Efficient Spherical Designs with Good Geometric Properties

Transcription

1 Efficient Spherical Designs with Good Geometric Properties Rob Womersley, 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, / 34

2 Outline 1 Spherical designs Spheres and point sets Aims Spherical polynomials Number of points 2 Characterizations Nonlinear equations Variational characterizations Examples Evaluating A t,n,ψ (X N ) Degrees of freedom for S 2 Numerical results 3 Geometric properties Mesh norm Separation Mesh ratio 4 Conclusions (Shanghai Jiao Tong University) Efficient spherical designs April, / 34

3 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 y = x i y i, x 2 = x x i=1 Distance Euclidean distance: x,y S d, x y 2 = 2(1 x y) 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, / 34

4 Spherical designs Spheres and point sets Geometric quality of point set Spherical cap centre z S d, radius α { } C(z;α) = x S d : dist(x,z) α Separation (twice packing radius): δ XN = min i j Mesh norm (covering radius): Mesh ratio: ρ XN = 2h X N δ XN 1 h XN = max x S d dist(x i,x j ) min dist(x,x j) j=1,...,n Desire: ρ XN c (Shanghai Jiao Tong University) Efficient spherical designs April, / 34

5 Spherical designs Aims Aims Numerical integration (cubature) Q N (f) := N w j f(x j ) I(f) := f(x)dω(x) S d 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 1 N N j=1 p(x j ) = 1 S d p(x)dω(x) p P t (S d ), S d N point, equal weight w j = Sd N Efficient: Low number of points N cubature rule, degree of precision t Good geometric properties: Quasi-uniform: Mesh ratio ρ X c (Shanghai Jiao Tong University) Efficient spherical designs April, / 34

6 Spherical designs Spherical polynomials Spherical Polynomials Space P t P t ( S d ) of spherical polynomials of degree at most t Dimension of space of homogeneous harmonic polynomials of degree l Z(d,0) = 1; Z(d,l) = (2l+d 1)Γ(l+d 1), Γ(d)Γ(l+1) Orthonormal basis Y l,k, l = 0,1,2,..., k = 1,...,Z(d,l) Dimension P t ( S d ) is D(d,t) = Z(d+1,t) t d Addition Theorem Z(d,l) k=1 Y l,k (x)y l,k (y) = Z(d,l) S d Normalized Gegenbauer polynomial P (d+1) l Jacobi polynomial P (α,β) l (z) for z [ 1,1] P (d+1) l (x y), 2,d 2 2 ) (z) P (d 2 2,d 2 2 ) l (1) P(d 2 l (z) = (Shanghai Jiao Tong University) Efficient spherical designs April, / 34

7 Spherical designs Number of points Spherical t-designs Number of points N Delsarte, Goethals and Seidel (1977)N point t-design on S d 2 ( d+m) N N d if t = 2m+1, (d,t) := ( d+m ) ( d + d+m 1 ) d if t = 2m. Positive weight cubature, degree of precision t = N dimp t/2 (S d ) 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 td (Shanghai Jiao Tong University) Efficient spherical designs April, / 34

8 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]: 2 n n p(z j ) = j=1 1 1 Equal weights = n = O(t 2 ) points Survey Gautschi (2004) p(z)dz p P t ([ 1,1]) Tensor product constructions based on 1-D existence result (Shanghai Jiao Tong University) Efficient spherical designs April, / 34

9 Spherical designs Number of points Evidence for S 2 Hardin and Sloane (1996) Summary of known results for S 2 Conjecture N = t2 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, / 34

10 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) d N (d,t) N D(d,t) 2 t 2 4 +t+o(1) (t+1)2 3 t t2 8 +O(t) t 3 3 +O(t2 ) 4 t t3 12 +O(t2 ) t O(t3 ) 5 t t O(t3 ) t O(t4 ) (Shanghai Jiao Tong University) Efficient spherical designs April, / 34

11 Spherical designs Number of points Spherical Harmonic Basis matrix Spherical Harmonic Basis matrix [ Y0,1 e T Y = Ŷ ] R D(d,t) N Rows = basis functions, Columns = points e = (1,1,...,1) T R N Consider case N D(d,t) Gram matrix Addition Theorem implies t G ii = G = Y T Y = Y 2 0,1 eet +ŶT Ŷ R N N l=0 Z(d, l) S d P (d+1) l (x i x i ) = D(d,t) S d Fixed diagonal elements so trace(g) = ND(d,t) S d constant (Shanghai Jiao Tong University) Efficient spherical designs April, / 34

12 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 r l,k (X N ) := N Y l,k (x j ) = 0 j=1 for k = 1,...,Z(d,l), l = 1,...,t. Constant (l = 0) polynomial Y 0,1 = 1/ S d not included in (12) Integral of all spherical harmonics of degree l 1 is zero Weyl sums: In matrix form r(x N ) := Ŷe = 0 e = (1,...,1) T R N Ŷ R D(d,t) 1 N, Spherical harmonic basis matrix excluding first row (Shanghai Jiao Tong University) Efficient spherical designs April, / 34

13 Characterizations Variational characterizations Polynomials with positive Legendre coefficients Polynomial ψ t P t [ 1,1] with positive coefficients ψ t (z) := t l=1 a t,l P (τ,τ) l (z), a t,l > 0 for l = 1,...,t. P (τ,τ) l (z) for z [ 1,1] Jacobi polynomial, parameter τ = d ψ t(z)dz = 0 Variational form A t,n,ψ (X N ) := 1 N 2 N i=1 j=1 N ψ t (x i x j ) (Shanghai Jiao Tong University) Efficient spherical designs April, / 34

14 Characterizations Variational characterizations Spherical designs variational characterizations t 1, X N = {x 1,...,x N } S d, Then 0 A t,n,ψ (X N ) t a t,l = ψ t (1) 1 A t,n,ψ := ( S d ) N A t,n,ψ (x 1,...,x N )dω(x 1 ) dω(x N ) = ψ t(1) S d S d N X N is a spherical design if and only if l=1 A t,n,ψ (X N ) = 0. Weighted sum of squares, strictly positive coefficients A t,n,ψ (X N ) = Sd N 2 t l=1 Z(d,l) a t,l (r l,k (X N )) 2 Z(d, l) 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, / 34 k=1

15 Characterizations Examples Examples Grabner and Tichy (1993) Cohn and Kumar (2007) Sloan and W. (2009) P (1,0) t ψ t (z) = z t +z t 1 a t,0 { 1 t t odd, a t,0 = 1 t+1 t even. ψ t (z) = (1+z) t 2t t+1. ψ t (z) = 1 4π P(1,0) t (z) 1 = Jacobi polynomial t Z(d,l)P l (z) (Shanghai Jiao Tong University) Efficient spherical designs April, / 34 l=1

16 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 Diagonal matrix D of weights r := Ŷe = 0, Ψ = S d ŶT DŶ ( ) at,l D = diag,k = 1,...,Z(d,l),l = 1,...,t Z(d,l) Any symmetric positive definite D possible Minimize A t,n,ψ (X N ) = 1 N 2eT Ψe = Sd N 2 et Ŷ T DŶe = Sd N 2 rt Dr (Shanghai Jiao Tong University) Efficient spherical designs April, / 34

17 Characterizations Evaluating A t,n,ψ (X N ) Evaluating A t,n,ψ (X N ) using Ψ N by N matrix Ψ ij = ψ t (x i x j ) Constant diagonal elements ψ t (1) = t l=1 a t,l Matrix Ψ for a t,l = Z(d,l) D = I Advantages: simple, (trivially) parallel Issue: cancelation errors in summing off diagonal elements (Shanghai Jiao Tong University) Efficient spherical designs April, / 34

18 Characterizations Degrees of freedom for S 2 Degrees of freedom for S 2 Spherical parametrization, normalization = n = 2N 3 variables m = dim(p t ) 1 = (t+1) 2 1 equations Threshold n m = N N(2,t) := (t+1) 2 )/2 +1 N less than twice the DGS lower bound N 2N (2,t) N(2,t) = t, Sum of squares for t = 19, varying N, N(2,19) = Residual SSQ r T r for t = Number of points N (Shanghai Jiao Tong University) Efficient spherical designs April, / 34

19 Characterizations Degrees of freedom for S 2 Symmetric designs Exploit symmetry to reduce conditions: Sobolev (1962) Symmetric: N even, x X N x X N Equal weights, l odd = Y l,k integrated exactly Constraints from even degrees t, t odd m = (t 1)/2 k=1 2(2k)+1 = (t 1)(t+2) 2 N = 2K points = 2K 3 = N 3 degrees of freedom Degrees of freedom number of equations = t 2 +t+4 N N(2,t) := 2 4 Slightly less than N(2,t) 2N (2,t) N(2,t) = 3 2 t { 3 2 if mod (t,4) = 1, 1 2 if mod (t,4) = 3. (Shanghai Jiao Tong University) Efficient spherical designs April, / 34

20 Characterizations Degrees of freedom for S 2 Degrees of freedom S d S d R d+1 : Spherical parametrization = d variables N points x j,j = 1,...,N = Nd variables Orthogonal invariance = Qx j so d(d+1)/2 zero elements Number of variables n = Nd d(d+1)/2 Number of equations for t-design m = t Z(d,l) = Z(d+1,t) 1 l=1 Number of variables number of conditions = N(d,t) Symmetric point set (both x j, x j in set) automatically integrates odd degree polynomial = N(d,t) (Shanghai Jiao Tong University) Efficient spherical designs April, / 34

21 Characterizations Degrees of freedom for S 2 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) N(d,t) ensures n m N(d,t) ensures symmetric point set has n m Ratio of leading terms of D(d,t)/ N(d,t) = d d N (d,t) N(d,t) N(d,t) D(d,t) 2 t 2 4 +t+o(1) t t 2 +O(1) t 2 2 +t+o(1) (t+1)2 3 t t2 8 +O(t) t t2 3 +O(t) t t2 2 +O(t) t 3 3 +O(t2 ) 4 t t3 12 +O(t2 ) t t3 8 +O(t2 ) t t3 6 +O(t2 ) t O(t3 ) 5 t t O(t3 ) t t4 30 +)(t3 ) t t4 24 +)(t3 ) t O(t4 ) (Shanghai Jiao Tong University) Efficient spherical designs April, / 34

22 Characterizations Degrees of freedom for S 2 Least squares Spherical parametrization of variables x R n Residuals r : R n R m, m = t l=1 Z(d,l) Number of points chosen so n = m or n = m+1. Sum of squares objective f(x) = r(x) T r(x) = m i=1 [r i(x)] 2 Jacobian A(x) R m n Gradient f(x) = 2A(x)r(x) Hessian 2 f(x) = 2A(x) T A(x)+2 m i=1 r i(x) 2 r i (x) r(x ) = 0 and A(x ) rank n = strict (isloated) global minimum n > m have some freedom Algorithm: Levenberg-Marquardt (A T A+λI)d = A T r Issues Singular Jacobians A(x) Stuck with f(x) = 0 but f(x) > 0, perhaps small Different solutions: depends on starting point, algorithm parameters,... (Shanghai Jiao Tong University) Efficient spherical designs April, / 34

23 Characterizations Numerical results Spherical designs - numerical results Use N = N(2,t), = t odd, n = m, t even, n = m+1 Rounding error limits achievable accuracy in A t,n Both A t,n,ψ (X N ), r T r order of rounding error = what confidence? t = 180 = N(2,t) = 16382, m = 32760, n = Residual ssq for spherical t-designs Degree t (Shanghai Jiao Tong University) Efficient spherical designs April, / 34

24 Characterizations Numerical results Spherical designs - rate of convergence Rate of convergence: t = 45, N = 1059, m = 2115, n = 2115 SSQ for t = 45, N = 1059, m = 2115, n = Sum of squares Iterations (Shanghai Jiao Tong University) Efficient spherical designs April, / 34

25 Characterizations Numerical results Spherical designs - Jacobian singular values Jacobian singular values: t = 45, N = 1059, m = 2115, n = Jacobian singular values: t = 45, N = 1059, m = 2115, n = (Shanghai Jiao Tong University) Efficient spherical designs April, / 34

26 Characterizations Numerical results Symmetric spherical designs - numerical results For t odd, use N = N(2,t) mod (t,4) = 3 = n = m, mod (t,4) = 1 = n = m+1 t = 277 = N(2,t) = 38506, m = 38502, n = Symmetric spherical t-designs, SSQ Degree t (Shanghai Jiao Tong University) Efficient spherical designs April, / 34

27 Geometric properties Mesh norm Mesh norm Mesh norm (covering radius) h XN = max x S 2 min dist(x,x j) c cov j=1,...,n N Yudin (1995) Mesh norm h given by largest zero z t = cos(h) of P (1,0) (z) Reimer (2003) extended to any positive weight cubature rule with degree of precision t Spherical t desgins, Mesh norm Mesh norm h(x N ) N Number of points N (Shanghai Jiao Tong University) Efficient spherical designs April, / 34

28 Geometric properties Separation Separation Separation (twice packing radius) δ XN = min i j dist(x i,x j ) c pack N Union of two spherical t-designs is a spherical t-design X N QX N is 2N point spherical t-design with arbitrary separation N < 2N, N < 2N so cannot occur If N sufficiently small get separation? Spherical t desgins, Separation Separation δ(x N ) N Number of points N (Shanghai Jiao Tong University) Efficient spherical designs April, / 34

29 Geometric properties Mesh ratio Mesh ratio Mesh ratio ρ XN = 2h X N Covering radius = δ XN Packing radius 1 ρ XN bounded = X N quasi-uniform Spherical t-designs with N points 2 Spherical t-designs, Mesh ratio Degree t (Shanghai Jiao Tong University) Efficient spherical designs April, / 34

30 Geometric properties Mesh ratio Mesh ratio - Symmetric t-designs Mesh ratio for symmetric t-designs with N points 2 Symmetric spherical t-designs, Mesh ratio Degree t Converge to different spherical designs from different starting points If n = m, r = 0 and σ n (A ) > 0 = strict global minimizer If n = m+1 use freedom to reduce mesh ratio (Shanghai Jiao Tong University) Efficient spherical designs April, / 34

31 Geometric properties Mesh ratio Reducing mesh ratio - one degree of freedom Symmetric 53-design: n v = 1431, n c = 1430, tangent to r = 0, SSQ Symmetric 53-design: n v = 1431, n c = 1430, tangent to r = 0, RHO (Shanghai Jiao Tong University) Efficient spherical designs April, / 34

32 Geometric properties Mesh ratio Reducing mesh ratio - isolated zero Symmetric 55-design, n v = 1539, n c = 1539, σ 1538 = 1.46e-01, SSQ Symmetric 55-design, n v = 1539, n c = 1539, σ 1538 = 1.46e-01, RHO (Shanghai Jiao Tong University) Efficient spherical designs April, / 34

33 Geometric properties Mesh ratio Starting points Close to spherical design (low sums of squares) Good mesh ratio Generalized spiral points Equal area points 9000 Symmetric equal areas points: SSQ 2 Symmetric equal areas points: Mesh ratio Degree t (Shanghai Jiao Tong University) Efficient spherical designs April, / 34

34 Conclusions Conclusions Efficient sets of (numerical) t-designs for S 2 Equal weight cubature rule, degree of precision t with N = (t 2 +2t)/2+O(1) points for t = 1,...,180 Symmetric equal weight cubature rule, degree of precision t with N = (t 2 +t)/2+o(1) points for t = 1,...,277 Good geometric properties: mesh norm, separation, mesh ratio < 1.8 Larger N: Use extra degrees of freedom to satisfy other criteria Issues Rounding errors in evaluating criteria, speed of extended precision Convergence difficulties with close to singular Jacobians No proof of nearby exact spherical designs when N < (t+1) 2 No proof of existence for all t There exist t-designs with N < N(d,t); special symmetries Calculation by optimization for each t, N Finding points sets with better mesh ratio ad-hoc Point sets X N not nested Higher dimensions d (Shanghai Jiao Tong University) Efficient spherical designs April, / 34