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Spherical tǫ-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

1

Spherical t-designs

2

Computational existence proofs

3

Spherical tǫ-designs

4

Interval analysis of spherical tǫ-designs

5

Worst-case errors of numerical integration using spherical tǫ-designs

6

Polynomial approximation on the sphere using spherical tǫ-designs l2 − l1 regularized weighted polynomial approximation Numerical experiments

7

Conclusions

• X. Chen (PolyU)

Spherical tǫ-Designs April 2015, SJTU 2 / 47

On the unit sphere

Sd = {x ∈ Rd+1 : x2 = 1 } Pt: the linear space of restrictions of polynomials of degree ≤ t in d + 1 variables to Sd For d = 2, Area |S2| = 4π, dim Pt = (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 XN = {x1, . . . , xN} ⊂ Sd such that 1 N

N

• j=1

p(xj) = 1 |Sd|

• Sd p(x)dωd(x)

(1) for every polynomial p ∈ Pt, where dωd(x) denotes the surface measure and |Sd| is the area of Sd. The average value of p ∈ Pt on the whole sphere equals the average value of p on the set. The equally weighted cubature rule is exact for all p ∈ Pt. 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 N ≥        d + t/2 d

• +

d + t/2 − 1 d

• for t even,

2 d + ⌊t/2⌋ d

• for t odd.

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(td) on Sd. In 1996, Hardin, Sloane conjectured that there exist spherical t-designs on S2 with N = 1

2t2 + o(t2).

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 ≥ Cdtd 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

• f 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

Pt can be spanned by an orthonormal set of real spherical harmonics with degree ℓ and

• rder k,

{ Yℓ,k | k = 1, . . . , 2ℓ + 1, ℓ = 0, 1, . . . , t}. Let XN = {x1, . . . , xN} ⊂ S2 be a set of N-points on the sphere. The Gram matrix is defined as Gt(XN) = Y(XN)T Y(XN), where Y(XN) ∈ R(t+1)2×N and the j-th column of Y(XN) is given by Yℓ,k(xj), k = 1, . . . , 2ℓ + 1, ℓ = 0, 1, . . . , t. The Gram matrix Gt is a function of a set of N-points XN.

• X. Chen (PolyU)

Spherical tǫ-Designs April 2015, SJTU 7 / 47

Computational existence proofs

Reformulation I, Parametrization

N = (t + 1)2, m = 2N − 3, XN = {x1, . . . , xN} ⊂ S2. Represent xi ∈ XN ⊂ S2 using polar coordinates with angles θi, φi. x1 =   1   , x2 =   sin(θ2) cos(θ2)   , xi =   sin(θi) cos(φi) sin(θi) sin(φi) cos(θi)   , i = 3, . . . , N Fix x1 on the north pole and x2 on the zero meridian, θ1 = φ1 = φ2 = 0. Let x = [xT

θ , xT φ ]T ∈ Rm with xθ = [θ2, . . . , θN]T ,

xφ = [φ3, . . . , φN]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 p0(u) = 1 p1(u) = u ℓpℓ(u) = (2ℓ − 1)upℓ−1(u) − (ℓ − 1)pℓ−2(u) for ℓ = 2, . . . , t, u ∈ [−1, 1]. Define the Jacobi polynomials Jt(u) =

t

• ℓ=0

(2ℓ + 1)pℓ(u). Define the Gram matrix Gt(x) ∈ RN×N Gi,j(XN(x)) = Jt(xi(x)T xj(x)), where XN(x) = {x1(x), . . . , xN(x)}, and x = [xT

θ , xT φ ]T ∈ Rm,

xθ = [θ2, . . . , θN]T , xφ = [φ3, . . . , φN]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 Gt(XN(x∗)) be nonsingular. Then XN(x∗) is a spherical t-design with (t + 1)2 points if and only if x∗ is a solution of c(XN(x)) = EGt(XN(x))e = 0 (⇔ Gt(XN(x))e = conste) where E =       1 −1 . . . 1 −1 ... . . . . . . . . . ... ... 1 . . . −1       ∈ R(N−1)×N, e =    1 . . . 1    ∈ RN. c : Rm → Rn, n = N − 1, m = 2N − 3, N = (t + 1)2

• X. Chen (PolyU)

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Computational existence proofs

Computational method

• 1. Find an approximate zero

x of c(x) by using the Gauss-Newton method with an extremal system x0 as an initial guess. x0 = argmax det(Gt(XN(x))).

• 2. Use

x to construct a narrow interval z and show 2.1 z contains a solution x∗ of c(XN(x)) = 0 2.2 Gt(XN(x)) is non-singular for all x ∈ z. The interval z contains a fundamental spherical t-design. XN = {x1, . . . , xN} is a fundamental system if the zero polynomial is the only member

• f Pt which vanishes at each xi, i = 1, . . . , N.

Equivalent to Gt(XN(x)) is nonsingular.

• X. Chen (PolyU)

Spherical tǫ-Designs April 2015, SJTU 11 / 47

Computational existence proofs

Sufficient conditions

c(XN(x)) = 0, det(G(XN(x))) > 0 ⇒ XN(x) Spherical t-Design c(X4(x)) = 0 c(X4(x)) = 0 c(X4(x)) = 0 det(G1(X4(x))) =

1 π4

det(G1(X4(x))) = 0 det(G1(X4(x))) = 0 X4(x) S 2-D X4(x) S 1-D X4(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 IRm, zi = [xi, ¯ xi] Interval matrix A in IRm×m ai,j = [ai,j, ¯ ai,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 : Rn → Rn be continuously differentiable. Let z ∈ IRn and A ∈ IRn×n such that

• x ∈ z and f ′(x) ∈ A for all x ∈ z. Let V ∈ Rn×n be nonsingular.

The Krawczyk operator K( x, V, z, A) = x − V f( x) + (I − V A)(z − x) Existence Theorem K( x, V, z, A) ⊆ z ⇒ f has a zero x∗ in K( 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 ∈ IRn×n be an interval matrix, and let H ∈ Rn×n be a non-singular matrix. I − HG < 1 ⇒ all matrices G ∈ G are non-singular. In computation, we choose H = (midG)−1 . where ”mid” is the midpoint (midG)i,j = 1 2( ¯ Gi,j + Gi,j)

• X. Chen (PolyU)

Spherical tǫ-Designs April 2015, SJTU 15 / 47

Computational existence proofs

Compute K(ˆ x, V, x, A)

Define a function f : Rn → Rn by choosing n variables from m = 2(t + 1)2 − 3 = 2(n + 1) − 3 variables. QR-decomposition with column pivoting c′(ˆ x)P = Q[R|∗], P ∈ Rm×m, Q, R ∈ Rn×n P is a permutation matrix, Q is orthogonal and R is upper triangular. P arises from a greedy strategy to obtain maximum diagonal elements in R. Taking the index set B as those components which are permuted to the first n position by P, we get x = (xB, xN ) and f(xB) := c(xB, ˆ xN ) = 0.

• X. Chen (PolyU)

Spherical tǫ-Designs April 2015, SJTU 16 / 47

Computational existence proofs

Compute narrow enclosures for Jt(u)

• X. Chen (PolyU)

Spherical tǫ-Designs April 2015, SJTU 17 / 47

Computational existence proofs

Compute narrow enclosures for c(XN(x))

• X. Chen (PolyU)

Spherical tǫ-Designs April 2015, SJTU 18 / 47

Computational existence proofs

Complexity of Algorithm

The time complexity of our algorithm is O(t6).

• X. Chen (PolyU)

Spherical tǫ-Designs April 2015, SJTU 19 / 47

Computational existence proofs

Percent of cost in the various steps

• X. Chen (PolyU)

Spherical tǫ-Designs April 2015, SJTU 20 / 47

Computational existence proofs

An extremal system with (100 + 1)2 points

From Rob Womersley’s website

• X. Chen (PolyU)

Spherical tǫ-Designs April 2015, SJTU 21 / 47

Computational existence proofs

Spherical 100-design

Chen-Frommer-Lang, Numer. Math. (2011)

• X. Chen (PolyU)

Spherical tǫ-Designs April 2015, SJTU 22 / 47

Spherical tǫ-designs

Spherical tǫ-designs

Definition 3 (spherical tǫ-design) A spherical tǫ-design with 0 ≤ ǫ < 1 on Sd is a set of points Xǫ

N := {xǫ 1, . . . , xǫ N} ⊂ Sd such

that the quadrature rule with weights w = (w1, . . . , wN)T satisfying |Sd| N (1 − ǫ) ≤ wi ≤ |Sd| N (1 − ǫ)−1, i = 1, . . . , N, (2) is exact for all spherical polynomials p of degree at most t, that is,

N

• i=1

wip(xǫ

i) =

• Sd p(x)dωd(x).

(3) A spherical t-design is a spherical t0-design with ǫ = 0. By letting p(x) ≡ 1 in (3) we can

• btain N

i=1 wi = |Sd| and thus 0 < wi < |Sd| for i = 1, . . . , N.

Xǫ

N := {xǫ 1, . . . , xǫ N} ⊂ S2 is a spherical tǫ-design if and only if

Y(Xǫ

N)T w −

√ 4πe1 = 0 and 4π(1 − ǫ) N e ≤ w ≤ 4π(1 − ǫ)−1 N e, (4) where e1 = (1, 0, . . . , 0)T ∈ R(L+1)2 and e = (1, . . . , 1)T ∈ RN.

• X. Chen (PolyU)

Spherical tǫ-Designs April 2015, SJTU 23 / 47

Spherical tǫ-designs

(a) spherical 9-design X100 = {x1, . . . , x100} ⊂ S2 (b) neighborhood

• f

spherical 9-design X100 = {C(xi, γi) ⊂ S2, i = 1, . . . , 100}

Any point set chosen in a suffciently small neighborhood of a spherical t-design can be a spherical tǫ-design.

• X. Chen (PolyU)

Spherical tǫ-Designs April 2015, SJTU 24 / 47

Spherical tǫ-designs

Spherical tǫ-Design for Numerical Approximations on the Sphere, II

Yang Zhou

Department of Applied Mathematics The Hong Kong Polytechnic University

April 2015, Shanghai Jiaotong University

• Y. Zhou (PolyU)

Spherical tǫ-Designs, II April 2015, SJTU 25 / 47

Spherical tǫ-designs

Distances among points and point sets

For point sets XN = {x1, . . . , xN} ⊂ S2 and X′

N = {x′ 1, . . . , x′ N} ⊂ S2:

Separation distance of XN: ρ(XN) = min

i=j cos−1(xi · xj),

Least distance of a point x from a set XN: dist(x, XN) = min

1≤i≤N dist(x, xi) =

min

1≤i≤N cos−1(x · xi),

Hausdorff distance between two point sets XN and X′

N:

σ(XN, X′

N) = max{ max 1≤i≤N dist(x′ i, XN),

max

1≤i≤N dist(xi, X′ N)}.

Neighborhood of a point set XN C(XN, σ∗) = {X′

N ⊂ S2, σ(XN, X′ N) ≤ σ∗}

(5)

• Y. Zhou (PolyU)

Spherical tǫ-Designs, II April 2015, SJTU 26 / 47

Spherical tǫ-designs

One to one correspondence of xi ∈ XN and x′

j ∈ X′ N

For two point sets XN and X′

N, if σ(XN, X′ N) < 1 2ρ(XN), then for each xi ∈ XN there

exists a unique x′

j ∈ X′ N satisfying x′ j ∈ C(xi, 1 2ρ(XN)), where

C(xi, 1 2ρ(XN)) = {x ∈ S2 | cos−1(x · xi) ≤ 1 2ρ(XN)}.

• Y. Zhou (PolyU)

Spherical tǫ-Designs, II April 2015, SJTU 27 / 47

Spherical tǫ-designs

Way out: matrix perturbation theory

By the inequality (2), the lower bound of ǫ can be derived by ǫ 1 − ǫ ≥ N

4π w − e∞

using the relationship:    Y(X0

N)T e −

√ 4πe1 = 0 if X0

N is a spherial t − design,

Y(Xǫ

N)T w −

√ 4πe1 = 0 if Xǫ

N is a spherial tǫ − design.

By the well known matrix perturbation theory we have w − 4π

N e∞

4π

N e∞

≤ (Y(X0

N)T )−1∞ (Y(XN) − Y(X0 N))T ∞

1 − (Y(X0

N)T )−1∞(Y(XN) − Y(X0 N))T ∞ ,

if I − Y(X0

N))−1Y(XN)T ∞ ≤ Y(X0 N)−11Y(X0 N) − Y(XN)1 < 1.

• Y. Zhou (PolyU)

Spherical tǫ-Designs, II April 2015, SJTU 28 / 47

Spherical tǫ-designs

Spherical tǫ-designs: neighborhood of spherical t-designs

Lemma 4 Let X0

N be a fundamental spherical t-design and N = (t + 1)2. Then any point set

XN ∈ C(X0

N, σ∗) is a fundamental spherical tǫ-design with

τσ∗Y(X0

N)−11

1 − τσ∗Y(X0

N)−11

≤ ǫ < 1, (6) where σ∗ < 1 2 min

• 1

τY(X0

N)−11

, ρ(X0

N)

• ,

(7) with τ =

• 2t+1

4π (t + 1)3.

Corollary 5 Let X0

N be a fundamental spherical t-design and N = (t + 1)2. For any 0 ≤ ǫ < 1, if

σ(XN, X0

N) < min

• 1

2 ρ(X0

N),

ǫ τ(1 + ǫ)Y(X0

N)−11

• ,

(8) then XN is a fundamental spherical tǫ-design.

• Y. Zhou (PolyU)

Spherical tǫ-Designs, II April 2015, SJTU 29 / 47

Interval analysis of spherical tǫ-designs

Set of spherical caps

XN = {[x]i = C(ˆ xi, γi) ⊂ S2, i = 1, . . . , N} (9) Set of center points: ˆ XN = {ˆ x1, . . . , ˆ xN} ⊂ S2. Radius of XN: rad(XN) = max

1≤i≤N γi,

Separation distance of XN: ρ(XN) = min

i = j xi ∈ [x]i, xj ∈ [x]j ,

dist(xi, xj). xi ∈ [x]i and xi / ∈ [x]j for i = 1, . . . , N and i = j ⇒ XN ∈ XN.

• Y. Zhou (PolyU)

Spherical tǫ-Designs, II April 2015, SJTU 30 / 47

Interval analysis of spherical tǫ-designs

Set of spherical caps containing a fundamental spherical t-design

Assumption 6 Let XN defined by (9) be a set of intervals. Assume that

1

there exists a spherical t-design X0

N ∈ XN;

2

Y( ˆ XN) is nonsingular. Theorem 7 Under Assumption 6, any point set XN ∈ XN is a fundamental spherical tǫ-design with 2τrad(XN)Y( ˆ XN)−11 1 − 4τrad(XN)Y( ˆ XN)−11 ≤ ǫ < 1, (10) if rad(XN) < min

• 1

4 ρ(XN), ǫ 2(1 + 2ǫ)τY( ˆ XN)−11

• .

(11)

• Y. Zhou (PolyU)

Spherical tǫ-Designs, II April 2015, SJTU 31 / 47

Interval analysis of spherical tǫ-designs

Set of Spherical rectangles

ZN = {[z]1, . . . , [z]N} with [z]i =      sin([θ]i) cos([ϕ]i) sin([θ]i) sin([ϕ]i) cos([θ]i)      , i = 1, . . . , N, where [θ]i = [θ i, ¯ θi], [ϕ]i = [ϕi, ¯ ϕi]. Cap-cover of ZN ˜ XN = {C(˜ xi, γi), i = 1, . . . , N} with ˜ xi =       sin( 1

2 (¯

θi + θ i)) cos( 1

2 ( ¯

ϕi + ϕi)) sin( 1

2 (¯

θi + θ i)) sin( 1

2 ( ¯

ϕi + ϕi)) cos( 1

2 (¯

θi + θ i))       , γi = max{dist(˜ xi, xi,1) , dist(˜ xi, xi,3)}, with xi,1, xi,3 defined by(θ i, φi), (¯ θi, φi).

• Y. Zhou (PolyU)

Spherical tǫ-Designs, II April 2015, SJTU 32 / 47

Interval analysis of spherical tǫ-designs

Set of spherical caps containing a fundamental spherical t-design

Corollary 8 Let ZN be a set of spherical rectangle and its cap-cover ˜ XN satisfy Assumption 6. Then any point set XN ∈ ZN is a fundamental spherical tǫ-design with 2τrad(XN)Y( ˜ XN)−11 1 − 4τrad(XN)Y( ˜ XN)−11 ≤ ǫ < 1, (12) if rad(ZN) < min 1 4ρ(ZN), ǫ 2(1 + 2ǫ)τY( ˜ XN)−11

• .

(13)

• Y. Zhou (PolyU)

Spherical tǫ-Designs, II April 2015, SJTU 33 / 47

Interval analysis of spherical tǫ-designs

Lower bounds of ǫ for sets of spherical rectangles in 1

ǫ: the lower bound of ǫ derived from (12).

20 40 60 80 100 10

−14

10

−12

10

−10

10

−8

10

−6

10

−4

10

−2

10 t ǫ 10−

14.4 (t + 1)6.9

Figure: ǫ for t = 2, . . . , 100

• 1X. Chen, A. Frommer, and B. Lang. Computational existence proofs for spherical t-designs.
• Numer. Math., 117(2):289–305, 2011.
• Y. Zhou (PolyU)

Spherical tǫ-Designs, II April 2015, SJTU 34 / 47

Interval analysis of spherical tǫ-designs

Remark

To obtain an approximate spherical t-design as accurate as possible, in 2 the radius of the set of interval enclosures rad(ZN) is computed to a small scale around 10−10. With the introduction of the concept spherical tǫ-designs, it has been shown that any point set selected in ZN is a fundamental spherical tǫ-design with small enough rad(ZN) and ǫ is increasing in rad(ZN). As a result, to reduce the difficulty of computing ZN, one may relax rad(ZN) to a larger scale with keeping ǫ < 1 holding. Then all point sets selected in ZN are still fundamental spherical tǫ-designs with 0 ≤ ǫ < 1.

• 2X. Chen, A. Frommer, and B. Lang. Computational existence proofs for spherical t-designs.
• Numer. Math., 117(2):289–305, 2011.
• Y. Zhou (PolyU)

Spherical tǫ-Designs, II April 2015, SJTU 35 / 47

Worst-case errors of numerical integration using spherical tǫ-designs

Worst-case errors (WCEs) of numerical integration using spherical tǫ-designs

Define Q[Xǫ

N, w](f) := N

• j=1

wif(xǫ

j),

I(f) :=

• Sd f(x)dωd(x),

(14) and the worse-case error of Q[Xǫ

N, w] by

Es,d(Q[Xǫ

N, w]) := sup

• |Q[Xǫ

N, w](f) − I(f)| : f ∈ Hs(Sd), fHs ≤ 1

• .

(15) For simplicity denote Es(Q[Xǫ

N, w]) = Es,2(Q[Xǫ N, w]).

Computation of WCEs for numerical integration using spherical t-designs in Sobolev spaces is studied in [J. Brauchart, E. Saff, I. Sloan, and R. Womersley, QMC designs:

• ptimal order quasi monte carlo integration schemes on the sphere, Math. Comp., 83

(2014), pp. 2821-2851].

• Y. Zhou (PolyU)

Spherical tǫ-Designs, II April 2015, SJTU 36 / 47

Worst-case errors of numerical integration using spherical tǫ-designs

WCEs of spherical tǫ-designs on Hs(S2)

1 < s ≤ 2 Es(Q[XN, w]) =  

N

• i=1

N

• j=1

wiwj(V2−2s(S2) − |xi − xj|2s−2)  

1 2

, (16) s > 2 Es(Q[XN, w]) =  

N

• i=1

N

• j=1

wiwj

• QL(xi · xj) + (−1)L+1|xi − xj|2s−2

−(−1)L+1V2−2s(S2) 2 . (17) where V2−2s(S2) := 22s−1 Γ(3/2)Γ(s) √πΓ(1 + s) , QL(x · y) :=

L

• ℓ=0

((−1)L+1−ℓ − 1)a(s)

ℓ

(2ℓ + 1)Pℓ(x · y), x, y ∈ S2.

• Y. Zhou (PolyU)

Spherical tǫ-Designs, II April 2015, SJTU 37 / 47

Worst-case errors of numerical integration using spherical tǫ-designs

Numerical results for WCEs of spherical t0.1-designs

Spherical tǫ-designs can be obtained by finding a minimizer of the least square form (4) using a smoothing trust-region filter method proposed in [5]. Spherical t-designs applied here are approxiamte spherical t-designs calculated by Sloan and Womersley.

10

1

10

2

10

3

10

−1.9

10

−1.7

10

−1.5

10

−1.3

10

−1.1

N Es s= 1.5 spherical tε−design spherical t−design

(a) s = 1.5

10

1

10

2

10

3

10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10 N Es s = 5.5 spherical t−design spherical tε−design

(b) s = 5.5

Figure: WCEs for Hs(S2) with different s

• Y. Zhou (PolyU)

Spherical tǫ-Designs, II April 2015, SJTU 38 / 47

Polynomial approximation on the sphere using spherical tǫ-designs l2 − l1 regularized weighted polynomial approximation

Regularized weighted polynomial approximation using spherical tǫ-designs

l2 − l1 regularized weighted discrete least squares polynomial approximation min

αℓ,k∈R

1 2

N

• j=1

µj(

L

• ℓ=0

2k+1

• k=1

αℓ,kYℓ,k(xj) − f δ(xj))2 + λ

L

• ℓ=0

2ℓ+1

• k=1

|βℓ,kαℓ,k| (18) µj > 0, λ > 0, βℓ,k ≥ 0, j = 1, . . . , N, ℓ = 0, . . . , L, k = 1, . . . , 2ℓ + 1. ⇒ min

α∈R(L+1)2

1 2Λ

1 2 (YLα − f δ)2

2 + λDα1,

(19) where Λ =    µ1 ... µN    ∈ RN×N, and D is a diagonal matrix satisfying Dℓ2+k,ℓ2+k = βℓ,k with βℓ,k ≥ 0.

• Y. Zhou (PolyU)

Spherical tǫ-Designs, II April 2015, SJTU 39 / 47

Polynomial approximation on the sphere using spherical tǫ-designs l2 − l1 regularized weighted polynomial approximation

Solution of (18)

Theorem 9 Let XN be a spherical tǫ-design and w be the vector of weights satisfying (2) and (3) with respect to XN. Assume t ≥ 2L. For model (19) set µj = wj for j = 1, . . . , N. Then HL = YT

LΛYL = I(L+1)2,

(20) and (19) has the unique solution αℓ,k = max{0, sℓ,k − λβℓ,k} + min{0, sℓ,k + λβℓ,k}, (21) for ℓ = 0, . . . , L, k = 1, . . . , 2k + 1, where sℓ,k = N

i=1 wiYℓ,k(xi)f δ(xi).

Proposition 10 Denote the approximation residual as A(α) = N

j=1 µj(pL,N(xj) − f δ(xj))2. Let α∗(λ)

be the optimal solution of (19) with different regularization parameters λ. Let Xǫ

N be a

spherical tǫ-design with t ≥ 2L and µj = wj for j = 1, . . . , N. Then A(α∗(λ)) is increasing in λ.

• Y. Zhou (PolyU)

Spherical tǫ-Designs, II April 2015, SJTU 40 / 47

Polynomial approximation on the sphere using spherical tǫ-designs Numerical experiments

Numerical experiments

Models: l2 − l1 model (19), l2 − l2 model (22). min

α∈R(L+1)2

1 2Λ

1 2 (YLα − f δ)2

2 + λDα2 2.

(22) Target functions: low-degree spherical polynomials, Franke function, “Franke plus cap” function. Sets of points: spherical t0.1-designs, spherical t-designs. Noise: uniform distribution in [−δ, δ].

• Y. Zhou (PolyU)

Spherical tǫ-Designs, II April 2015, SJTU 41 / 47

Polynomial approximation on the sphere using spherical tǫ-designs Numerical experiments

Errors for restoring 18-degree polynomial using spherical 370.1-design with δ = 0.1

10

−20

10

−15

10

−10

10

−5

10 10

5

10

−2

10

−1

10 10

1

λ Uniform errors l2−l2 l2−l1

(a) Uniform Errors with different λ

10

−20

10

−15

10

−10

10

−5

10 10

5

10

−1

10 10

1

10

2

λ L2 errors l2−l2 l2−l1

(b) L2 errors with different λ

• Y. Zhou (PolyU)

Spherical tǫ-Designs, II April 2015, SJTU 42 / 47

Polynomial approximation on the sphere using spherical tǫ-designs Numerical experiments

Errors for restoring 18-degree polynomial using spherical 370.1-design with different δ

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10

−1

10 δ Uniform errors l2−l2 l2−l1

(c) Uniform errors with the best λ

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10

−1

10 δ L2 errors l2−l2 l2−l1

(d) L2 errors with the best λ

• Y. Zhou (PolyU)

Spherical tǫ-Designs, II April 2015, SJTU 43 / 47

Polynomial approximation on the sphere using spherical tǫ-designs Numerical experiments

Errors for approximating Franke function with different scales of spherical t0.1-designs and spherical t-designs

100 200 300 400 500 600 10

−1

10 N Uniform errors spherical t0.1−designs spherical t−designs

(e) Uniform Errors

100 200 300 400 500 600 10

−3

10

−2

N L2 errors spherical t0.1−designs spheical t−designs

(f) L2 errors

• Y. Zhou (PolyU)

Spherical tǫ-Designs, II April 2015, SJTU 44 / 47

Polynomial approximation on the sphere using spherical tǫ-designs Numerical experiments

Restoration of “Franke plus cap” function (f2) using spherical 370.1-design with l2 − l1 and l2 − l2 models

(g) f2 (h) fδ

2 with δ = 0.5

(i) l2 − l1 restoration (j) l2 − l2 restoration (k) l2 − l1 restoration errors (l) l2 − l2 restoration errors

• Y. Zhou (PolyU)

Spherical tǫ-Designs, II April 2015, SJTU 45 / 47

Conclusions

Conclusions

A set of points in a sufficiently small neighborhood of a fundamental spherical t-design can be a spherical tǫ-design. Any point set arbitrarily chosen in a set of small enough interval enclosures containing a fundamental spherical t-design can be a spherical tǫ-design. Worst-case errors for numerical integration using spherical tǫ-designs are studied, which can be improved with the increasing of ǫ. An l2 − l1 regularized polynomial approximation using spherical tǫ-designs is established with good accuracy and efficiency.

• Y. Zhou (PolyU)

Spherical tǫ-Designs, II April 2015, SJTU 46 / 47

Conclusions

References

• 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.

• 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.

• E. Bannai, On tight spherical designs, J. Comb. Theory Ser. A, 26 (1979), pp. 38–47.
• E. Bannai and E. Bannai, A survey on spherical designs and algebraic combinatorics on spheres,

European J. Combin., 30 (2009), 1392-1425.

• X. Chen, S. Du, and Y. Zhou, A smoothing trust region filter algorithm for nonsmooth nonconvex least

squares problems, tech. report, SIAM Annual Meeting 2014, Chicago, July, 711 2014.

• X. Chen, A. Frommer and B. Lang, Computational Existence Proofs for Spherical t-Designs , Numer.

Math., 117(2011), 289-305.

• X. Chen and R. S. Womersley, Existence of solutions to systems of underdetermined equations and

spherical designs , SIAM J. Numer.l Anal., 44(2006) 2326-2341.

• Y. Zhou, and X. Chen. Spherical tǫ-Designs for Approximations on the Sphere, arXiv preprint

arXiv:1502.03562 (2015).

• Y. Zhou (PolyU)

Spherical tǫ-Designs, II April 2015, SJTU 47 / 47