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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

1 Notations and definitions 2 Numerical construction for well conditioned spherical designs 1 Optimization problem 2 Interval method 3 Open Problem 3 Interpolation, Hyperinterpolation and Filtered hyperinterpolation on S2 4 Regularized least squares approximation on S2 by using spherical designs 1 Solutions 2 Numerical experiments With exact data With noisy data

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

S2 =

• [x, y, z]T ∈ R3 |x2 + y2 + z2 = 1
• XN = {x1, . . . ,xN} ⊂ S2

L Degree of polynomials PL = { spherical polynomials of degree ≤ L} with dL := dim(PL) = (L + 1)2 N Number of points ||f||C(S2) := supx∈S2 |f(x)|. Lebesgue constant: ||U||C(S2) := supf=0

||Uf||C(S2) ||f||C(S2)

EL(f) := minp∈PL ||p − f||C(S2) Basis for PL: orthonormal spherical harmonics { Yℓ,k : ℓ = 0, 1, . . . , L, k = 1, . . . , 2ℓ + 1}. Coefficient α = (αℓ,k) ∈ R(L+1)2. Laplace-Beltrami operator ∆∗: ∆∗Yℓ,k(x) = −ℓ(ℓ + 1)Yℓ,k(x) YL ∈ R(L+1)2×N [Yℓ,k(xj)], ℓ = 0, 1, . . . , L, k = 1, . . . , 2ℓ + 1, j = 1, . . . , N. (L + 1)2 by (L + 1)2 matrix HL(XN) := YLYT

L.

(1)

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

Spherical coordinates

x1 =    1    , x2 =    sin(θ2) cos(θ2)    , xi =    sin(θi) cos(φi) sin(θi) sin(φi) cos(θi)    , i = 3, . . . , N.

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

Franke function

f1 (x, y, z) = 0.75 exp(−(9x − 2)2/4 − (9y − 2)2/4 − (9z − 2)2/4) +0.75 exp(−(9x + 1)2/49 − (9y + 1)/10 − (9z + 1)/10) +0.5 exp(−(9x − 7)2/4 − (9y − 3)2/4 − (9z − 5)2/4) −0.2 exp(−(9x − 4)2 − (9y − 7)2 − (9z − 5)2), (x, y, z) ∈ S2, which is in C∞(S2).

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

Definition of Spherical t−design

Definition (Spherical t−design) The set XN = {x1, . . . , xN} ⊂ S2 is a spherical t-design if 1 N

N

• j=1

p (xj) = 1 4π

• S2 p(x)dω(x)

∀p ∈ Pt, (2) where dω(x) denotes surface measure on S2.

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 ≥ ct2 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 XN is bounded from below (Delsarte, Goethals, Seidel 1977 [9]), N ≥           

(t+1)(t+3) 4

if t is odd

(t+2)2 4

if t is even 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 Sd 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 Ct(XN) = 0

Let N ≥ (t + 1)2, define Ct : (S2)N → RN−1, Ct(XN) = EGt(XN)e, (3) where E = [1, −IN−1] ∈ R(N−1)×N, 1 = [1, . . . , 1]T ∈ RN−1, Gt (XN) := Yt (XN)T Yt (XN) ∈ RN×N, e =        1 1 . . . 1        ∈ RN. Theorem (An-Chen-Sloan-Womersley,[1]) Let N ≥ (t + 1)2. Suppose that XN = {x1, . . . , xN} is a fundamental system for Pt. Then XN is a spherical t-design if and only if Ct (XN) = 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 XN = {x1, . . . , xN} ⊂ S2 is a fundamental system for Pt if the zero polynomial is the only member of Pt that vanishes at each point xi, 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 XN = {x1, . . . , xN} ⊂ S2 of N ≥ (t + 1)2 points is an extremal spherical t-design if the determinant of the matrix Ht(XN) := Yt(XN)Yt(XN)T ∈ R(t+1)2×(t+1)2 is maximal subject to the constraint that XN 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 (Ht(XN)) XN ⊂ S2 subject to Ct (XN) = 0. (4) ⇓

Well conditioned spherical t-design.

The log of the determinant is bounded above by logdet(HL(XN)) ≤ (t + 1)2 log N 4π

• .

(5)

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 ˜

XN of Ct(XN) = 0 starting from X 0

N.

2 Use a nonlinear programming method to find

ˆ XN ≈ arg max{log det(Ht(XN)) | Ct(XN) = 0} starting from ˜ XN. N = (t + 1)2,we choose the extremal system as the starting point set. Interval method below to find an interval that contains ˆ XN 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(Gt(XN)) = det(Ht(XN)). 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(Gt(XN ))] and diameters of [log det(Gt(XN ))]

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(HL(XN)) subject to XN is a spherical t − design.

Mini-Workshop on Spherical Designs and Related Topics Interpolation, Hyperinterpolation and Filtered Hyperinterpolation on S2

Part III

Interpolation, Hyperinterpolation and Filtered Hyperinterpolation on S2

Mini-Workshop on Spherical Designs and Related Topics Interpolation, Hyperinterpolation and Filtered Hyperinterpolation on S2

Interpolation on S2

Let N = dL = (L + 1)2. For a given degree L, given a fundamental system XdL = {x1, . . . , xdL} for PL and an arbitrary f ∈ C(S2), we denote by ΛLf the unique polynomial in the space PL that interpolates f at each point

• f the fundamental system, that is

ΛLf ∈ PL, ΛLf (xj) = f (xj) , j = 1, . . . , dL. (6)

Mini-Workshop on Spherical Designs and Related Topics Interpolation, Hyperinterpolation and Filtered Hyperinterpolation on S2

Lagrange expression

The Lagrange basis polynomials {ℓ1, . . . , ℓdL} are defined, as usual, by ℓj ∈ PL, ℓj(xi) = δji, i, j = 1, . . . , dL. ℓj(x) = det

• GL
• x1, . . . , xj−1, x, xj+1, . . . , xdL
• det
• GL
• x1, . . . , xj−1, xj, xj+1, . . . , xdL

. (7) For given f ∈ C

• S2

ΛLf =

dL

• j=1

f (xj) ℓj. (8)

Mini-Workshop on Spherical Designs and Related Topics Interpolation, Hyperinterpolation and Filtered Hyperinterpolation on S2

Lebesgue constant for interpolation

ΛLC(S2) := sup

f∈C(S2)\{0}

ΛLf C(S2) f C(S2) = max

x∈S2 dL

• j=1

|ℓj(x)| , (9) Can ΛLC(S2) < c (10) by choosing points on the unit sphere? We can not prove this! :(

Mini-Workshop on Spherical Designs and Related Topics Interpolation, Hyperinterpolation and Filtered Hyperinterpolation on S2

Lebesgue constant for interpolation

ΛLC(S2) := sup

f∈C(S2)\{0}

ΛLf C(S2) f C(S2) = max

x∈S2 dL

• j=1

|ℓj(x)| , (9) Can ΛLC(S2) < c (10) by choosing points on the unit sphere? We can not prove this! :(

Mini-Workshop on Spherical Designs and Related Topics Interpolation, Hyperinterpolation and Filtered Hyperinterpolation on S2

Numerical results

Minimal energy system min

XN⊂S2 N

• i=j

1 xi − xj2 (11)

Mini-Workshop on Spherical Designs and Related Topics Interpolation, Hyperinterpolation and Filtered Hyperinterpolation on S2

The Lebesgue constants for interpolation with N = (t + 1)2

Mini-Workshop on Spherical Designs and Related Topics Interpolation, Hyperinterpolation and Filtered Hyperinterpolation on S2

Uniform interpolation error for Franke function with N = (t + 1)2

Mini-Workshop on Spherical Designs and Related Topics Interpolation, Hyperinterpolation and Filtered Hyperinterpolation on S2

Hyperinterpolation(Sloan, 1995[16])

Using a spherical t-design XN = {x1, . . . , xN} ⊂ S2 for t ≥ 2L, we define the semi-inner product (·, ·)N of two continuous functions f, g ∈ C(S2) by (f, g)N := 4π N

N

• j=1

f(xj)g(xj), j = 1, . . . , N. (12) It is clear that (p, q)N = (p, q)L2 =

• S2 p(x)q(x)dω(x),

p, q ∈ PL, because pq ∈ P2L and t ≥ 2L. We note that for f ∈ C(S2), (f, f)N = 0 implies f(xj) = 0, j = 1, . . . , N, but does not imply f ≡ 0.

Mini-Workshop on Spherical Designs and Related Topics Interpolation, Hyperinterpolation and Filtered Hyperinterpolation on S2

Let GL(x, y) := Kℓ(x · y) =

L

• ℓ=0

Yℓ,k(x)Yℓ,k(y). (13) Definition (Hyperinterpolant, Sloan, 1995) Let {Yℓ,k, k = 1, . . . , 2ℓ + 1, ℓ = 0, . . . , L} be the orthonormal spherical harmonics of degree up to L and order k . The hyperinterpolant of a function f ∈ C(S2) is defined by LLf(x) :=

L

• ℓ=0

2ℓ+1

• k=1

(f, Yℓ,k)NYℓ,k(x) = (f, GL(x, ·))N, x ∈ S2. (14)

Mini-Workshop on Spherical Designs and Related Topics Interpolation, Hyperinterpolation and Filtered Hyperinterpolation on S2

Why hyperinterpolation is charming?

As tight spherical 2L-design for S2 do not exist for L ≥ 2, the number of points of hyperinterpolation will exceed the dimension of PL (see [4]). Theorem (1995 Sloan[16]) Let LLf be the hyperintepolation approximation (14), with an N-point spherical 2L-design. If L ≥ 2, then N > dL = (L + 1)2.

Mini-Workshop on Spherical Designs and Related Topics Interpolation, Hyperinterpolation and Filtered Hyperinterpolation on S2

Lebesgue constant for hyperinterpolation

The Lebesgue constant of the operator LL, defined by ||LL||C(S2) := sup

f∈C(S2)\{0}

||LLf||C(S2) ||f||C(S2) . (15) ||LL||C(S2) satisfies[18](Sloan-Womersley,2000) c √ L + 1 ≤ ||LL||C(S2) ≤ c1 √ L + 1, L = 0, 1, . . . , (16)

Mini-Workshop on Spherical Designs and Related Topics Interpolation, Hyperinterpolation and Filtered Hyperinterpolation on S2

Filtered hyperinterpolation

In the method of filtered hyperinterpoltion [19](Sloan and Womersley, 2012) the kernel GL in (13) is replaced by a “filtered” kernel HL(x, y) = HL(x · y) =

L−1

• ℓ=0

h ℓ L

• Kℓ(x · y) =

L−1

• ℓ=0

h ℓ L 2ℓ + 1 4π Pℓ(x · y), (17) with h : R+ → R+ a function with at least C1(R+) smoothness, satisfying h(x) =    1, x ∈ [0, 1/2], 0, x ∈ [1, ∞). (18)

Mini-Workshop on Spherical Designs and Related Topics Interpolation, Hyperinterpolation and Filtered Hyperinterpolation on S2

Thus the filtered hyperinterpolant FLf ∈ PL−1 is FLf(x) := (f, HL(x, ·))N = 4π N

N

• j=1

f(xj)HL(x, xj) =

L−1

• ℓ=0

(2ℓ + 1)h ℓ L 1 N

N

• j=1

Pℓ(x · xj)f(xj). (19) Define the Lebesgue constant for FL by ||FL||C(S2) := sup

f∈C(S2)\{0}

||FLf||C(S2) ||f||C(S2) . (20)

Mini-Workshop on Spherical Designs and Related Topics Interpolation, Hyperinterpolation and Filtered Hyperinterpolation on S2

Filter function ⇒ Uniform convergence

Let[2](An-Chen-Sloan-Womersley, 2012) h(x) :=          1, x ∈ [0, 1/2], sin2 πx, x ∈ [1/2, 1], 0, x ∈ [1, ∞). (21) Then we have [2](An-Chen-Sloan-Womersley,2012),[19](Sloan-Womersley,2012) sup

L≥0

||FLf||C(S2) =: c1 < ∞. (22) Consequently, by using h(x) to FLwe have FLp = p ∀p ∈ P⌊L/2⌋ (23) and hence [2][19] ||FLf − f||C(S2) ≤ cE⌊L/2⌋(f), (24) where ⌊·⌋ denotes the floor function and c is a constant independent of L.

Mini-Workshop on Spherical Designs and Related Topics Interpolation, Hyperinterpolation and Filtered Hyperinterpolation on S2

Consistence from spherical designs[3](An, 2011)

Theorem (An, 2011) Let LLF and FLf be hyperinterpolant defined by (14) and filtered hyperinterpolant defined by (17) and (18). Then

• S2 LLf(x)dω(x) =
• S2 FLf(x)dω(x) = 4π

N

N

• j=1

f(xj). More precisely, we obtain that

N

• j=1

LLf(xj) =

N

• j=1

FLf(xj) =

N

• j=1

f(xj). Yet we note that in general LLf, FLf and f are all different at the quadrature points.

Mini-Workshop on Spherical Designs and Related Topics Regularized Least Squares Approximation on S2

Part IV

Regularized Least Squares Approximation on S2, An-Chen-Sloan-Womersley, 2012 [2]

Mini-Workshop on Spherical Designs and Related Topics Regularized Least Squares Approximation on S2

Regularized least-squares problem

Let f ∈ C(S2). Study the discrete regularized least-squares

min

p∈PL

{

N

• j=1

(p(xj) − f(xj))2 + λ

N

• j=1

(Rp(xj))2}, λ > 0, (25) where R is rotationally invariant. RLp(x) :=

L

• ℓ=0

βℓ

2ℓ+1

• k=1

Yℓ,k(x)(Yℓ,k, p)L2 =

L

• ℓ=0

βℓ

• S2

(2ℓ + 1) 4π Pℓ(x · y)p(y)dω(y), (26) where β0, β1, . . . , βL are arbitrary non-negative numbers, which may depend on L. 1 0 operator⇒interpolation, quasi-interpolation, hyperinterpolation [18]; 2 Related to (−∆∗)s, s > 0⇒recover function from noisy data; 3 Related to filter functions⇒ uniformly convergence when t ≥ 2L.

Mini-Workshop on Spherical Designs and Related Topics Regularized Least Squares Approximation on S2

Solution of regularized least squares

As we fix the basis as Yℓ,k, (25) can be reduced to the convex

• ptimization problem

min

α∈R(L+1)2 YT Lα − f2 2 + RL(XN)T α2 2,

(27) where f = [f(x1), . . . , f(xN)]T ∈ RN, RL(XN) = BLYL ∈ R(L+1)2×N, BL := diag(β0, β1, β1, β1

• 3

, . . . , βL, . . . , βL

• 2L+1

) ∈ R(L+1)2×(L+1)2. (28) Our aim is to find “best” polynomial approximations pL,XN =

L

• ℓ=0

2ℓ+1

• k=1

αℓkYℓ,k. (29) In (27), α satisfies the first order condition TLα = YLf, (30)

Mini-Workshop on Spherical Designs and Related Topics Regularized Least Squares Approximation on S2

The solution for t ≥ 2L

Theorem (Diagonalization [2]) Assume f ∈ C(S2). Let L ≥ 0 be given, and let XN = {x1, . . . , xN} be a spherical t-design on S2 with t ≥ 2L. Then

HL(XN) = N 4π I(L+1)2 ∈ R(L+1)2×(L+1)2. (32) while (27) has a unique solution αℓk = 4π N(1 + β2

ℓ ) N

• j=1

Yℓ,k(xj)f(xj), (33) and the unique minimizer of (25) is given by pL,XN (x) = 4π N

L

• ℓ=0

2ℓ+1

• k=1

Yℓ,k(x) 1 + β2

ℓ N

• j=1

Yℓ,k(xj)f(xj) (34)

Mini-Workshop on Spherical Designs and Related Topics Regularized Least Squares Approximation on S2

Relationship with continuous approximation

Due to the convergence [2] of spherical designs, we have Theorem ([2]) Let f ∈ C(S2), and let L ≥ 0 be given. Assume that the sets X (t)

N(t) = {x(t) 1 , . . . , x(t) N(t)} for t = 1, 2, . . . form a sequence of spherical

t-designs with t → ∞. Then for t ≥ L the unique minimizer pL,XN(t) ∈ PL of (25) has the uniform limit pL, that is lim

t→∞pL,XN(t) − pL∞ = 0,

(35) where pL ∈ PL denotes the unique minimizer of the continuous regularized least-squares problem min

p∈PL {f − p2 L2 + Rp2 L2}.

(36)

Mini-Workshop on Spherical Designs and Related Topics Regularized Least Squares Approximation on S2

Numerical results: Testing functions

We choose two test functions for our numerical experiments. The first function is the Franke function f1 (x, y, z) = 0.75 exp(−(9x − 2)2/4 − (9y − 2)2/4 − (9z − 2)2/4) +0.75 exp(−(9x + 1)2/49 − (9y + 1)/10 − (9z + 1)/10) +0.5 exp(−(9x − 7)2/4 − (9y − 3)2/4 − (9z − 5)2/4) −0.2 exp(−(9x − 4)2 − (9y − 7)2 − (9z − 5)2), (x, y, z) ∈ S2, which is in C∞(S2). f2 = f1 + fcap, (37) where fcap(x) =    ρ cos π arccos(xc · x) 2r

• ,

x ∈ C(xc, r), 0,

• therwise,

Mini-Workshop on Spherical Designs and Related Topics Regularized Least Squares Approximation on S2

Testing functions

Mini-Workshop on Spherical Designs and Related Topics Regularized Least Squares Approximation on S2

Numerical results: Error

Figure: Uniform and L2 errors for hyperinterpolation and filtered hyperinterpolation with t = 2L, N = (t + 1)2 and L = 1, . . . , 30

Mini-Workshop on Spherical Designs and Related Topics Regularized Least Squares Approximation on S2

Numerical results: Error

Figure: Uniform and L2 errors for hyperinterpolation and filtered hyperinterpolation with t = 2L, N = (t + 1)2 and L = 1, . . . , 30

Mini-Workshop on Spherical Designs and Related Topics Regularized Least Squares Approximation on S2

Numerical results: Error

Figure: Errors

Mini-Workshop on Spherical Designs and Related Topics Regularized Least Squares Approximation on S2

Numerical results: Recover from contaminated data, with different λ

Figure: Using Laplace-Beltrami regularization operator approximate function f2 with contaminated data

Mini-Workshop on Spherical Designs and Related Topics Regularized Least Squares Approximation on S2

Numerical results: Recover from contaminated data, with different λ

Figure: Using Laplace-Beltrami regularization operator to approximate function f2 with contaminated data

Mini-Workshop on Spherical Designs and Related Topics Regularized Least Squares Approximation on S2

Thank you very much!

Mini-Workshop on Spherical Designs and Related Topics REFERENCES

REFERENCES

Mini-Workshop on Spherical Designs and Related Topics REFERENCES [1]

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ArXiv:1009.4407v1[math.MG], 2011. [7]

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spherical designs, SIAM J. Numer. Anal., 44 (2006), pp. 2326–2341. [8]

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(1995), pp. 238–254. [17]

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• I. H. Sloan and R. S. Womersley, Constructive polynomial approximation on the sphere, J. Approx.

Theory, 103 (2000), pp. 91-98. [19]

• I. H. Sloan and R. S. Womersley, Filtered hyperinterpolation: A constructive polynomial approximation
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