ratio of leading terms of D ( d, t ) /N ∗ ( d, t ) = 2 d Efficient if N < D ( d, t ) N ∗ ( d, t ) d N D ( d, t ) t 2 ( t + 1) 2 2 4 + t + O (1) t 3 24 + 3 t 2 t 3 3 + O ( t 2 ) 3 8 + O ( t ) 192 + t 3 t 4 t 4 12 + O ( t 2 ) 12 + O ( t 3 ) 4 1920 + 5 t 4 t 5 t 5 384 + O ( t 3 ) 60 + O ( t 4 ) 5 (Shanghai Jiao Tong University) Efficient spherical designs April, 2015 10 / 34
Spherical designs Number of points Spherical Harmonic Basis matrix Spherical Harmonic Basis matrix � � Y 0 , 1 e T ∈ R D ( d,t ) × N Y = � Y Rows = basis functions, Columns = points e = (1 , 1 , . . ., 1) T ∈ R N Consider case N ≤ D ( d, t ) Gram matrix G = Y T Y = Y 2 0 , 1 ee T + � Y T � Y ∈ R N × N Addition Theorem implies t � Z ( d, ℓ ) ( x i · x i ) = D ( d, t ) | S d | P ( d +1) G ii = ℓ | S d | ℓ =0 Fixed diagonal elements so trace ( G ) = ND ( d,t ) constant | S d | (Shanghai Jiao Tong University) Efficient spherical designs April, 2015 11 / 34
Characterizations Nonlinear equations Spherical designs – nonlinear equations Delsarte, Goethals and Seidel (1977) X N = { x 1 , . . . , x N } ⊂ S d is a spherical t -design if and only if N � r ℓ,k ( X N ) := Y ℓ,k ( x j ) = 0 j =1 for k = 1 , . . . , Z ( d, ℓ ) , ℓ = 1 , . . . , t . � | S d | not included in (12) Constant ( ℓ = 0 ) polynomial Y 0 , 1 = 1 / Integral of all spherical harmonics of degree ℓ ≥ 1 is zero Weyl sums: In matrix form r ( X N ) := � Y e = 0 e = (1 , . . . , 1) T ∈ R N � Y ∈ R D ( d,t ) − 1 × N , Spherical harmonic basis matrix excluding first row (Shanghai Jiao Tong University) Efficient spherical designs April, 2015 12 / 34
Characterizations Variational characterizations Polynomials with positive Legendre coefficients Polynomial ψ t ∈ P t [ − 1 , 1] with positive coefficients t � a t,ℓ P ( τ,τ ) ψ t ( z ) := ( z ) , ℓ ℓ =1 a t,ℓ > 0 for ℓ = 1 , . . . , t. P ( τ,τ ) ( z ) for z ∈ [ − 1 , 1] Jacobi polynomial, parameter τ = d − 2 ℓ � 1 2 − 1 ψ t ( z ) dz = 0 Variational form N N � � 1 A t,N,ψ ( X N ) := ψ t ( x i · x j ) N 2 i =1 j =1 (Shanghai Jiao Tong University) Efficient spherical designs April, 2015 13 / 34
Characterizations Variational characterizations Spherical designs – variational characterizations t ≥ 1 , X N = { x 1 , . . . , x N } ⊂ S d , Then t � 0 ≤ A t,N,ψ ( X N ) ≤ a t,ℓ = ψ t (1) ℓ =1 � � 1 S d A t,N,ψ ( x 1 , . . . , x N ) dω ( x 1 ) · · · dω ( x N ) = ψ t (1) A t,N,ψ := S d · · · ( | S d | ) N N X N is a spherical design if and only if A t,N,ψ ( X N ) = 0 . Weighted sum of squares, strictly positive coefficients Z ( d,ℓ ) t � � A t,N,ψ ( X N ) = | S d | a t,ℓ ( r ℓ,k ( X N )) 2 N 2 Z ( d, ℓ ) ℓ =1 k =1 A t,N,ψ ( X N ) = 0 ⇐ ⇒ X N spherical t -design Global min A t,N,ψ ( X N ) > 0 = ⇒ no spherical t -design with N points (Shanghai Jiao Tong University) Efficient spherical designs April, 2015 14 / 34
Characterizations Examples Examples Grabner and Tichy (1993) ψ t ( z ) = z t + z t − 1 − a t, 0 � 1 t odd , t a t, 0 = 1 t even . t +1 Cohn and Kumar (2007) 2 t ψ t ( z ) = (1 + z ) t − t + 1 . Sloan and W. (2009) t � ψ t ( z ) = 1 4 πP (1 , 0) ( z ) − 1 = Z ( d, ℓ ) P ℓ ( z ) t ℓ =1 P (1 , 0) Jacobi polynomial t (Shanghai Jiao Tong University) Efficient spherical designs April, 2015 15 / 34
Characterizations Evaluating A t,N,ψ ( X N ) Evaluating A t,N,ψ ( X N ) Matrix Ψ : Ψ ij = ψ t ( x i · x j ) , i, j = 1 , . . . , N Spherical t -design ⇐ ⇒ D ( d, t ) − 1 equations r := � Ye = 0 , Diagonal matrix D of weights Y T D � Ψ = | S d | � Y � � a t,ℓ D = diag Z ( d, ℓ ) , k = 1 , . . . , Z ( d, ℓ ) , ℓ = 1 , . . . , t Any symmetric positive definite D possible Minimize N 2 e T Ψe = | S d | Ye = | S d | 1 N 2 e T � Y T D � N 2 r T Dr A t,N,ψ ( X N ) = (Shanghai Jiao Tong University) Efficient spherical designs April, 2015 16 / 34
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