Numerical Integration over unit sphere by using spherical t-designs

Transcription

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

2 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

3 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

4 Spherical coordinates x 1 = 0 0 1, x2 = sin(θ 2) 0 cos(θ 2), xi = sin(θ i) cos(φ i) sin(θ i) sin(φ i) cos(θ i), i = 3,..., N. (1)

5 Background on spherical t designs Part I Background on spherical t designs

6 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 1 N p (x j) = 1 p(x)dω(x) p P t, (2) N 4π S 2 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].

7 Background on spherical t designs Real Spherical harmonics Real Spherical harmonics[14] Y lk : k = 1,..., 2l + 1, l = 0, 1,..., t Basis P t=span{y lk : k = 1,..., 2l + 1, l = 0, 1,..., t} Orthonormality with respect to L 2 inner product (p, q) L2 = p(x)q(x)dω(x), S 2 Normalization Y 0,1 = 1 4π dim P t = (t + 1) 2 2l+1 Addition Theorem k=1 Y l,k (x)y l,k (y) = 2l+1 4π P l (x y), x, y S2

8 Background on spherical t designs Spherical harmonic basis matrix For t 1, and N dim(p t) = (t + 1) 2, let Y 0 t N matrix defined by be the ((t + 1) 2 1) by Yt 0 (X N ) := [Y l,k (x j)], k = 1,..., 2l + 1, l = 1,..., t; j = 1,..., N, (3) [ ] Y t(x N ) := where e = [1,..., 1] T R N. 1 4π e T Y 0 t (X N ) R (t+1)2 N, (4) 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.

9 Background on spherical t designs Nonlinear system C t (X N ) = 0 Let N (t + 1) 2, define C t : (S d ) N R, where the N N Gram matrix G t for X N S 2 C t(x N ) = EG t(x N )e (5) e = G t (X N ) = Y t (X N ) T Y t (X N ) R N, E = [1, I] R (N 1) N, 1 = [1,..., 1] T R N 1 (6)

10 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.

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

12 Numerical Integration over unit sphere by using 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.

13 Numerical Integration over unit sphere by using t-designs Well conditioned spherical designs Optimization Problem on S 2 max log det (H t(x N )) X N S 2 subject to C t (X N ) = 0. (7) 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) 4π

14 Numerical Verification method III Numerical Verification method

15 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 = 2rad [a], 5 F : D R n R n be a continuously differentiable function.let [df] IR n 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].

16 Numerical Verification method Krawczyk operator Definition (Krawczyk operator,[11]) Given a nonsingular matrix B L R n n, ž [z] D and [df] IR n n, the Krawczyk operator [11] is defined by: k F (ž, [z], B L, [df]) := ž B LF(ž) + (I n B L [df])([z] ž). (10) It is known that k F (ž, [z], B L, [df]) is an interval extension of the function ψ(z) := z B LF(z) over [z], that is, z B LF(z) k F (ž, [z], B L, [F]) for all z [z].

17 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] IR n, ž [z] D, an invertible matrix B L R n n and [df] IR n n such that F (ξ) [df] for all ξ [z]. Assume that k F (ž, [z], B L, [df]) [z]. Then F has a zero z in k F (ž, [z], B L, [df]).

18 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.

19 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[df]) 1 we can verify the existence of a fixed point of z B LF(z), which is a solution of F(z) = 0.

20 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) ( ) NΠ 2 Let β =. Then j=1 Ujj 0 < β(1 r) N det (A) β(1 + r) N, for A [A] and A T = A a. (12) 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

21 Numerical Verification method Proof. We consider a symmetric matrix A [A]. Noting that U T AU preserves the symmetric structure, we denote its (real) eigenvalues by λ i(u T AU). Since ( ) max 1 1 i N λi(ut AU) = ρ I n U T AU I n U T AU r, where ρ is the spectral radius, we have 0 < 1 r λ i(u T AU) 1 + r, i = 1,..., N. Hence, ( ) (1 r) N det U T AU (1 + r) N. Noting that det (U) det ( U ) ( ) NΠ 2 T = = β 1, from j=1 Ujj 0 < (1 r) N β 1 det (A) (1 + r) N, we obtain (12).

22 Numerical Verification method In practical computation for H t 1 Choose a preconditioning matrix U s.t (U 1 ) T U 1 = mid[h t] 2 Conduct all operations in machine interval arithmetic and get an interval enclosing I n U T [H t]u. I n U T [H t]u = U T ((U 1 ) T U 1 [H t])u = U T (mid(h t) [H t])u U T rad(h t) U < 1, (13a) (13b) (13c) 3 [log det (H t(x N ))] [ b, b ] (14) for all X N [X N ], where b = log β + N log (1 r) and b = log β + N log (1 + r).

23 Numerical results of verification method IV Numerical results of verification method For N = (t + 1) 2, det(g t(x N )) = det(h t(x N )). Using an Extremal system 1 as a initial point set. Based on the MATLAB toolbox INTLAB I. H. Sloan and R. S. Womersley, Extremal systems of points and numerical integration on the sphere, Adv. Comput. Math., 21, pp (2004). 2 X. Chen, A. Frommer and B. Lang, Computational existence proof for spherical t-designs, Numer. Math., 117 (2011), pp S. M. Rump, INTLAB INTerval LABoratory, in Developments in Reliable Computing, T. Csendes, ed., Dordrecht, Kluwer Academic Publishers, pp , 1999.

24 Numerical results of verification method For t = 1,..., 151 with N = (t + 1) 2 1 max diam([x N ]) represents the maximum diameter of all computed enclosures for the parametrization of the respective spherical t-design. 2 [log det(g t(x N ))] is over 10 4 for the largest t.

25 Numerical results of verification method Figure: The diameters of [X N ]

26 Numerical results of verification method Figure: Middle point values and diameters of [log det(g t (X N ))]

27 Numerical results of verification method Geometry Separation distance well separated spherical t-design δ XN := min dist (xi, xj) π x i,x j X N,i j 2t π 2 N. Figure: The separation of X N with N = (t + 1) 2

28 Numerical results of verification method Existence of well separated spherical t-designs For each even N C d t d, there exists of a well separated spherical t-design in the sphere S d consisting of N points, where C d is a constant depending only on d 4. 4 Bondarenko. A., Radchenko, D. Viazovska, M, Well-Separated Spherical Designs,Constructive Approximation February 2015, Volume 41, Issue 1, pp

29 Numerical results of verification method Geometry Mesh norm h XN := max min dist(y, x i) , y S 2 x i X N t Figure: The mesh norm of X N with N = (t + 1) 2

30 Numerical results of verification method Geometry Mesh ratio ρ XN := 2h X N δ XN 1 Figure: The mesh ratio of extremal spherical t-designs with N = (t + 1) 2

31 Numerical results of verification method Conjecture on S 2 Let C δ, C h be constants. A lower bound on the separation of well conditioned spherical t-designs for δ XN C δ N 1 2, combined with the known upper bounds on mesh norm would give the uniform bound h XN C h N 1 2 ρ XN 2C h C δ independent of t, N. (15)

32 Numerical results of verification method An example Figure: Well conditioned 49 design with 2500 points

33 Numerical results of verification method A question Can we verify Womersley s efficient spherical t-designs successfully by using Interval analysis?

34 Numerical Integrations over unit sphere V Numerical Integrations over unit sphere 1 Bivariate trapezoidal rule 5, with q = Well conditioned spherical t-designs 3 Equal area points 6 5 K. Atkinson. Quadrature of singular integrands over surfaces. Electron. Trans. Numer. Anal., 7: , EA. Rakhmanov, EB. Saff, and Y. Zhou, Minimimal discrete energy on the sphere. Math. Res. Lett., 11(6): , 1994

35 Numerical Integrations over unit sphere Bivariate trapezoidal rule For the problem of approximate I(f) = f(x)dω(x) S 2 in which f is several times continuously differentiable over the unit sphere S 2, we can use spherical coordinates to rewrite it as I(f) =. π 2π 0 0 f(cos φ sin θ, sin φ sin θ, cos θ) sin θdφdθ

36 Numerical Integrations over unit sphere We use a transformation L : S 2 S 2 With respect to spherical coordinates on S 2 L : x = (cos φ sin θ, sin φ sin θ, cos θ) (16) x = (cos φ sinq θ, sin φ sin q θ, cos θ) = L(θ, φ). (17) cos2 θ + sin 2q θ In this transformation, q 1 is a grading parameter, The north and south poles of S 2 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.

37 Numerical Integrations over unit sphere The integral I(f) becomes I(f) = f(lx)j L ( x)dω( x) S 2 with J L ( x) the jacobian of the mapping L, J L ( x) = D φ L(θ, φ) D θ L(θ, φ) = sin2q 1 θ(q cos 2 θ + sin 2 θ). (cos 2 θ + sin 2q θ) 3 2 In spherical coordinates, I(f) = π 0 sin 2q 1 θ(q cos 2 θ + sin 2 θ) (cos 2 θ + sin 2q θ) 3 2 2π (ξ, η, ζ) = (cos φ sinq θ, sin φ sin q θ, cos θ) cos2 θ + sin 2q θ 0 f(ξ, η, ζ)dφdθ, (18)

38 Numerical Integrations over unit sphere For n 1, let h = π/n, and φ j = θ j = jh π 2π 0 n 1 h 2 Error satisfies 0 2n k=1 j=1 g(sin θ, cos θ, sin φ, cos φ) dφ dθ g(sin θ k, cos θ k, sin φ j, cos φ j ) I n, g = sin2q 1 θ(q cos 2 θ + sin 2 θ) f(ξ, η, ζ). (cos 2 θ + sin 2q θ) 3 2 I I n = O(h k ) f C k (S 2 )

39 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 T j each of which has the same area T j = 4π N, j = 1,..., N, and diam(t j) c N, Then we obtain the equal weight rule I N f := 4π N N f(x j). j=1 We have If I N f 4πσc N

40 Numerical Integrations over unit sphere Geometry of different nodes (a) Bi. trapezoidal rule (b) Equal area points (c) spherical t-design

41 Numerical Integrations over unit sphere Franke1 function f 1(x, y, z) =0.75 exp( (9x 2) 2 /4 (9y 2) 2 /4 (9z 2) 2 /4) exp( (9x + 1)/49 (9y + 1)/10 (9z + 1)/10) exp( (9x 7) 2 /4 (9y 3) 2 /4 (9z 5) 2 /10) exp( (9x 4) 2 (9y 7) 2 (9z 5) 2 /10)

42 Numerical Integrations over unit sphere C 0 function f 2(x) = sin 2 (1 + x 1)/10 is not continuously differentiable at points where any component of x is zero. Figure: f 2

43 Numerical Integrations over unit sphere Nearby singular function f 3(x, y, z) = 1/( z) is analytic over S 2, it has a pole just off the surface of the sphere at x = (0, 0, 1.01), in other word, f 3((0, 0, 1.01)) = inf. Figure: f 3

44 Numerical Integrations over unit sphere Cap function with R = 1 3, center x 0 = (0, 0, 1) T. cos 2 ( π dist(x, x0)/r) if dist(x, x0) < R 2 f 4(x) = 0 if dist(x, x 0) R, Figure: f 4

45 Performance of Numerical Integrations VI Performance of Numerical Integrations

46 Performance of Numerical Integrations Performance of Numerical Integrations function exact integration values f f f 3 f 4 π log 201/ Absolute error = If I n f (19)

47 Performance of Numerical Integrations Figure: f 1

48 Performance of Numerical Integrations Figure: f 2

49 Performance of Numerical Integrations Figure: f 3

50 Performance of Numerical Integrations Figure: f 4

51 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...

52 Performance of Numerical Integrations Thank you very much!

53 REFERENCES REFERENCES

54 REFERENCES [1] 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 [2] C. An, X, Chen, I. H. Sloan, R. S. Womersley, Regularized least squares approximation on sphere using spherical designs, SIAM J. Numer. Anal. Vol. 50, No. 3, pp. 1513õ1534. [3] C.An, Distributing points on the unit sphere and spherical designs, Ph.D Thesis, The Hong Kong Polytechnic University, [4] E. Bannai, R.M. Damerell, Tight spherical designs I. Math Soc Japan 31(1): ,1979. [5] R. Bauer, Distribution of points on a sphere with application to star catalogs, J. Guidance, Control, and Dynamics, 23 (2000), pp [6] A. Bondarenko, D. Radchenko, and M. Viazovska, Optimal asymptotic bounds for spherical designs. ArXiv: v1[math.MG], [7] A. Bondarenko, D. Radchenko, and M. Viazovska,Well separated spherical designs. ArXiv: [math.mg],2013. [8] X. Chen and R. S. Womersley, Existence of solutions to systems of undetermined equations and spherical designs, SIAM J. Numer. Anal., 44 (2006), pp [9] X. Chen, A. Frommer and B. Lang, Computational existence proof for spherical t-designs, Numer. Math., 117 (2011), pp [10] P. Delsarte, J. M. Goethals and J. J. Seidel, Spherical codes and designs, Geom. Dedicata, 6 (1977), pp [11] R. Krawczyk, Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken, Computing, 4, pp (1969).

55 [12] R. E. Moore, A test for existence of solutions to nonlinear systems, SIAM J. Numer. Anal., 14, pp (1977). [13] S. M. Rump, INTLAB INTerval LABoratory, in Developments in Reliable Computing, T. Csendes, ed., Dordrecht, Kluwer Academic Publishers, pp , [14] C. Müller, Spherical Harmonics, vol. 17 of Lecture Notes in Mathematics, Springer Verlag, Berlin, New-York, [15] R. J. Renka, Multivariate interpolation of large sets of scattered data, ACM Trans. Math. Software, 14 (1988), pp [16] P.D. Seymour and T. Zaslavsky,Averaging sets: A generalization of mean values and spherical designs. Advances in Mathematics, vol. 52, pp ,1984. [17] I. H. Sloan, Polynomial interpolation and hyperinterpolation over general regions, J. Approx. Theory, 83 (1995), pp [18] I. H. Sloan and R. S. Womersley, Extremal systems of points and numerical integration on the sphere, Adv. Comput. Math., 21, pp (2004). [19] I. H. Sloan and R. S. Womersley, Constructive polynomial approximation on the sphere, J. Approx. Theory, 103 (2000), pp [20] I. H. Sloan and R. S. Womersley, Filtered hyperinterpolation: A constructive polynomial approximation on the sphere, Int. J. Geomath., 3, pp. 95õ117, 2012.