Spherical t -Design for Numerical Approximations on the Sphere, I - PowerPoint PPT Presentation

Spherical t ǫ -Design for Numerical Approximations on the Sphere, I Xiaojun Chen Department of Applied Mathematics The Hong Kong Polytechnic University April 2015, Shanghai Jiaotong University X. Chen (PolyU) Spherical tǫ -Designs April 2015, SJTU 1 / 47

Outline Spherical t -designs 1 Computational existence proofs 2 Spherical t ǫ -designs 3 Interval analysis of spherical t ǫ -designs 4 Worst-case errors of numerical integration using spherical t ǫ -designs 5 Polynomial approximation on the sphere using spherical t ǫ -designs 6 l 2 − l 1 regularized weighted polynomial approximation Numerical experiments Conclusions 7 X. Chen (PolyU) Spherical tǫ -Designs April 2015, SJTU 2 / 47

On the unit sphere S d = { x ∈ R d +1 : � x � 2 = 1 } P t : the linear space of restrictions of polynomials of degree ≤ t in d + 1 variables to S d For d = 2 , Area | S 2 | = 4 π , dim P t = ( t + 1) 2 X. Chen (PolyU) Spherical tǫ -Designs April 2015, SJTU 3 / 47

Spherical t -designs Spherical t -designs Definition 1 (Delsarte-Goethals-Seidel 1977) A spherical t -design is a set of N points X N = { x 1 , . . . , x N } ⊂ S d such that � N � 1 1 p ( x j ) = S d p ( x )d ω d ( x ) (1) N | S d | j =1 for every polynomial p ∈ P t , where d ω d ( x ) denotes the surface measure and | S d | is the area of S d . The average value of p ∈ P t on the whole sphere equals the average value of p on the set. The equally weighted cubature rule is exact for all p ∈ P t . No answer to what is the number of points needed to construct a spherical t -design for any t ≥ 1 ? Can we guarantee the existence of spherical t -designs with ( t + 1) 2 points for d = 2 ? X. Chen (PolyU) Spherical tǫ -Designs April 2015, SJTU 4 / 47

Spherical t -designs Existence of spherical t -designs In 1977, Delsarte, Goethals and Seidel gave the lower bound on N that  � d + t/ 2 � � d + t/ 2 − 1 �   + for t even ,  d d � d + ⌊ t/ 2 ⌋ � N ≥   2 for t odd .  d In 1979, Bannai and Damerell showed that tight spherical designs with d ≥ 2 do not exist except for t = 1 , 2 , 3 , 4 , 5 , 7 , 11 . Moreover, if t = 11 , then d = 23 and hence N = 196560 . In 1984 Seymour and Zaslavsky showed the existence of spherical t -designs for any t . In 1993, Korevaar and Meyyers proved the existence of spherical t -designs of size O ( t d ) on S d . In 1996, Hardin, Sloane conjectured that there exist spherical t -designs on S 2 with 2 t 2 + o ( t 2 ) . N = 1 In 2009, Womersley numerically obtained spherical t -designs for t ≤ 263 via a new variational characterization of spherical designs. In 2013, Bondarenko, Radchenko, Viazovska proved the existence of spherical t -designs with N ≥ C d t d for d ≥ 2 . X. Chen (PolyU) Spherical tǫ -Designs April 2015, SJTU 5 / 47

Computational existence proofs Our approach ( d = 2 ) Reformulate the problem of finding a spherical t -design with ( t + 1) 2 points as a system of nonlinear equations. Prove the existence of solutions of the nonlinear equations by Krawczyk’s method and interval arithmetic. X. Chen and R. Womersley: Existence of solutions to systems of underdetermined equations and spherical desings, SIAM J. Numer. Anal. (2006). 2326-2341. Provide narrow interval enclosures containing spherical t -designs with mathematical certainty for t up to 100 . X. Chen, A. Frommer and B. Lang, Computational existence proofs for spherical t -designs, Numer. Math., 117(2011), 289-205. Applications to integration and interpolation C. An, X. Chen, I.H. Sloan and R.S. Womersley, Well-conditioned spherical designs for integration and interpolation on the two-sphere, SIAM J. Numer. Anal., 48(2010), 2135-2157. C. An, X. Chen, I.H. Sloan and R.S. Womersley, Regularized least squares approximations on the sphere using spherical designs, SIAM J. Numer. Anal., 50(2012), 1513-1534. Condition number of Gram matrix X. Chen, R. Womersley and J. Ye, Minimizing the Condition Number of a Gram Matrix, SIAM J. Optim., 21(2011), 127-148. Challenging highly nonlinear and large-scale systems. X. Chen (PolyU) Spherical tǫ -Designs April 2015, SJTU 6 / 47

Computational existence proofs Gram matrix P t can be spanned by an orthonormal set of real spherical harmonics with degree ℓ and order k , { Y ℓ,k | k = 1 , . . . , 2 ℓ + 1 , ℓ = 0 , 1 , . . . , t } . Let X N = { x 1 , . . . , x N } ⊂ S 2 be a set of N -points on the sphere. The Gram matrix is defined as G t ( X N ) = Y ( X N ) T Y ( X N ) , where Y ( X N ) ∈ R ( t +1) 2 × N and the j -th column of Y ( X N ) is given by Y ℓ,k ( x j ) , k = 1 , . . . , 2 ℓ + 1 , ℓ = 0 , 1 , . . . , t. The Gram matrix G t is a function of a set of N -points X N . X. Chen (PolyU) Spherical tǫ -Designs April 2015, SJTU 7 / 47

Computational existence proofs Reformulation I, Parametrization N = ( t + 1) 2 , X N = { x 1 , . . . , x N } ⊂ S 2 . m = 2 N − 3 , Represent x i ∈ X N ⊂ S 2 using polar coordinates with angles θ i , φ i .       0 sin( θ 2 ) sin( θ i ) cos( φ i )  ,  ,  ,    x 1 = 0 x 2 = 0 x i = sin( θ i ) sin( φ i ) i = 3 , . . . , N 1 cos( θ 2 ) cos( θ i ) Fix x 1 on the north pole and x 2 on the zero meridian, θ 1 = φ 1 = φ 2 = 0 . φ ] T ∈ R m with x θ = [ θ 2 , . . . , θ N ] T , x φ = [ φ 3 , . . . , φ N ] T . Let x = [ x T θ , x T X. Chen (PolyU) Spherical tǫ -Designs April 2015, SJTU 8 / 47

Computational existence proofs Reformulation II, Gram matrix Define the Legendre polynomials by the recurrence p 0 ( u ) = 1 p 1 ( u ) = u ℓp ℓ ( u ) = (2 ℓ − 1) up ℓ − 1 ( u ) − ( ℓ − 1) p ℓ − 2 ( u ) u ∈ [ − 1 , 1] . for ℓ = 2 , . . . , t, Define the Jacobi polynomials � t J t ( u ) = (2 ℓ + 1) p ℓ ( u ) . ℓ =0 Define the Gram matrix G t ( x ) ∈ R N × N G i,j ( X N ( x )) = J t ( x i ( x ) T x j ( x )) , φ ] T ∈ R m , where X N ( x ) = { x 1 ( x ) , . . . , x N ( x ) } , and x = [ x T θ , x T x θ = [ θ 2 , . . . , θ N ] T , x φ = [ φ 3 , . . . , φ N ] T . X. Chen (PolyU) Spherical tǫ -Designs April 2015, SJTU 9 / 47

Computational existence proofs Reformulation III, Nonlinear equations Theorem 2 (Chen-Womersley 2006) Let G t ( X N ( x ∗ )) be nonsingular. Then X N ( x ∗ ) is a spherical t -design with ( t + 1) 2 points if and only if x ∗ is a solution of c ( X N ( x )) = EG t ( X N ( x )) e = 0 ( ⇔ G t ( X N ( x )) e = const e ) where   − 1 1 0 . . . 0   1 .  ...  .   − 1 1 0 .  .  ∈ R ( N − 1) × N ,  ∈ R N .   E = e = .    . . . ... ... . .   . . 0 1 − 1 1 0 . . . 0 c : R m → R n , N = ( t + 1) 2 n = N − 1 , m = 2 N − 3 , X. Chen (PolyU) Spherical tǫ -Designs April 2015, SJTU 10 / 47

Computational existence proofs Computational method 1. Find an approximate zero � x of c ( x ) by using the Gauss-Newton method with an extremal system x 0 as an initial guess. x 0 = argmax det( G t ( X N ( x ))) . 2. Use � x to construct a narrow interval z and show 2.1 z contains a solution x ∗ of c ( X N ( x )) = 0 2.2 G t ( X N ( x )) is non-singular for all x ∈ z . The interval z contains a fundamental spherical t -design. X N = { x 1 , . . . , x N } is a fundamental system if the zero polynomial is the only member of P t which vanishes at each x i , i = 1 , . . . , N . Equivalent to G t ( X N ( x )) is nonsingular. X. Chen (PolyU) Spherical tǫ -Designs April 2015, SJTU 11 / 47

Computational existence proofs Sufficient conditions c ( X N ( x )) = 0 , det( G ( X N ( x ))) > 0 ⇒ X N ( x ) Spherical t-Design c ( X 4 ( x )) = 0 c ( X 4 ( x )) = 0 c ( X 4 ( x )) = 0 1 det( G 1 ( X 4 ( x ))) = det( G 1 ( X 4 ( x ))) = 0 det( G 1 ( X 4 ( x ))) = 0 π 4 X 4 ( x ) S 2-D X 4 ( x ) S 1-D X 4 ( x ) not S t-D X. Chen (PolyU) Spherical tǫ -Designs April 2015, SJTU 12 / 47

Computational existence proofs Interval arithmetic Interval vector z in IR m , z i = [ x i , ¯ x i ] Interval matrix A in IR m × m a i,j = [ a i,j , ¯ a i,j ] Interval extension F of a function f is an interval operator such that { f ( x ) : x ∈ z } ⊆ F ( z ) Outward rounding is used to guarantee that the computed interval always contains the true result. INTLAB toolbox for MATLAB; C ++ class library C-XSC X. Chen (PolyU) Spherical tǫ -Designs April 2015, SJTU 13 / 47

Computational existence proofs Krawczyk Operator Let f : R n → R n be continuously differentiable. Let z ∈ IR n and A ∈ IR n × n such that x ∈ z and f ′ ( x ) ∈ A for all x ∈ z . Let V ∈ R n × n be nonsingular. � The Krawczyk operator x − V f ( � x ) + ( I − V A )( z − � K ( � x, V, z , A ) = � x ) Existence Theorem f has a zero x ∗ in K ( � K ( � x, V, z , A ) ⊆ z ⇒ x, V, z , A ) K ( � x, V, z , A ) ∩ z = ∅ ⇒ f has no zero in z f has a zero x ∗ in z x ∗ ∈ K ( � ⇒ x, V, z , A ) ∩ z X. Chen (PolyU) Spherical tǫ -Designs April 2015, SJTU 14 / 47

Computational existence proofs Verify the non-singularity of Gram matrix Let G ∈ IR n × n be an interval matrix, and let H ∈ R n × n be a non-singular matrix. � I − H G � < 1 ⇒ all matrices G ∈ G are non-singular. In computation, we choose H = (mid G ) − 1 . where ”mid” is the midpoint (mid G ) i,j = 1 2( ¯ G i,j + G i,j ) X. Chen (PolyU) Spherical tǫ -Designs April 2015, SJTU 15 / 47