Hyperuniformity on the Sphere

Transcription

1 Hyperuniformity on the Sphere Peter Grabner Institute for Analysis and Number Theory Graz University of Technology Problem Session, ICERM, February 12, 2018

2 Two point distributions

3 Quantify evenness For every point set X N = {x 1,..., x N } of distinct points, we assign several qualitative measures that describe aspects of even distribution.

4 Quantify evenness For every point set X N = {x 1,..., x N } of distinct points, we assign several qualitative measures that describe aspects of even distribution. Then we can try to minimise or maximise these measures for given N.

5 Combinatorial measures discrepancy D N (X N ) = sup C 1 N N χ C (x n ) σ(c) n=1

6 Combinatorial measures discrepancy D N (X N ) = sup C 1 N N χ C (x n ) σ(c) n=1 covering radius δ N (X N ) = sup x S d min k x x k

7 Combinatorial measures discrepancy D N (X N ) = sup C 1 N N χ C (x n ) σ(c) n=1 covering radius δ N (X N ) = sup x S d min k x x k separation N (X N ) = min i j x i x j

8 Analytic measures error in numerical integration 1 N I N (f, X N ) = f(x n ) f(x) dσ d (x) N S d n=1

9 Analytic measures error in numerical integration 1 N I N (f, X N ) = f(x n ) f(x) dσ d (x) N S d n=1 Worst-case error for integration in a normed space H: wce(x N, H) = sup I N (f, X N )), f H f =1

10 L 2 -discrepancy and energy L 2 -discrepancy: π 0 S d 1 N N χ C(x,t) (x n ) σ d (C(x, t)) n=1 2 dσ d (x) dt

11 L 2 -discrepancy and energy L 2 -discrepancy: π 0 S d 1 N N χ C(x,t) (x n ) σ d (C(x, t)) n=1 2 dσ d (x) dt (generalised) energy: E g (X N ) = N g( x i, x j ) = i,j=1 i j N g( x i x j ), i,j=1 i j where g denotes a positive definite function.

12 L 2 -discrepancy and energy L 2 -discrepancy: π 0 S d 1 N N χ C(x,t) (x n ) σ d (C(x, t)) n=1 2 dσ d (x) dt (generalised) energy: E g (X N ) = N g( x i, x j ) = i,j=1 i j N g( x i x j ), i,j=1 i j where g denotes a positive definite function. L 2 -discrepancy and the worst case error (for many function spaces) turn out to be generalised energies of the underlying point configuration.

13 Hyperuniformity in R d Heuristic Hyperuniformity = asymptotically uniform + extra order Counting points in test sets, e.g. balls B R N R := N i=1 1 BR (X i ), where (X 1,..., X N ) ρ (N) V

14 Hyperuniformity in R d Heuristic Hyperuniformity = asymptotically uniform + extra order Counting points in test sets, e.g. balls B R N R := N i=1 1 BR (X i ), where (X 1,..., X N ) ρ (N) V

15 Hyperuniformity in R d Heuristic Hyperuniformity = asymptotically uniform + extra order Counting points in test sets, e.g. balls B R N R := N i=1 1 BR (X i ), where (X 1,..., X N ) ρ (N) V The expected number of points in B R is E [N R ] th. ρ B R

16 Hyperuniformity in R d The variance measures the rate of convergence. Example: (X i ) i i.i.d. V[N R ] th. ρ B R. Definition (ρ (N) ) N N hyperuniform lim th. V[N R ] B R for large R Remarks:

17 Hyperuniformity in R d The variance measures the rate of convergence. Example: (X i ) i i.i.d. V[N R ] th. ρ B R. Definition (ρ (N) ) N N hyperuniform lim th. V[N R ] B R for large R Remarks: If (ρ (N) ) N N hyperuniform, i.e. R d -term of lim th. V [N R ] vanishes R d 1 -term cannot vanish.

18 Hyperuniformity in R d The variance measures the rate of convergence. Example: (X i ) i i.i.d. V[N R ] th. ρ B R. Definition (ρ (N) ) N N hyperuniform lim th. V[N R ] B R for large R Remarks: If (ρ (N) ) N N hyperuniform, i.e. R d -term of lim th. V [N R ] vanishes R d 1 -term cannot vanish. Hyperuniformity is a long-scale property.

19 Hyperuniformity on the sphere Definition (Hyperuniformity) Let (X N ) N N be a sequence of point sets on the sphere S d. The number variance of the sequence for caps of opening angle φ is given by A sequence is called V (X N, φ) = V x # (X N C(x, φ)). (1)

20 Hyperuniformity on the sphere Definition (Hyperuniformity) Let (X N ) N N be a sequence of point sets on the sphere S d. The number variance of the sequence for caps of opening angle φ is given by A sequence is called V (X N, φ) = V x # (X N C(x, φ)). (1) hyperuniform for large caps if for all φ (0, π 2 ) ; V (X N, φ) = o (N) as N (2)

21 Hyperuniformity on the sphere (continued) Definition (continued)

22 Hyperuniformity on the sphere (continued) Definition (continued) hyperuniform for small caps if V (X N, φ N ) = o (Nσ(C(, φ N ))) as N (3) and all sequences (φ N ) N N such that

23 Hyperuniformity on the sphere (continued) Definition (continued) hyperuniform for small caps if V (X N, φ N ) = o (Nσ(C(, φ N ))) as N (3) and all sequences (φ N ) N N such that 1 lim N φ N = 0

24 Hyperuniformity on the sphere (continued) Definition (continued) hyperuniform for small caps if V (X N, φ N ) = o (Nσ(C(, φ N ))) as N (3) and all sequences (φ N ) N N such that 1 lim N φ N = 0 2 lim N Nσ(C(, φ N )) =.

25 Hyperuniformity on the sphere (continued) Definition (continued) hyperuniform for small caps if V (X N, φ N ) = o (Nσ(C(, φ N ))) as N (3) and all sequences (φ N ) N N such that 1 lim N φ N = 0 2 lim N Nσ(C(, φ N )) =. hyperuniform for caps at threshold order, if lim sup V (X N, tn 1 d ) = O(t d 1 ) as t. (4) N

26 How to get hyperuniform sequences? jittered sampling

27 How to get hyperuniform sequences? jittered sampling determinantal point processes

28 How to get hyperuniform sequences? jittered sampling determinantal point processes t-designs of minimal order

29 How to get hyperuniform sequences? jittered sampling determinantal point processes t-designs of minimal order QMC-designs

30 Determinantal point process in S2 Figure: sampled points from an i.i.d. process and a DPP, resp. P. Grabner Hyperuniformity on the Sphere

31 Open questions Find relations with other measures of uniformity: discrepancy, error of integration, energy...

32 Open questions Find relations with other measures of uniformity: discrepancy, error of integration, energy... Find explicit deterministic constructions for hyperuniform point sets for any N.

33 Open questions Find relations with other measures of uniformity: discrepancy, error of integration, energy... Find explicit deterministic constructions for hyperuniform point sets for any N. Find explicit deterministic constructions for point sets achieving the best possible discrepancy bound (or even a bound better than N 1 2 )