NEEDLET APPROXIMATION FOR ISOTROPIC RANDOM FIELDS ON THE SPHERE

Transcription

1 NEEDLET APPROXIMATION FOR ISOTROPIC RANDOM FIELDS ON THE SPHERE Quoc T. Le Gia Ian H. Sloan Yu Guang Wang Robert S. Womersley School of Mathematics and Statistics University of New South Wales, Australia ANZIAM Symposium 2015, Wollongong Dedicated to Professor Jim Hill s 70th Birthday

2 Random fields on the sphere Applications

3 Random fields on the sphere Applications Cosmology

4 Random fields on the sphere Applications Cosmology Climate model

5 Random fields on the sphere Probability measure space (Ω, P) Unit sphere S 2 of R 3 Definition An F B(S 2 )-measurable function T : Ω S 2 R is said to be a real-valued random field on the sphere S 2. Notation: T (ω, x) or T (x) or T (ω)

6 Two-weakly isotropic random fields on the sphere If for all x S 2, E [ T (x) 2] < + and if for x 1, x 2 S 2 and for any rotation ρ SO(3), E [T (x 1 )] = E [T (ρx 1 )] E [T (x 1 )T (x 2 )] = E [T (ρx 1 )T (ρx 2 )].

7 An example: Gaussian random field on S 2, scaling factor δ = 1/5 Fig. One realisation of T 1/5,300, s = 1 T δ,m (ω, x) = M 2l+1 l=1 m=1 a lm (ω)y l,m (x), where a l,m N (0, σ 2 ) with { σ 2 A0, l = 0, := A l /2, l 1, where A l := 1/(1 + δl) 2s+2. Lang & Schwab, 2015.

8 An example: Gaussian random field on S 2, scaling factor δ = 1/20 Fig. One realisation of T 1/20,300, s = 1 T δ,m (ω, x) = M 2l+1 l=1 m=1 a lm (ω)y l,m (x), where a l,m N (0, σ 2 ) with { σ 2 A0, l = 0, := A l /2, l 1, where A l := 1/(1 + δl) 2s+2. Lang & Schwab, 2015.

9 What is a needlet approximation of a random field? For a random field T on S 2, J = 0, 1,..., Definition (Semidiscrete needlet approximation) N J j VJ need (T ; ω, x) := (T (ω), ψ jk ) L2 (S 2 ) ψ jk(x), x S 2, ω Ω, j=0 k=1 where (T (ω), ψ jk ) L2 (S 2 ) := S 2 T (ω, y)ψ jk (y) dσ 2 (y).

10 Needlet ψ jk, j = 9, k = 10 Fig. ψ jk with a C 5 (R +)-filter, tensor rule (Gauss-Legendre Equal)

11 Quadrature rule for needlets for ψ jk, j = 5 Fig. A symmetric spherical 31-design Nodes x jk S 2, k = 1,..., 498. Exact for polynomials P of degree up to = 31 S 2 P (x) dσ 2 (x) = 1 N Womersley, N P (x jk ). k=1

12 Localised zonal polynomial: needlet ψ jk, j = 9, k = 10 Estimate Fig. ψ jk with a C 5 -needlet filter, tensor rule (Gauss-Legendre Equal)

13 Dubai Tower A needlet with lower resolution

14 Tight frame Proposition (Almost orthogonal) The needlets ψ jk, ψ j k are L 2-orthogonal if j j 2: ( ψjk, ψ j k ) L 2 (S 2 ) = 0.

15 Tight frame Proposition (Almost orthogonal) The needlets ψ jk, ψ j k are L 2-orthogonal if j j 2: ( ψjk, ψ j k ) L 2 (S 2 ) = 0. Proposition (Parseval indentity) N j (f, ψjk ) 2 L2 (S 2 ) = f L2 (S 2 ), f L 2 (S 2 ). j=0 k=1 Narcowich et al., 2006.

16 Does the needlet approximation converge in L 2 ( Ω S 2 )? Theorem The needlet approximation with smooth filter of a two-weakly isotropic random field on S 2 converges in L 2 ( Ω S 2 ) -norm. If h C κ (R + ), κ 3, J (T ) T L2 (Ω S 2 ) = 0. lim V need J

17 Does the needlet approximation converge in L 2 ( Ω S 2 )? Theorem The needlet approximation with smooth filter of a two-weakly isotropic random field on S 2 converges in L 2 ( Ω S 2 ) -norm. If h C κ (R + ), κ 3, lim V need J J (T ) T L2 (Ω S 2 ) = 0. Equivalently, [ V lim E need J (T ) T ] 2 J L 2 = 0. (S 2 )

18 How about the needlet approximation of smooth random fields? Theorem The needlet approximation with smooth filter of a two-weakly isotropic random field on S 2 converges at rate 2 Js in L 2 ( Ω S 2 ) -norm if T W s 2 (S2 ) P a.s., s > 0. If h C κ (R + ), κ 3, VJ need (T ) T L2 (Ω S 2 ) C 2 Js, [ ] where C := c h,s E T 2. W s 2 (S2 )

19 How to implement needlet approximations? Given J = 0, 1,... Discretisation quadrature Q N := {(W i, y i ) : i = 1,..., N} Exact for polynomials of degree 3 2 J 1 1 Need NOT be the same as needlet quadrature Approximate needlet coefficient by quadrature Q N (T (ω), ψ jk ) L2 (S 2 ) S = T (ω, y)ψ jk (y) dσ 2 (y) d N W i T (ω, y i ) ψ jk (y i ) =: (T (ω), ψ jk ) QN i=1

20 How to implement needlet approximations? Given J = 0, 1,... Discretisation quadrature Q N := {(W i, y i ) : i = 1,..., N} Exact for polynomials of degree 3 2 J 1 1 Need NOT be the same as needlet quadrature Definition (Fully discrete needlet approximation) V need J,N (T ; ω, x) := N J j (T (ω), ψ jk ) QN ψ jk (x), ω Ω, x S 2. j=0 k=1

21 Discretisation does not lower the convergence rate Given J = 0, 1,..., let Q N be a discretisation quadrature exact for degree 3 2 J 1 1. Theorem The discrete needlet approximation with smooth filter and Q N of a two-weakly isotropic random field on S 2 converges at rate 2 Js in L 2 ( Ω S 2 ) -norm if T W s 2 (S2 ) P a.s., s > 0. If h C κ (R + ), κ 3, VJ,N need (T ) T L2 (Ω S 2 ) C 2 Js, where C := c h,s E [ ] T 2. W s 2 (S2 )

22 Discrete needlet approximation for one realisation of T 1/5,300 (a) Original (b) Discrete needlet approximation Fig. Discrete needlet approximation VJ,N need for one realisation of the scaled Gaussian random field T δ,m, δ = 1/5, M = 300, s = 1, J = 7, h C 5 (R +)

23 Pointwise errors Fig. Discrete needlet approximation VJ,N need for one realisation of the scaled Gaussian random field T δ,m, δ = 1/5, M = 300, s = 1, J = 7, h C 5 (R +)

24 Relative L 2 -errors [ ] err rela := E T VJ,N need(t ) 2 L 2 (S 2 ) [ ]. E T 2 W s 2 (S2 )

25 Relative L 2 -errors Relative Errors s = 1 2.9e J All samples s = 2 2.3e J All samples Order J (a) δ = s = 1 2.9e J All samples s = 2 7.7e J All samples Order J (b) δ = 1/5 Fig. Relative L 2-errors for discrete needlet approximations of T δ,m, M = 300, J = 0,..., 7, h C 5 (R +), T δ,m W s 2(S 2 ) P a.s., s = 1, 2

26 More facts... Pointwise errors: L 2 (S 2 )-errors of VJ need (T ) and VJ,N need (T ) converge asymptotically at rate 2 Jr, r < s, P-almost surely. Approximations are stable: Variances of errors converge to zero as order J. All results can be generalised to S d, d 3. Convergence of semidiscrete needlet approximation can be generalised to L p ( Ω S d ), 1 p, d 2.

27 References F. Narcowich, P. Petrushev, and J. Ward. Decomposition of Besov and Triebel-Lizorkin spaces on the sphere. J. Funct. Anal., 238(2): , F. J. Narcowich, P. Petrushev, and J. D. Ward. Localized tight frames on spheres. SIAM J. Math. Anal., 38(2): , A. A. Lang and Ch. C. Schwab. Isotropic Gaussian random fields on the sphere: regularity, fast simulation, and stochastic partial differential equations. Ann. Appl. Probab., In press. Q. T. Le Gia, I. H. Sloan, Y. G. Wang and R. S. Womersley. Needlet approximation for isotropic random fields on the sphere. Preprint. Y. G. Wang, Q. T. Le Gia, I. H. Sloan and R. S. Womersley. Fully discrete needlet approximation on the sphere. Submitted. R. S. Womersley. Efficient spherical designs with good geometric properties

28 The End Thank you!

29 Gaussian random fields on spheres We say T is a Gaussian random field (GRF) on S d if the vector (T (x 1 ),..., T (x k )) follows the multivariate Gaussian distribution for all k 1 and x 1,..., x k S d.

30 Gaussian random fields on spheres We say T is a Gaussian random field (GRF) on S d if the vector (T (x 1 ),..., T (x k )) follows the multivariate Gaussian distribution for all k 1 and x 1,..., x k S d. Remark (i) If the law of T is determined by its moments, then T is -weakly isotropic if and only if T is strongly isotropic and E [ T (x) ν ] < + for every integer ν 2 and each x S d ; (ii) Let T be a GRF on S d. Then, T is strongly isotropic if and only if T is 2-weakly isotropic.

31 Needlet filter Given κ 0, a needlet filter h is a real function on R + satisfying (i) (Compact support.) h C κ (R + ), supp h [1/2, 2]; (ii) (Partition of unity.) For all t 1, j=0 [ ( )] t 2 h 2 j = 1.

32 Localisation of needlets Let d 2 and h C κ (R + ) with κ 1. ψ jk (x) c d,h 2 jd (1 + 2 j dist(x, x jk )) κ, x S d. [Mhaskar. J. Approx. Theory 2004] [Petrushev, Xu. J. Fourier Anal. Appl. 2005] [Narcowich, Petrushev, Ward. SIAM J. Math. Anal., J. Funct. Anal. 2006] Back