Discrete Minimum Energy Problems

Transcription

1 Discrete Minimum Energy Problems D.P. Hardin Department of Mathematics Vanderbilt University TU Graz, July 3, 2009

2 Reisz s energy The Riesz s-energy of ω N = {x 1, x 2,..., x N } R p is, for s > 0, E s(ω N ) := NX X k s(x i, x j ) i=1 j i where ( x y s, s > 0 k s(x, y) := log ( x y ), s = 0 Note: x y s 1 s log x y as s 0.

3 Constrained optimization problem: Given a compact set A R p. Minimize the objective function E s(ω N ) := NX X k s(x i, x j ) i=1 j i subject to the constraint ω N = {x 1,..., x N } A.

4 Constrained optimization problem: Given a compact set A R p. Minimize the objective function E s(ω N ) := NX X k s(x i, x j ) i=1 j i subject to the constraint ω N = {x 1,..., x N } A. Let ω N := {x 1,N,..., x N,N } denote an optimal configuration and let E s(a, N) := E(ω N).

5 Cases s = p 1 and s The function k p 1 (, y) is harmonic on R p \ {y}.

6 Cases s = p 1 and s As s and fixed N, 0 X 1 A x i x j s i j 1/s 1 min{ x i x j, i j}. Thus, minimal energy configurations become best-packing configurations, i.e., they maximize the minimum pairwise distance between N points on A as s.

7 Example: A = S 2 Describe optimal 2 point configurations.

8 Example: A = S 2 Describe optimal 3 point configurations.

9 Example: A = S 2 Describe optimal 4 point configurations.

10 Example: A = S 2 Describe optimal 5 point configurations.

11 Example: A = S 2 For a configuration of points ω N = {x 1,..., x N } S 2, let ν i denote the number of nearest neighbors in ω N to x i (i.e., the number of edges for the Voronoi cell for x i ). A simple application of Euler s characteristic formula NX gives: (6 ν i ) = 12. i=1

12 Example: A = S 2 ; N = 174; s = 0, 1, 2 Matthew Calef: Vanderbilt PhD student

13 Example: A = S 2 ; N = 174; s = 0, 1, 2 N = 174; s = 2

14 Example: A = S 2 ; N = 174; s = 0, 1, 2 N = 174; s = 1

15 Example: A = S 2 ; N = 174; s = 0, 1, 2 N = 174; s = 0

16 Red = heptagon, Green = hexagon, Blue = pentagon N = 1600, s = 4

17 Spherical crystallography of colloidosomes Spherical crystallography of colloidosomes! Adsorb, say, latex spheres onto lipid bilayer vesicles or water droplets! Useful for encapsulation of flavors and fragrances, drug delivery [H. Aranda-Espinoza e.t al. Science 285, 394 (1999)]!Strength of colloidal armor plating influenced by defects in shell.! For water droplets, surface tension prevents buckling. Colloidosome = colloids of radius a coating water droplet (radius R) -- Weitz Laboratory Ordering on a sphere! a minimum of 12 5-fold disclinations, as in soccer balls and fullerenes -- what happens for R/a >> 1? Confocal image: P. Lipowsky, & A. Bausch

18 Questions from physics How does long range order (crystalline structure) arise out of simple pairwise interactions? How does the structure depend on the geometry of the world A (dimension, curvature,...) in which the particles live. How does the structure depend on the interaction? How does the order break down as we move away from the ground state?

19 Asymptotics of configurations as N Problem: What is the asymptotic behavior of E s(a, N) and of ω N as N? Q1: How are minimal s-energy configurations for A distributed for large N? Q2: How does the asymptotic behavior of E s(a, N) depend on A and s?

20 Connections to Potential Theory Let A R p be compact with Hausdorff dimension d = dim H(A). M A := {all Borel probability measures µ on A}. For µ M A, let Z Z I s(µ) := 1 x y s dµ(y)dµ(x). Frostman (1935): For s < d, there exists a unique equilibrium measure µ s in M A such that I s(µ s) I s(ν) for all ν M A and I s(ν) = for s d and all ν M A. The s-capacity of A is cap s (A) = I s(µ s) 1. Points in an optimal configuration are also called Fekete points.

21 Connection Between Continuous & Discrete Problems Theorem (Fekete, 1923; Pólya and Szegő, 1931) Let A R p be compact, s < d := dim H(A), and µ s denote the Riesz s-equilibrium measure on A. Then lim N E s(a, N) N(N 1) = Is(µs) and minimal s-energy configurations ωn = ωn(a, s) satisfy in the weak-star topology ν N := 1 X δ x µ s as N. N x ω N Remark: Weak-star convergence of ν N to µ s means for any f C(A). 1 N NX Z f (x) f dµ s x ω N

22 Sketch of Proof Step 1: First observe that Then E s(a, N) = E s(ω N) = 1 N 2 τ N := Es(A, N) N(N 1) NX E s (ωn \ {x k,n }) N Es(A, N 1). N 2 k=1 showing that τ N is increasing with N. Es(A, N 1) N(N 1) N N 2 = τ N 1 Let E s(a, N) τ := lim N N(N 1)

23 Sketch of Proof Step 2: Show τ I s(µ s). E s(a, N) NX i j 1 x i x j s, x 1,..., x N A. Then Z E s(a, N) A Z A NX i j 1 x i x j s dµs(x 1) dµ s(x N )

24 Sketch of Proof Step 2: Show τ I s(µ s). E s(a, N) NX i j 1 x i x j s, x 1,..., x N A. Then E s(a, N) NX Z i j A Z 1 A x i x j dµs(x 1) dµ s(x s N )

25 Sketch of Proof Step 2: Show τ I s(µ s). Then E s(a, N) E s(a, N) NX i j i j 1 x i x j s, x 1,..., x N A. NX Z Z A A 1 x i x j s dµs(x i)dµ s(x j )

26 Sketch of Proof Step 2: Show τ I s(µ s). Then E s(a, N) E s(a, N) NX i j i j 1 x i x j s, x 1,..., x N A. NX Z Z A A NX I s(µ s) i j 1 x i x j s dµs(x i)dµ s(x j )

27 Sketch of Proof Step 2: Show τ I s(µ s). Then E s(a, N) E s(a, N) NX i j i j 1 x i x j s, x 1,..., x N A. NX Z Z A A 1 x i x j s dµs(x i)dµ s(x j ) NX I s(µ s) = N(N 1)I s(µ s) i j

28 Sketch of Proof Step 2: Show τ I s(µ s). Then E s(a, N) E s(a, N) NX i j i j 1 x i x j s, x 1,..., x N A. NX Z Z A A 1 x i x j s dµs(x i)dµ s(x j ) NX I s(µ s) = N(N 1)I s(µ s) i j and so: τ N I s(µ s) τ I s(µ s).

29 Sketch of Proof Step 3: By Banach-Alaoglu Thm, ν N has a weak-star cluster point µ. Consider Z Z 1 I s(µ) = x y dµ(x)dµ(y) s Z Z j ff 1 = lim min M x y, M dµ(x)dµ(y) s Z Z j ff = lim lim 1 min M N x y, M dν s N (x)dν N (y) lim lim 1 {Es(A, N) + NM} M N N2 = τ I s(µ s). So µ = µ s and hence τ = I s(µ s) and ν N µ s.

30 N = 1000 points s =0.2 s =1.0 s =2.0 s =4.0

31 N = 4000 points s =0.2 s =1.0 s =2.0 s =4.0

32 Surfaces of revolution the case s = 0 For A in the right-half xy-plane, let Γ(A) denote the set in R 3 obtained by rotating A about the y-axis. Let µ 0,Gamma(A) denote the log energy equilibrium on Γ(A). Let A + denote the right-most portion of A.

33 Surfaces of revolution the case s = 0 For A in the right-half xy-plane, let Γ(A) denote the set in R 3 obtained by rotating A about the y-axis. Let µ 0,Gamma(A) denote the log energy equilibrium on Γ(A). Let A + denote the right-most portion of A. Theorem (H., Saff, and Stahl, 2006) Suppose A is a compact set in the right-half plane R + R. Then the support of the equilibrium measure µ 0,Γ(A) is contained in Γ(A +). J. Brauchart, H., and Saff (2008) also provide related results for 0 < s < 1.

34 Ed Saff, Vanderbilt

35 Herbert Stahl, Technische Fachhochschule Berlin

36 Johann Brauchart, (TU Graz PhD) Vanderbilt

37 What about s d? For s > d = dim A, I s(µ) = for any µ M A. Also E s(a) τ = lim =. N N 2

38 What about s d? For s > d = dim A, I s(µ) = for any µ M A. Also E s(a) τ = lim =. N N 2 So new methods are required for s d.

39 The case A = S d and s = d.

40 The case A = S d and s = d. Theorem (Kuijlaars & Saff, 1998) E d (S d, N) lim N N 2 log N = 1 Γ((d + 1)/2) = Vol(B d) d π Γ(d/2) Area(S d ) = H d(b d ) H d (S d ), and (Götz & Saff, 2001) d-energy optimal configurations are asymptotically uniformly distributed on S d as N. Here H d denotes d dimensional Hausdorff measure on R p appropriately normalized. Grabner and Damelin, 2003, give discrepancy bounds for d-energy optimal configurations on S d.

41 Compact sets in R d Theorem (H. & Saff, 2005) Suppose A is a compact set in R d. Then (a) (b) E d (A, N) lim N N 2 log N = H d(b d ) H d (A), E s(a, N) C s,d lim = N N 1+s/d [H d (A)], s > d. s/d If H d (A) > 0, then ν N := 1 N X δ x Hd A as N. x ω N where H A d = H d ( A) H d (A).

42 About the proof. An important step in the proof is to show the existence of the limit (2) for A = U d := [0, 1] d. The unit cube is self-similar with scaling 1/m for any m = 2, 3,... We use this to find bounds relating E s(a, N) and E s(a, m d N): =

43 d-rectifiable sets A R p is a d-rectifiable set if A is the image of a bounded set in R d under a Lipschitz mapping. If A is a d-rectifiable set, then A is almost the finite disjoint union of almost isometric images of compact sets in R d :

44 d-rectifiable sets A R p is a d-rectifiable set if A is the image of a bounded set in R d under a Lipschitz mapping. If A is a d-rectifiable set, then A is almost the finite disjoint union of almost isometric images of compact sets in R d : Lemma (Federer, 1969) If A is a d-rectifiable set then for every ɛ > 0 there exist compact sets K 1, K 2, K 3,... R d and bi-lipschitz mappings ψ i : K i R p with constant 1 + ɛ, i = 1, 2, 3,..., such that ψ 1 (K 1 ), ψ 2 (K 2 ), ψ 3 (K 3 ),... are disjoint subsets of A with H d (A \ [ ψ i (K i )) = 0. i Any compact subset of a smooth d dimensional manifold is a d-rectifiable set.

45 Minimum energy on d-rectifiable sets Theorem (H. & Saff, 2005; Borodachov, H., & Saff, 2007) Suppose s d and A R p is a d-rectifiable set. When s = d we further assume A is a subset of a d-dimensional C 1 manifold. Then E d (A, N) (a) lim N N 2 log N = H d(b d ) H d (A), E s(a, N) C s,d (b) lim = N N 1+s/d [H d (A)], s > d. s/d If H d (A) > 0, then ν N := 1 N X x ω N δ x H d( ) A H d (A) as N.

46 Sergiy Borodachov (Vanderbilt PhD), Towson University

47 Problem Definition and Motivation Potential Theory Hypersingular case: s d. Best packing in Rd

48 The constant C s,d d = 1: Since N-th roots of unity are optimal on unit circle, we obtain C s,1 = 2ζ(s) for s > 1, where ζ(s) is the classical Riemann zeta function.

49 The constant C s,d d = 1: Since N-th roots of unity are optimal on unit circle, we obtain C s,1 = 2ζ(s) for s > 1, where ζ(s) is the classical Riemann zeta function. For s > d, the Epstein zeta function for a d dimensional lattice Λ is ζ Λ (s) := X v s. 0 v Λ Summing over lattice configurations gives: C s,d ζ min d (s) := min Λ Λ s/d ζ Λ (s), s > d. (1)

50 The constant C s,d d = 1: Since N-th roots of unity are optimal on unit circle, we obtain C s,1 = 2ζ(s) for s > 1, where ζ(s) is the classical Riemann zeta function. For s > d, the Epstein zeta function for a d dimensional lattice Λ is ζ Λ (s) := X v s. 0 v Λ Summing over lattice configurations gives: C s,d ζd min (s) := min Λ s/d ζ Λ (s), s > d. (1) Λ It is almost surely true (although not proved) that C s,2 = Λ s/d ζ Λ (s) where Λ is the equilateral hexagonal lattice. It is conjectured (Cohn and Kumar, 2007) that the E 8 and Leech lattices play the same role in dimensions d = 8 and d = 24.

51 Sphere packing density in R d. Let d denote the largest sphere packing density in R d. Connection between C s,d and d (Borodachov, H., & Saff, 2007) 1 = 1, «1/d (C s,d ) 1/s Vol(Bd ) (1/2) as s,. d 2 = π/ (Fejes Toth, 1940 s), 3 = π/ (Hales, 2007). For d > 3, d is unknown, although extremely precise bounds are available for d = 2, 8 and 24 (Cohn and Elkies, 2003). Ratio of upper bound to lower bound is

52 Sphere packing density in R d. Let d denote the largest sphere packing density in R d. Connection between C s,d and d (Borodachov, H., & Saff, 2007) 1 = 1, «1/d (C s,d ) 1/s Vol(Bd ) (1/2) as s,. d 2 = π/ (Fejes Toth, 1940 s), 3 = π/ (Hales, 2007). For d > 3, d is unknown, although extremely precise bounds are available for d = 2, 8 and 24 (Cohn and Elkies, 2003). Ratio of upper bound to lower bound is In each of these dimensions the densest packing appears to be a lattice packing (the hexagonal lattice, E 8, Leech lattice, respectively).

53 Best packing in dimensions d = 2, 8, and 24 Cohn & Elkies bounds for d for d = 2, 8, 24 are based on the following: Theorem (Cohn & Elkies, 2003) Suppose f : R d R is an admissible function satisfying the following three conditions : (1) f (0) = ˆf (0) > 0, (2) f (x) 0 for x r, and (3) ˆf (t) 0 for all t. Then d Vol(B d )(r/2) d.

54 Fork in the road Fejes Toth s proof of best packing in R 2. Movie (Rob Womersley)

55 Densest packing in R 2. The fact that the largest sphere packing 2 = π/ 12 follows directly from the following: Theorem (Fejes Toth) Suppose Ω is a convex polygon in R 2 with six or fewer sides. Suppose Ω contains N pairwise disjoint open discs of radius r > 0. Then N A(Ω) r 2 a(6) where A(Ω) denotes the area of Ω and a(n) = n tan(π/n) denotes the area of the regular n-gon with inradius 1. Then Nπr 2 A(Ω) π a(6) = π/ 12

56 Sketch of proof. Proof. Let {B(x i, r)} be a collection of N non-overlapping discs in Ω and let {V i } denote the Voronoi decomposition of Ω associated with the centers {x i } N i=1. Then each V i is a ν i -gon containing a disc of radius r.

57 Sketch of proof. Proof. Let {B(x i, r)} be a collection of N non-overlapping discs in Ω and let {V i } denote the Voronoi decomposition of Ω associated with the centers {x i } N i=1. Then each V i is a ν i -gon containing a disc of radius r. A(V i ) r 2 a(ν i ).

58 Sketch of proof. Proof. Let {B(x i, r)} be a collection of N non-overlapping discs in Ω and let {V i } denote the Voronoi decomposition of Ω associated with the centers {x i } N i=1. Then each V i is a ν i -gon containing a disc of radius r. A(V i ) r 2 a(ν i ). Euler s formula for planar graphs: P N i=1 ν i 6N.

59 Sketch of proof. Proof. Let {B(x i, r)} be a collection of N non-overlapping discs in Ω and let {V i } denote the Voronoi decomposition of Ω associated with the centers {x i } N i=1. Then each V i is a ν i -gon containing a disc of radius r. A(V i ) r 2 a(ν i ). Euler s formula for planar graphs: P N i=1 ν i 6N. The function a(x) = x tan(π/x) is decreasing and strictly convex on the interval [3, ).

60 Sketch of proof. Proof. Let {B(x i, r)} be a collection of N non-overlapping discs in Ω and let {V i } denote the Voronoi decomposition of Ω associated with the centers {x i } N i=1. Then each V i is a ν i -gon containing a disc of radius r. A(V i ) r 2 a(ν i ). Euler s formula for planar graphs: P N i=1 ν i 6N. The function a(x) = x tan(π/x) is decreasing and strictly convex on the interval [3, ). A(Ω) = P N i=1 A(V i)

61 Sketch of proof. Proof. Let {B(x i, r)} be a collection of N non-overlapping discs in Ω and let {V i } denote the Voronoi decomposition of Ω associated with the centers {x i } N i=1. Then each V i is a ν i -gon containing a disc of radius r. A(V i ) r 2 a(ν i ). Euler s formula for planar graphs: P N i=1 ν i 6N. The function a(x) = x tan(π/x) is decreasing and strictly convex on the interval [3, ). A(Ω) = P N i=1 A(V i) r 2 P N i=1 a(ν i)

62 Sketch of proof. Proof. Let {B(x i, r)} be a collection of N non-overlapping discs in Ω and let {V i } denote the Voronoi decomposition of Ω associated with the centers {x i } N i=1. Then each V i is a ν i -gon containing a disc of radius r. A(V i ) r 2 a(ν i ). Euler s formula for planar graphs: P N i=1 ν i 6N. The function a(x) = x tan(π/x) is decreasing and strictly convex on the interval [3, ). A(Ω) = P N i=1 A(V i) r 2 P N i=1 a(ν i) Nr 2 P N i=1 a(ν i) 1 N

63 Sketch of proof. Proof. Let {B(x i, r)} be a collection of N non-overlapping discs in Ω and let {V i } denote the Voronoi decomposition of Ω associated with the centers {x i } N i=1. Then each V i is a ν i -gon containing a disc of radius r. A(V i ) r 2 a(ν i ). Euler s formula for planar graphs: P N i=1 ν i 6N. The function a(x) = x tan(π/x) is decreasing and strictly convex on the interval [3, ). A(Ω) = P N i=1 A(V i) r 2 P N i=1 a(ν i) Nr 2 P N i=1 a(ν i) 1 N Nr 2 P 1 a N N i=1 ν i

64 Sketch of proof. Proof. Let {B(x i, r)} be a collection of N non-overlapping discs in Ω and let {V i } denote the Voronoi decomposition of Ω associated with the centers {x i } N i=1. Then each V i is a ν i -gon containing a disc of radius r. A(V i ) r 2 a(ν i ). Euler s formula for planar graphs: P N i=1 ν i 6N. The function a(x) = x tan(π/x) is decreasing and strictly convex on the interval [3, ). A(Ω) = P N i=1 A(V i) r 2 P N i=1 a(ν i) Nr 2 P N i=1 a(ν i) 1 N Nr 2 P 1 a N N i=1 ν i Nr 2 a(6)

65 Sketch of proof. Proof. Let {B(x i, r)} be a collection of N non-overlapping discs in Ω and let {V i } denote the Voronoi decomposition of Ω associated with the centers {x i } N i=1. Then each V i is a ν i -gon containing a disc of radius r. A(V i ) r 2 a(ν i ). Euler s formula for planar graphs: P N i=1 ν i 6N. The function a(x) = x tan(π/x) is decreasing and strictly convex on the interval [3, ). A(Ω) = P N i=1 A(V i) r 2 P N i=1 a(ν i) Nr 2 P N i=1 a(ν i) 1 N Nr 2 P 1 a N N i=1 ν i Nr 2 a(6)

66 References S. Borodachov, D. Hardin, and E. Saff. Asymptotics for discrete weighted minimal energy problems on rectifiable sets. Trans. Amer. Math. Soc., 360(3): , M. Calef and D. Hardin. Riesz s-equilibrium measures on d-rectifiable sets as s approaches d. Potential Analysis, 30(4): , H. Cohn and N. Elkies. New upper bounds on sphere packings I. Ann. Math., 157: , H. Cohn and A. Kumar. Universally optimal distribution of points on spheres. J. Am. Math. Soc., 157:99 148, P. Grabner and S. Damelin. Energy functionals, numerical integration and asymptotic equidistribution on the sphere. Journal of Complexity 19(3): , T. Hales. A proof of the Kepler conjecture. Ann. Math., 162: , D. Hardin and E. Saff. Minimal Riesz energy point configurations for rectifiable d- dimensional manifolds. Adv. Math, 193: , 2005.