Applications of Spherical Designs to Numerical Analysis Congpei AN - PowerPoint PPT Presentation

Mini-Workshop on Spherical Designs and Related Topics Applications of Spherical Designs to Numerical Analysis Congpei AN Department of Mathematics Jinan University Guangzhou Shang Hai Jiao Tong University 2012.11.18-21

Mini-Workshop on Spherical Designs and Related Topics Outline 4 Regularized least squares approximation on S 2 by 1 Notations and definitions using spherical designs 2 Numerical construction for well 1 Solutions conditioned spherical designs 2 Numerical experiments Optimization problem 1 With exact data 2 Interval method With noisy data 3 Open Problem 3 Interpolation, Hyperinterpolation and Filtered hyperinterpolation on S 2

Mini-Workshop on Spherical Designs and Related Topics Points on the unit sphere

Mini-Workshop on Spherical Designs and Related Topics Notations and definitions Part I Notations and definitions

Mini-Workshop on Spherical Designs and Related Topics Notations and definitions Notations and definitions S 2 = � � [ x, y, z ] T ∈ R 3 | x 2 + y 2 + z 2 = 1 Basis for P L : orthonormal spherical harmonics { Y ℓ,k : ℓ = 0 , 1 , . . . , L, k = X N = { x 1 , . . . , x N } ⊂ S 2 1 , . . . , 2 ℓ + 1 } . L Degree of polynomials Coefficient α = ( α ℓ,k ) ∈ R ( L +1) 2 . P L = Laplace-Beltrami operator ∆ ∗ : { spherical polynomials of degree ≤ ∆ ∗ Y ℓ,k ( x ) = − ℓ ( ℓ + 1) Y ℓ,k ( x ) L } with Y L ∈ R ( L +1) 2 × N d L := dim( P L ) = ( L + 1) 2 N Number of points [ Y ℓ,k ( x j )] , ℓ = 0 , 1 , . . . , L, || f || C ( S 2 ) := sup x ∈ S 2 | f ( x ) | . k = 1 , . . . , 2 ℓ + 1 , j = 1 , . . . , N. Lebesgue constant: ||U f || C ( S 2) ( L + 1) 2 by ( L + 1) 2 matrix ||U|| C ( S 2 ) := sup f � = 0 || f || C ( S 2) H L ( X N ) := Y L Y T L . (1) E L ( f ) := min p ∈ P L || p − f || C ( S 2 )

Mini-Workshop on Spherical Designs and Related Topics Notations and definitions 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 )

Mini-Workshop on Spherical Designs and Related Topics Notations and definitions Franke function 0 . 75 exp( − (9 x − 2) 2 / 4 − (9 y − 2) 2 / 4 − (9 z − 2) 2 / 4) f 1 ( x, y, z ) = +0 . 75 exp( − (9 x + 1) 2 / 49 − (9 y + 1) / 10 − (9 z + 1) / 10) +0 . 5 exp( − (9 x − 7) 2 / 4 − (9 y − 3) 2 / 4 − (9 z − 5) 2 / 4) − 0 . 2 exp( − (9 x − 4) 2 − (9 y − 7) 2 − (9 z − 5) 2 ) , ( x, y, z ) ∈ S 2 , which is in C ∞ ( S 2 ) .

Mini-Workshop on Spherical Designs and Related Topics Notations and definitions 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 and Seidel,1977[9]. An positive equal-weight quadrature rule.

Mini-Workshop on Spherical Designs and Related Topics Notations and definitions Theoretical results For t ≥ 1 , the spherical t -designs exist for sufficiently large N (Seymour and Zaslavsky,1984 [15]), and the existence of a spherical t -design for all N ≥ ct 2 for some unknown c > 0 has been claimed in [6](Bondarenko, Radchenko and Viazovska,2011). Lower bound on cardinality of spherical t -design The cardinality of a spherical t − design X N is bounded from below (Delsarte, Goethals, Seidel 1977 [9]),  ( t +1)( t +3) if t is odd 4      N ≥    ( t +2) 2  if t is even  4 The spherical t − design is called tight if this bound is attained.

Mini-Workshop on Spherical Designs and Related Topics Notations and definitions Theoretical results Unfortunately, tight t -designs rarely exist. By the result of Bannai and Damerell in 1979 [4], tight t-design in S d with d ≥ 2 exists, then necessarily either t ≤ 5 , or t = 7 , 11 .

Mini-Workshop on Spherical Designs and Related Topics Numerical Construction for Well Conditioned Spherical Designs Part II Numerical Construction for Well Conditioned Spherical Designs

Mini-Workshop on Spherical Designs and Related Topics Numerical Construction for Well Conditioned Spherical Designs Nonlinear system C t ( X N ) = 0 Let N ≥ ( t + 1) 2 , define C t : ( S 2 ) N → R N − 1 , C t ( X N ) = EG t ( X N ) e , (3) where E = [ 1 , − I N − 1 ] ∈ R ( N − 1) × N , 1 = [1 , . . . , 1] T ∈ R N − 1 ,   1   1   G t ( X N ) := Y t ( X N ) T Y t ( X N ) ∈ R N × N , e = ∈ R N .   .   .   .   1 Theorem (An-Chen-Sloan-Womersley,[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 .

Mini-Workshop on Spherical Designs and Related Topics Numerical Construction for Well Conditioned Spherical Designs Definition of extremal spherical designs 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. Chen and Womersley [7], Chen, Frommer and Lang [8] verified that a spherical t -design exists in a neighborhood of an extremal system for t = 100 . This leads to the idea of extremal spherical t -designs , which first appeared in [7] 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 an 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.

Mini-Workshop on Spherical Designs and Related Topics Numerical Construction for Well Conditioned Spherical Designs Optimization Problem max log det ( H t ( X N )) X N ⊂ S 2 (4) 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 . (5) 4 π

Mini-Workshop on Spherical Designs and Related Topics Numerical Construction for Well Conditioned Spherical Designs Numerical strategy The following strategy is adopted. Choose a nonnegative integer t , N ≥ ( t + 1) 2 , and a fundamental system X 0 N as a starting point set. 1 Use the Gauss-Newton method to find an approximate solution ˜ X N of C t ( X N ) = 0 starting from X 0 N . 2 Use a nonlinear programming method to find ˆ X N ≈ arg max { log det( H t ( X N )) | C t ( X N ) = 0 } starting from ˜ X N . N = ( t + 1) 2 ,we choose the extremal system as the starting point set. Interval method below to find an interval that contains ˆ X N and a true spherical t -design.

Mini-Workshop on Spherical Designs and Related Topics Numerical Construction for Well Conditioned Spherical Designs Numerical verification results For N = ( t + 1) 2 , det( G t ( X N )) = det( H t ( X N )) . Using an Extremal system as a initial point set. Based on the MATLAB toolbox INTLAB.

Mini-Workshop on Spherical Designs and Related Topics Numerical Construction for Well Conditioned Spherical Designs Figure: The max diam of [ z ] max diam( [ z ] ) represents the maximum diameter of all computed enclosures for the parametrization of the

Mini-Workshop on Spherical Designs and Related Topics Numerical Construction for Well Conditioned Spherical Designs Figure: Middle point values of [log det( G t ( X N ))] and diameters of [log det( G t ( X N ))]

Mini-Workshop on Spherical Designs and Related Topics Numerical Construction for Well Conditioned Spherical Designs Open problems 1 There is no proof that spherical t -designs with N = ( t + 1) 2 exist for all t . 2 Finding spherical t -designs by other methods: SDP? 3 Min F ( H L ( X N )) subject to X N is a spherical t − design.

Mini-Workshop on Spherical Designs and Related Topics Interpolation, Hyperinterpolation and Filtered Hyperinterpolation on S 2 Part III Interpolation, Hyperinterpolation and Filtered Hyperinterpolation on S 2

Mini-Workshop on Spherical Designs and Related Topics Interpolation, Hyperinterpolation and Filtered Hyperinterpolation on S 2 Interpolation on S 2 Let N = d L = ( L + 1) 2 . For a given degree L , given a fundamental system X d L = { x 1 , . . . , x d L } for P L and an arbitrary f ∈ C ( S 2 ) , we denote by Λ L f the unique polynomial in the space P L that interpolates f at each point of the fundamental system, that is Λ L f ∈ P L , Λ L f ( x j ) = f ( x j ) , j = 1 , . . . , d L . (6)

Mini-Workshop on Spherical Designs and Related Topics Interpolation, Hyperinterpolation and Filtered Hyperinterpolation on S 2 Lagrange expression The Lagrange basis polynomials { ℓ 1 , . . . , ℓ d L } are defined, as usual, by ℓ j ∈ P L , ℓ j ( x i ) = δ ji , i, j = 1 , . . . , d L . � � �� ℓ j ( x ) = det x 1 , . . . , x j − 1 , x , x j +1 , . . . , x d L G L �� . (7) � � det G L x 1 , . . . , x j − 1 , x j , x j +1 , . . . , x d L S 2 � � For given f ∈ C d L � Λ L f = f ( x j ) ℓ j . (8) j =1