numerical integration over unit sphere{by using...
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Numerical Integration over unit sphere–by using spherical t-designs
Numerical Integration over unit
sphere–by using spherical t-designs
Congpei An1
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
XN = {x1, . . . , xN} ⊂ S2 ={x, y, z ∈ R3 |x2 + y2 + z2 = 1
}Pt = { spherical polynomials of degree ≤ t}
= {polynomials in x, y, z of degree ≤ t restricted to S2}
N = Number of points
t = Degree of polynomials
Numerical Integration over unit sphere–by using spherical t-designs
Spherical coordinates
x1 =
0
0
1
, x2 =
sin(θ2)
0
cos(θ2)
, xi =
sin(θi) cos(φi)
sin(θi) sin(φi)
cos(θi)
, i = 3, . . . , N.
(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 XN = {x1, . . . ,xN} ⊂ S2 is a spherical t-design if
1
N
N∑j=1
p (xj) =1
4π
∫S2p(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,
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`k : k = 1, . . . , 2` + 1, ` = 0, 1, . . . , tBasis
Pt=Span{Y`k : k = 1, . . . , 2`+ 1, ` = 0, 1, . . . , t}
Orthonormality with respect to L2 inner product
(p, q)L2 =
∫S2p(x)q(x)dω(x),
Normalization Y0,1 = 1√4π
dimPt = (t+ 1)2
Addition Theorem2`+1∑k=1
Y`,k(x)Y`,k(y) =2`+14π P` (x · y) , x, y ∈ S2
Numerical Integration over unit sphere–by using spherical t-designs
Background on spherical t−designs
Spherical harmonic basis matrix
For t ≥ 1, and N ≥ dim(Pt) = (t+ 1)2, let Y0t be the ((t+ 1)2 − 1) by
N matrix defined by
Y0t (XN ) := [Y`,k(xj)], k = 1, . . . , 2`+ 1, ` = 1, . . . , t; j = 1, . . . , N, (3)
Yt(XN ) :=
[1√4π
eT
Y0t (XN )
]∈ R(t+1)2×N , (4)
where e = [1, . . . , 1]T ∈ RN .
Gt (XN ) := Yt (XN )T Yt (XN ) ∈ RN×N ,
Ht (XN ) := Yt (XN )Yt (XN )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 Ct(XN) = 0
Let N ≥ (t+ 1)2, define Ct : (Sd)N → R,
Ct(XN ) = EGt(XN )e (5)
where the N ×N Gram matrix Gt for XN ⊂ S2
Gt (XN ) = Yt (XN )T Yt (XN )
e =
1
1...
1
∈ RN ,E = [1,−I] ∈ R(N−1)×N , 1 = [1, . . . , 1]T ∈ RN−1 (6)
Numerical Integration over unit sphere–by using spherical t-designs
Background on spherical t−designs
Nonlinear system Ct(XN) = 0
Theorem (ACSW2010,[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.
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.
Ht(XN ) is nonsingular ⇐⇒ XN is a fundamental system for Pt.Let N = (t+ 1)2, Gt(XN ) is nonsingular⇐⇒ XN is a fundamental system
for Pt.
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 XN = {x1, . . . ,xN} ⊂ S2 of N ≥ (t+ 1)2 points is a 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.
Numerical Integration over unit sphere–by using spherical t-designs
Well conditioned spherical designs
Optimization Problem on S2
max log det (Ht(XN ))
XN ⊂ S2
subject to Ct (XN ) = 0.
(7)
⇓
Well conditioned spherical t-design.
The log of the determinant is bounded above by
logdet(HL(XN )) ≤ (t+ 1)2 log
(N
4π
). (8)
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 IRn, denote [a] = [a, a], a, a ∈ Rn, a ≤ a
2 +,−, ∗, / can be extended from Rn to IRn and from Rn×n to IRn×n.
3 Let mid[a] = (a+ a)/2 in componentwise.
4 diam[a] = a− a = 2rad [a] ,
5 F : D ⊆ Rn → Rn be a continuously differentiable function.Let
[dF] ∈ IRn×n be an interval matrix containing F′(ξ) for all ξ ∈ [x],
i.e.
{F′(x) : x ∈ [x]} ⊆ [dF] ([x]). (9)
Such [dF] 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 BL ∈ Rn×n, z ∈ [z] ⊆ D and [dF] ∈ IRn×n, the
Krawczyk operator [11] is defined by:
kF(z, [z] ,BL, [dF]) := z−BLF(z) + (In −BL · [dF])([z]− z). (10)
It is known that kF(z, [z] ,BL, [dF]) is an interval extension of the function
ψ(z) := z−BLF(z) over [z], that is, z−BLF(z) ∈ kF(z, [z] ,BL, [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 ⊂ Rn → Rn be a continuously differentiable function. Choose
[z] ∈ IRn, z ∈ [z] ⊆ D, an invertible matrix BL ∈ Rn×n and [dF] ∈ IRn×n
such that F′ (ξ) ∈ [dF] for all ξ ∈ [z]. Assume that
kF (z, [z],BL, [dF]) ⊆ [z].
Then F has a zero z∗ in kF (z, [z],BL, [dF]).
Numerical Integration over unit sphere–by using spherical t-designs
Numerical Verification method
Deal with Ct(XN)
1 Represent the points xi on the sphere by spherical coordinates with φ, θ .
That is
[xi] = [sin([θ])cos([φi]), sin([θi])sin([φi]), cos([θi])]T , i = 1, . . . , N.
2 Ct(XN ) is redefined as a system of nonlinear equation
F(y) = 0.
The components of y are yi−1 = θi, i = 2, . . . , N ,
yN+i−3 = ϕi, 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 yN , then write
y := (z,yN ), and define a new function F(z) = F(z,yN ), where
F : RN−1 → RN−1.
2 Using the Krawczyk operator with BL = (mid[dF])−1 we can verify the
existence of a fixed point of z−BLF(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
‖In −UT [A]U‖∞ ≤ r < 1. (11)
Let β =
(N
Πj=1
Ujj
)−2
. Then
0 < β(1− r)N ≤ det (A) ≤ β(1 + r)N , for A ∈ [A] and AT = Aa. (12)
aC. 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.
Numerical Integration over unit sphere–by using spherical t-designs
Numerical Verification method
Proof. We consider a symmetric matrix A ∈ [A]. Noting that UTAU
preserves the symmetric structure, we denote its (real) eigenvalues by
λi(UTAU). Since
max1≤i≤N
| 1− λi(UTAU) |= ρ(In −UTAU
)≤ ‖In −UTAU‖∞ ≤ r,
where ρ is the spectral radius, we have
0 < 1− r ≤ λi(UTAU) ≤ 1 + r, i = 1, . . . , N.
Hence,
(1− r)N ≤ det(UTAU
)≤ (1 + r)N .
Noting that det (U) det(UT)
=
(N
Πj=1
Ujj
)2
= β−1, from
0 < (1− r)N ≤ β−1 det (A) ≤ (1 + r)N ,
we obtain (12). 2
Numerical Integration over unit sphere–by using spherical t-designs
Numerical Verification method
In practical computation for Ht
1 Choose a preconditioning matrix U s.t (U−1)TU−1 = mid[Ht]
2 Conduct all operations in machine interval arithmetic and get an interval
enclosing ‖In −UT [Ht]U‖∞.
‖In −UT [Ht]U‖∞ =‖UT ((U−1)TU−1 − [Ht])U‖∞ (13a)
=‖UT (mid(Ht)− [Ht])U‖∞ (13b)
≤‖UT ‖∞‖rad(Ht)‖∞‖U‖∞ < 1, (13c)
3
[log det (Ht(XN ))] ⊆[b, b
](14)
for all XN ∈ [XN ], where
b = log β +N log (1− r) and b = log β +N log (1 + r) .
Numerical Integration over unit sphere–by using spherical t-designs
Numerical results of verification method
IV
Numerical results of verification
methodFor N = (t+ 1)2, det(Gt(XN )) = det(Ht(XN )).
Using an Extremal system 1as a initial point set.
Based on the MATLAB toolbox INTLAB 2§3.
1I. H. Sloan and R. S. Womersley, Extremal systems of points and numerical
integration on the sphere, Adv. Comput. Math., 21 , pp. 107–125(2004).2X. Chen, A. Frommer and B. Lang, Computational existence proof for
spherical t-designs, Numer. Math., 117 (2011), pp. 289-2053S. M. Rump, INTLAB – INTerval LABoratory, in Developments in Reliable
Computing, T. Csendes, ed., Dordrecht, Kluwer Academic Publishers, pp.
77–104, 1999.
Numerical Integration over unit sphere–by using spherical t-designs
Numerical results of verification method
For t = 1, . . . , 151 with N = (t+ 1)2
1 max diam([XN ]) represents the maximum diameter of all computed
enclosures for the parametrization of the respective spherical t-design.
2 [log det(Gt(XN ))] is over 104 for the largest t.
Numerical Integration over unit sphere–by using spherical t-designs
Numerical results of verification method
Figure: The diameters of [XN ]
Numerical Integration over unit sphere–by using spherical t-designs
Numerical results of verification method
Figure: Middle point values and diameters of [log det(Gt(XN ))]
Numerical Integration over unit sphere–by using spherical t-designs
Numerical results of verification method
GeometrySeparation distance–well separated spherical t-design
δXN := minxi,xj∈XN ,i 6=j
dist (xi,xj) ≥π
2t≥ π
2√N.
Figure: The separation of XN with N = (t+ 1)2
Numerical Integration over unit sphere–by using spherical t-designs
Numerical results of verification method
Existence of well separated spherical t-designs
For each even N ≥ Cdtd, there exists of a well separated
spherical t-design in the sphere Sd consisting of N points,where Cd is a constant depending only on d4.
4Bondarenko. A., Radchenko, D. Viazovska, M, Well-Separated Spherical
Designs,Constructive Approximation February 2015, Volume 41, Issue 1, pp
93-112
Numerical Integration over unit sphere–by using spherical t-designs
Numerical results of verification method
GeometryMesh norm
hXN := maxy∈S2
minxi∈XN
dist(y,xi) ≤4.8097
t,
Figure: The mesh norm of XN with N = (t+ 1)2
Numerical Integration over unit sphere–by using spherical t-designs
Numerical results of verification method
GeometryMesh ratio ρXN :=
2hXNδXN
≥ 1
Figure: The mesh ratio of extremal spherical t-designs with N = (t+ 1)2
Numerical Integration over unit sphere–by using spherical t-designs
Numerical results of verification method
Conjecture on S2
Let Cδ, Ch be constants. A lower bound on the separation of well
conditioned spherical t-designs for
δXN ≥ CδN− 1
2 ,
combined with the known upper bounds on mesh norm
hXN ≤ ChN12
would give the uniform bound
ρXN ≤2ChCδ
independent of t, N. (15)
Numerical Integration over unit sphere–by using spherical t-designs
Numerical results of verification method
An example
Figure: Well conditioned 49 design with 2500 points
Numerical Integration over unit sphere–by using spherical t-designs
Numerical results of verification method
A question
Can we verify Womersley’s efficient sphericalt-designs successfully by using Interval analysis?
Numerical Integration over unit sphere–by using spherical t-designs
Numerical Integrations over unit sphere
V
Numerical Integrations over unit sphere
1 Bivariate trapezoidal rule5, with q = 2.5.
2 Well conditioned spherical t-designs
3 Equal area points6
5K. Atkinson. Quadrature of singular integrands over surfaces. Electron.
Trans. Numer. Anal., 7:133–150, 2004.6EA. Rakhmanov, EB. Saff, and Y. Zhou, Minimimal discrete energy on the
sphere. Math. Res. Lett., 11(6): 647–662, 1994
Numerical Integration over unit sphere–by using spherical t-designs
Numerical Integrations over unit sphere
Bivariate trapezoidal rule
For the problem of approximate
I(f) =
∫S2f(x)dω(x)
in which f is several times continuously differentiable over the
unit sphere S2, we can use spherical coordinates to rewrite it as
I(f) =
∫ π
0
∫ 2π
0
f(cosφ sin θ, sinφ sin θ, cos θ) sin θdφdθ
.
Numerical Integration over unit sphere–by using spherical t-designs
Numerical Integrations over unit sphere
We use a transformation L : S2 → S2 With respect to
spherical coordinates on S2
L : x = (cosφ sin θ, sinφ sin θ, cos θ) 7→ (16)
x =(cosφ sinq θ, sinφ sinq θ, cos θ)√
cos2 θ + sin2q θ= L(θ, φ). (17)
In this transformation, q ≥ 1 is a ‘grading parameter’, The
north and south poles of S2 remain fixed, while the region
around them is distorted by the mapping. if we chose a higher
q, the area near two poles will have more points and equator
area are more sparser.
Numerical Integration over unit sphere–by using spherical t-designs
Numerical Integrations over unit sphere
The integral I(f) becomes
I(f) =
∫S2f(Lx)JL(x)dω(x)
with JL(x) the jacobian of the mapping L,
JL(x) = |DφL(θ, φ)×DθL(θ, φ)| =sin2q−1 θ(q cos2 θ + sin2 θ)
(cos2θ + sin2q θ)32
.
In spherical coordinates,
I(f) =
∫ π
0
sin2q−1 θ(q cos2 θ + sin2 θ)
(cos2θ + sin2q θ)32
∫ 2π
0
f(ξ, η, ζ)dφdθ,
(ξ, η, ζ) =(cosφ sinq θ, sinφ sinq θ, cos θ)√
cos2 θ + sin2q θ(18)
Numerical Integration over unit sphere–by using spherical t-designs
Numerical Integrations over unit sphere
For n ≥ 1, let h = π/n, and
φj = θj = jh∫ π
0
∫ 2π
0
g(sin θ, cos θ, sinφ, cosφ) dφ dθ
≈ h2n−1∑k=1
2n∑j=1
g(sin θk, cos θk, sinφj, cosφj) ≡ In,
g =sin2q−1 θ(q cos2 θ + sin2 θ)
(cos2θ + sin2q θ)32
f(ξ, η, ζ).
Error satisfies
I − In = O(hk) f ∈ Ck(S2)
Numerical Integration over unit sphere–by using spherical t-designs
Numerical Integrations over unit sphere
Equal-Area Points
The equal-area points aim to achieve a partition T of the sphere into a
user-chosen number of N of subsets Tj each of which has the same area
|Tj | =4π
N, j = 1, . . . , N,
and
diam(Tj) ≤c√N,
Then we obtain the equal weight rule
INf :=4π
N
N∑j=1
f(xj).
We have
|If − INf | ≤4πσc√N
Numerical Integration over unit sphere–by using spherical t-designs
Numerical Integrations over unit sphere
Geometry of different nodes
(a) Bi. trapezoidal rule (b) Equal area points (c) spherical t-design
Numerical Integration over unit sphere–by using spherical t-designs
Numerical Integrations over unit sphere
Franke1 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)/49− (9y + 1)/10− (9z + 1)/10)
+ 0.5 exp(−(9x− 7)2/4− (9y − 3)2/4− (9z − 5)2/10)
+ 0.2 exp(−(9x− 4)2 − (9y − 7)2 − (9z − 5)2/10)
Figure: f1
Numerical Integration over unit sphere–by using spherical t-designs
Numerical Integrations over unit sphere
C0 function
f2(x) = sin2(1 + ||x||1)/10
is not continuously differentiable at points where any component of x is zero.
Figure: f2
Numerical Integration over unit sphere–by using spherical t-designs
Numerical Integrations over unit sphere
Nearby singular function
f3(x, y, z) = 1/(101− 100z)
is analytic over S2, it has a pole just off the surface of the sphere at
x = (0, 0, 1.01), in other word, f3((0, 0, 1.01)) = inf.
Figure: f3
Numerical Integration over unit sphere–by using spherical t-designs
Numerical Integrations over unit sphere
Cap function with R = 13 , center x0 = (0, 0, 1)T .
f4(x) =
cos2(π2
dist(x, x0)/R) if dist(x, x0) < R
0 if dist(x, x0) ≥ R,
Figure: f4
Numerical Integration over unit sphere–by using spherical t-designs
Performance of Numerical Integrations
VI
Performance of Numerical
Integrations
Numerical Integration over unit sphere–by using spherical t-designs
Performance of Numerical Integrations
Performance of Numerical Integrations
function exact integration values
f1 6.6961822200736179523
f2 0.45655373990000
f3 π log 201/50∗
f4 0.103351∗
Absolute error = |If − Inf | (19)
Numerical Integration over unit sphere–by using spherical t-designs
Performance of Numerical Integrations
Figure: f1
Numerical Integration over unit sphere–by using spherical t-designs
Performance of Numerical Integrations
Figure: f2
Numerical Integration over unit sphere–by using spherical t-designs
Performance of Numerical Integrations
Figure: f3
Numerical Integration over unit sphere–by using spherical t-designs
Performance of Numerical Integrations
Figure: f4
Numerical Integration over unit sphere–by using spherical t-designs
Performance of Numerical Integrations
Final Remark
1 Well conditioned Spherical t-design is a useful tool to deal
with numerical integration over the sphere.
2 Can we give a sharp error analysis for numerical
integration (with different nodes) over the sphere as
results on [−1, 1]?
3 How to set up a efficient quadrature rule when integrand
with special properties? such as highly oscillatory
integrals, potential integrals....
Numerical Integration over unit sphere–by using spherical t-designs
Performance of Numerical Integrations
Thank you very much!
Numerical Integration over unit sphere–by using spherical t-designs
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