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Numerical Integration over unit sphere–by using spherical t -designs Numerical Integration over unit sphere–by using spherical t -designs An 1 Congpei 1,Institute of Computational Sciences, Department of Mathematics, Jinan University Spherical Design and Numerical Analysis 2015, SJTU 2015 c 4 � 23 F

Numerical Integration over unit sphere–by using spherical t -designs Outline 1 Well conditioned spherical designs 2 Numerical verification methods 3 Numerical results of verification methods 4 Numerical integration over unit sphere 5 Performance of Numerical Integrations

Numerical Integration over unit sphere–by using spherical t -designs Notations X N = { x 1 , . . . , x N } ⊂ S 2 = x, y, z ∈ R 3 | x 2 + y 2 + z 2 = 1 � � P t = { spherical polynomials of degree ≤ t } = { polynomials in x, y, z of degree ≤ t restricted to S 2 } N = Number of points t = Degree of polynomials

Numerical Integration over unit sphere–by using spherical t -designs Spherical coordinates       0 sin( θ 2 ) sin( θ i ) cos( φ i )       x 1 =  , x 2 =  , x i =  , i = 3 , . . . , N. 0 0 sin( θ i ) sin( φ i )          1 cos( θ 2 ) cos( θ i ) (1)

Numerical Integration over unit sphere–by using spherical t -designs Background on spherical t − designs Part I Background on spherical t − designs

Numerical Integration over unit sphere–by using spherical t -designs Background on spherical t − designs Definition of Spherical t − design Definition (Spherical t − design) The set X N = { x 1 , . . . , x N } ⊂ S 2 is a spherical t -design if N 1 p ( x j ) = 1 � � S 2 p ( x ) dω ( x ) ∀ p ∈ P t, (2) N 4 π j =1 where dω ( x ) denotes surface measure on S 2 . The definition of spherical t − design was given by Delsarte, Goethals, Seidel in 1977 [10].

Numerical Integration over unit sphere–by using spherical t -designs Background on spherical t − designs Real Spherical harmonics Real Spherical harmonics[14] Y ℓk : k = 1 , . . . , 2 ℓ + 1 , ℓ = 0 , 1 , . . . , t Basis P t =Span { Y ℓk : k = 1 , . . . , 2 ℓ + 1 , ℓ = 0 , 1 , . . . , t } Orthonormality with respect to L 2 inner product � ( p, q ) L 2 = S 2 p ( x ) q ( x ) dω ( x ) , 1 Normalization Y 0 , 1 = √ 4 π dim P t = ( t + 1) 2 2 ℓ +1 Y ℓ,k ( x ) Y ℓ,k ( y ) = 2 ℓ +1 � 4 π P ℓ ( x · y ) , x , y ∈ S 2 Addition Theorem k =1

Numerical Integration over unit sphere–by using spherical t -designs Background on spherical t − designs Spherical harmonic basis matrix t be the (( t + 1) 2 − 1) by For t ≥ 1 , and N ≥ dim( P t ) = ( t + 1) 2 , let Y 0 N matrix defined by Y 0 t ( X N ) := [ Y ℓ,k ( x j )] , k = 1 , . . . , 2 ℓ + 1 , ℓ = 1 , . . . , t ; j = 1 , . . . , N, (3) � � 1 4 π e T √ ∈ R ( t +1) 2 × N , Y t ( X N ) := (4) Y 0 t ( X N ) where e = [1 , . . . , 1] T ∈ R N . G t ( X N ) := Y t ( X N ) T Y t ( X N ) ∈ R N × N , H t ( X N ) := Y t ( X N ) Y t ( X N ) T ∈ R ( t +1) 2 × ( t +1) 2 .

Numerical Integration over unit sphere–by using spherical t -designs Background on spherical t − designs Nonlinear system C t ( X N ) = 0 Let N ≥ ( t + 1) 2 , define C t : ( S d ) N → R , C t ( X N ) = EG t ( X N ) e (5) where the N × N Gram matrix G t for X N ⊂ S 2 G t ( X N ) = Y t ( X N ) T Y t ( X N )   1   1 ∈ R N , E = [ 1 , − I ] ∈ R ( N − 1) × N , 1 = [1 , . . . , 1] T ∈ R N − 1   e = (6)   .  .  .     1

Numerical Integration over unit sphere–by using spherical t -designs Background on spherical t − designs Nonlinear system C t ( X N ) = 0 Theorem (ACSW2010,[1]) Let N ≥ ( t + 1) 2 . Suppose that X N = { x 1 , . . . , x N } is a fundamental system for P t . Then X N is a spherical t -design if and only if C t ( X N ) = 0 . Definition (Fundamental system) A point set X N = { x 1 , . . . , x N } ⊂ S 2 is a fundamental system for P t if the zero polynomial is the only member of P t that vanishes at each point x i, i = 1 , . . . , N. H t ( X N ) is nonsingular ⇐ ⇒ X N is a fundamental system for P t . Let N = ( t + 1) 2 , G t ( X N ) is nonsingular ⇐ ⇒ X N is a fundamental system for P t .

Numerical Integration over unit sphere–by using spherical t -designs Well conditioned spherical designs II Well conditioned spherical designs

Numerical Integration over unit sphere–by using spherical t -designs Well conditioned spherical designs Definition Chen and Womersley [8], Chen, Frommer and Lang [9] verified that a spherical t -design exists in a neighborhood of an extremal system. This leads to the idea of extremal spherical t -designs , which first appeared in [8] in N = ( t + 1) 2 . We here extend the definition to N ≥ ( t + 1) 2 . Definition (Extremal spherical designs[1]) A set X N = { x 1 , . . . , x N } ⊂ S 2 of N ≥ ( t + 1) 2 points is a extremal spherical t -design if the determinant of the matrix H t ( X N ) := Y t ( X N ) Y t ( X N ) T ∈ R ( t +1) 2 × ( t +1) 2 is maximal subject to the constraint that X N is a spherical t -design.

Numerical Integration over unit sphere–by using spherical t -designs Well conditioned spherical designs Optimization Problem on S 2 max log det ( H t ( X N )) X N ⊂ S 2 (7) subject to C t ( X N ) = 0 . ⇓ Well conditioned spherical t -design . The log of the determinant is bounded above by � N � logdet( H L ( X N )) ≤ ( t + 1) 2 log (8) . 4 π

Numerical Integration over unit sphere–by using spherical t -designs Numerical Verification method III Numerical Verification method

Numerical Integration over unit sphere–by using spherical t -designs Numerical Verification method Notations on Interval method 1 By IR n , denote [ a ] = [ a, a ] , a, a ∈ R n , a ≤ a 2 + , − , ∗ , / can be extended from R n to IR n and from R n × n to IR n × n . 3 Let mid [ a ] = ( a + a ) / 2 in componentwise. 4 diam [ a ] = a − a = 2 rad [ a ] , 5 F : D ⊆ R n → R n be a continuously differentiable function.Let [ d F ] ∈ IR n × n be an interval matrix containing F ′ ( ξ ) for all ξ ∈ [ x ] , i.e. { F ′ ( x ) : x ∈ [ x ] } ⊆ [ d F ] ([ x ]) . (9) Such [ d F ] can be obtained by an interval arithmetic evaluation of (expressions for) the Jacobian F ′ at the interval vector [ x ] .

Numerical Integration over unit sphere–by using spherical t -designs Numerical Verification method Krawczyk operator Definition (Krawczyk operator,[11]) Given a nonsingular matrix B L ∈ R n × n , ˇ z ∈ [ z ] ⊆ D and [ d F ] ∈ IR n × n , the Krawczyk operator [11] is defined by: k F (ˇ z , [ z ] , B L , [ d F ]) := ˇ z − B L F (ˇ z ) + ( I n − B L · [ d F ])([ z ] − ˇ z ) . (10) It is known that k F (ˇ z , [ z ] , B L , [ d F ]) is an interval extension of the function ψ ( z ) := z − B L F ( z ) over [ z ] , that is, z − B L F ( z ) ∈ k F (ˇ z , [ z ] , B L , [ F ]) for all z ∈ [ z ] .

Numerical Integration over unit sphere–by using spherical t -designs Numerical Verification method Verification Theorem Theorem (Krawczyk 1969 [11], Moore 1977[12]) Let F : D ⊂ R n → R n be a continuously differentiable function. Choose z ∈ [ z ] ⊆ D , an invertible matrix B L ∈ R n × n and [ d F ] ∈ IR n × n [ z ] ∈ IR n , ˇ such that F ′ ( ξ ) ∈ [ d F ] for all ξ ∈ [ z ] . Assume that k F (ˇ z , [ z ] , B L , [ d F ]) ⊆ [ z ] . Then F has a zero z ∗ in k F (ˇ z , [ z ] , B L , [ d F ]) .

Numerical Integration over unit sphere–by using spherical t -designs Numerical Verification method Deal with C t ( X N ) 1 Represent the points x i on the sphere by spherical coordinates with φ, θ . That is [ x i ] = [ sin ([ θ ]) cos ([ φ i ]) , sin ([ θ i ]) sin ([ φ i ]) , cos ([ θ i ])] T , i = 1 , . . . , N. 2 C t ( X N ) is redefined as a system of nonlinear equation ˜ F ( y ) = 0 . The components of y are y i − 1 = θ i , i = 2 , . . . , N , y N + i − 3 = ϕ i , i = 3 , . . . , N .

Numerical Integration over unit sphere–by using spherical t -designs Numerical Verification method 1 Use a QR-factorization method at each step to determine the N − 2 least important components of y , which we label collectively by y N , then write y := ( z , y N ) , and define a new function F ( z ) = ˜ F ( z , y N ) , where F : R N − 1 → R N − 1 . 2 Using the Krawczyk operator with B L = ( mid [ d F ]) − 1 we can verify the existence of a fixed point of z − B L F ( z ) , which is a solution of F ( z ) = 0 .

Numerical Integration over unit sphere–by using spherical t -designs Numerical Verification method The estimate on determinant Theorem (ACSW2010,[1]) Let U be a nonsingular upper triangular matrix. Assume that � I n − U T [ A ] U � ∞ ≤ r < 1 . (11) � N � − 2 Let β = j =1 U jj Π . Then 0 < β (1 − r ) N ≤ det ( A ) ≤ β (1 + r ) N , A ∈ [ A ] and A T = A a . (12) for a C. An, X, Chen, I. H. Sloan, R. S. Womersley , Well Conditioned Spherical Designs for integration and interpolation on the two-Sphere , SIAM J. Numer. Anal. Vo. 48, Issue 6, pp. 2135-2157.