Quadrature using sparse grids on products of spheres

Transcription

1 Quadrature using sparse grids on products of spheres Paul Leopardi Mathematical Sciences Institute, Australian National University. For presentation at ANZIAM NSW/ACT Annual Meeting, Batemans Bay. Joint work with Markus Hegland, ANU. December 2009

2 Topics Weighted tensor product spaces on spheres Component-by-component construction Weighted tensor product quadrature Numerical results Accomplishments and next steps

3 Weighted tensor product spaces on spheres A product of two circles is a torus... ( Torus cycles - Wikipedia, 2005)

4 Weighted tensor product spaces on spheres Polynomials on the unit sphere Sphere S 2 := {x R 3 3 k=1 x2 k = 1}. P µ : spherical polynomials of degree at most µ. H l : spherical harmonics of degree l, dimension 2l + 1. P µ = µ l=0 H l has spherical harmonic basis {Y l,k l 0... µ, k l + 1}.

5 Weighted tensor product spaces on spheres Reproducing kernel Hilbert space H on X A reproducing kernel Hibert space H of real functions on a manifold X is a Hilbert space with inner product, and a kernel K : X X R, such that for all x X, if k x is defined by k x (y) := K(x, y) for all y X, then k x H and k x, f = f(x) for all f H.

6 Weighted tensor product spaces on spheres KS function space H (r) 1,γ on a single sphere For f L 2 (S 2 ), f(x) 2l+1 ˆf l=0 k=1 l,k Y l,k (x). For positive weight γ, Reproducing Kernel Hilbert Space H (r) 1,γ := {f : S2 R f 1,γ < }, where f 1,γ := f, f 1/2 γ and f, g 1,γ := ˆf 0,0 ĝ 0,0 + γ 1 (Kuo and Sloan, 2005) 2l+1 l=1 k=1 ( l(l + 1) )r ˆf l,k ĝ l,k.

7 Weighted tensor product spaces on spheres Reproducing kernel of H (r) 1,γ This is K (r) 1,γ (x, y) := 1 + γa r(x y), where for z [ 1, 1], 2l + 1 A r (z) := ( ) r P l (z), l(l + 1) l=1 where P is a Legendre polynomial. (Kuo and Sloan, 2005)

8 Weighted tensor product spaces on spheres The weighted tensor product space H (r) d,γ For γ := (γ 1,..., γ d ), on (S 2 ) d define the tensor product space H (r) d,γ := d j=1 H(r) 1,γ j. Reproducing kernel of H (r) d,γ is K d,γ (x, y) := d j=1 K (r) 1,γ j (x j, y j ) = d ( 1 + γj A r (x j y j ) ). j=1 (Kuo and Sloan, 2005)

9 Weighted tensor product spaces on spheres Equal weight quadrature error on H (r) d,γ Worst case error of equal weight quadrature Q m,d with m points: e 2 m,d (Q m,d) := sup f H (r) d,γ ((I Q m,d )f) 2 = m m K m 2 d,γ (x i, x h ) = m 2 m E(e 2 m,d ) = 1 ( 1 + m (Kuo and Sloan, 2005) i=1 h=1 m i=1 h=1 j=1 d ( 1 + γj A r (x i,j x h,j ) ), d ( 1 + γj A r (1) )) j=1 1 m exp ( A r (1) d ) γ j. j=1

10 Component-by-component construction Spherical designs on S 2 A spherical design of strength t on S 2 is an equal weight quadrature rule Q with m points (x 1,..., x m ), Qf := m k=1 f(x k), such that, for all p P t (S 2 ), Q p = p(y) dω(y)/ S 2. S 2 The linear programming bounds give t = O(m 1/2 ). Spherical designs of strength t are known to exist for m = O(t 3 ) and conjectured for m = (t + 1) 2. Spherical t -designs have recently been found numerically for m (t + 1) 2 /2 + O(1) for t up to 126. (Delsarte, Goethals and Seidel, 1977; Hardin and Sloane, 1996; Chen and Womersley, 2006; Womersley, 2008)

11 Component-by-component construction Construction using permutations The idea of Hesse, Kuo and Sloan, 2007 for quadrature on (S 2 ) d is to use a spherical design z = (z 1,..., z m ) of strength t for the first sphere and then successively permute the points of the design to obtain the coordinates for each subsequent sphere. The algorithm chooses permutations Π 1,..., Π d : 1... m 1... m, giving x i = (z Π1 (i),..., z Πd (i)) to ensure that the resulting squared worst case quadrature error is better than the average E(e 2 m,d ). (Hesse, Kuo and Sloan, 2007)

12 Component-by-component construction Error estimate for permutation construction Hesse, Kuo and Sloan prove that if (z 1,..., z m ) is a spherical t -design with m = O(t 2 ) or if r > 3/2 and m = O(t 3 ) for t large enough, then D 2 m := e2 m,1 γ1 =1 = 1 m 2 m A r(1) m. i=1 h=1 m A r (z Πj (i) z Πj (h)) This ensures that for m large enough, Mm,d 2, the average squared worst case error over all permutations, satisfies M 2 m,d E(e2 m,d ). (Hesse, Kuo and Sloan, 2007)

13 Weighted tensor product quadrature Weighted Korobov spaces Consider s = 1. H (1,r) 1,γ is a RKHS on the unit circle S 1, H (1,r) d,γ is a RKHS on the d -torus. This is a weighted Korobov space of periodic functions on [0, 2π) d. The Hesse, Kuo and Sloan construction in these spaces gives a rule with the same 1-dimensional projection properties as a lattice rule: the points are equally spaced. (Wasilkowski and Woźniakowski, 1999; Hesse, Kuo and Sloan, 2007)

14 Weighted tensor product quadrature The Smolyak construction on S 1 The Smolyak construction and variants have been well studied on unweighted and weighted Korobov spaces. Smolyak construction (unweighted case): For H (1,r) 1,1, define Q 1, 1 := 0 and define a sequence of equal weight rules Q 1,0, Q 1,1,... on [0, 2π), exact for trigonometric polynomials of degree t 0 = 0 < t 1 <.... Define q := Q 1,q Q 1,q 1 and for H (1,r) d,1, define Q d,q := 0 a a d q a1... ad. (Smolyak, 1963; Wasilkowski and Woźniakowski, 1995; Gerstner and Griebel, 1998)

15 Weighted tensor product quadrature The WTP variant of Smolyak on H (1,r) d,γ The WTP algorithm of Wasilkowski and Woźniakowski (1999) generalizes Smolyak by treating spaces of non-periodic functions, by allowing optimal weights, and by allowing other choices for the index sets a. For H (1,r) d,γ, define W d,n := a P n,d (γ) a1... ad, where P 1,d (γ) P 2,d (γ) N d, P n,d (γ) = n. W and W (1999) suggests to define P n,d (γ) by including the n rules a1... ad with largest norm. (Wasilkowski and Woźniakowski, 1999)

16 Weighted tensor product quadrature Optimal weight for one quadrature point (Illustration by Osborn, 2009)

17 Weighted tensor product quadrature Optimal weights for two quadrature points (Illustration by Osborn, 2009)

18 Weighted tensor product quadrature WTP rules using spherical designs For H (r) d,γ we can define a WTP rule based on spherical designs. Define a sequence of optimal weight rules Q 0, Q 1,... using unions of spherical designs of increasing strength t 0 = 0 < t 1 <... and cardinality m 0 = 1 < m 1 <.... The WTP construction then proceeds similarly to S 1. One difference between S 1 and S 2 is that the spherical designs themselves cannot be nested in general. (Wasilkowski and Woźniakowski, 1999)

19 Weighted tensor product quadrature Generic WTP algorithm for S 2 1. Begin with a sequence of spherical designs X 1, X 2,... X L, with increasing cardinality, nondecreasing strength. 2. For each h, form the optimal weight rule Q h from the point set h i=1 X i, and the difference rule h = Q h Q h Form products of the difference rules and rank them in decreasing norm (possibly weighted by the number of additional points). 4. Form WTP rules by adding product difference rules in rank order.

20 Numerical results The Hesse, Kuo and Sloan example space In Hesse, Kuo and Sloan, a numerical example is given with r = 3, γ j = 0.9 j. In other words, K d,γ (x, y) := where A 3 (z) = d j=1 l=1 K (3) 1,0.9 j (x j, y j ) = 2l + 1 ( l(l + 1) ) 3 P l (z). d ( j A 3 (x j y j ) ), j=1

21 Numerical results Error of WTP rule for S Error Error 2 Error 4 Error 8 Error Points

22 Numerical results Estimated upper bound of error of WTP rule log 10 error WTP algorithm: Upper bound on cost of given error: d=2 k, p γ =0.001, ν= log 10 cost

23 Numerical results Comparisons for 441, 961 and 2601 points 1.0 Worst case error of WTP for fixed number of points 0.8 Error HKS Dimension

24 Numerical results Weights for two quadrature points reconsidered (Illustration by Osborn, 2009)

25 Accomplishments and next steps Accomplishments Formulation of WTP algorithm for products of S 2. Implementation of WTP algorithm. Numerical results for d to 30 and up to points. Estimate for upper bound of error of WTP rule.

26 Accomplishments and next steps To do More error estimates for WTP rules. Lower bounds on error; initial rate of convergence. Improvement of WTP algorithm to obtain better initial rate of convergence. Best rate of increase of strength of spherical designs. Should it double very step? Best index sets. What is the best way to take weights into account? More numerical experiments.