Computing upper bounds for densest packings with congruent copies of - - PowerPoint PPT Presentation

Computing upper bounds for densest packings with congruent copies of a convex body

Fernando Mario de Oliveira Filho (FU Berlin) Frank Vallentin (TU Delft, CWI Amsterdam)

ERC Workshop: High complexity discrete geometry, FU Berlin October 26, 2011

The beginning of the story

image credit: Torquato, Jiao

tetrahedra tetris

the search for the densest packing of tetrahedra

This talk: Let’s start the race!

Overview

• 1. Packing problems in discrete geometry
• 2. Spectral bounds & convex optimization
• 3. Computational strategies

• 1. Packing problems

α(G) = 4

vertices: edges: Sn−1 = {x ∈ Rn : x · x = 1} x ∼ y if α < x · y < 1

pack spherical caps into Sn−1 to maximize fraction of covered space

Rn x ∼ y if 0 < x − y < 2 vertices: edges:

pack balls into Rn to maximize fraction of covered space

image credit: Torquato, Jiao

Rn SO(n)

pack bodies into Rn to maximize fraction of covered space

vertices: edges: (x, A) ∼ (y, B) if (x + AKo) ∩ (y + BKo) = ∅

• 2. Spectral bounds &

convex optimization

I ⊆ V independent set ≤ |I| max

x∈V K(x, x)

K : V × V → R be a symmetric matrix such that (1) K is positive semidefinite (2) K(x, y) ≤ 0 if {x, y} ∈ E Then,

Hoffman (1970) Delsarte (1973) Lov´ asz (1979)

G = (V, E) be finite graph, (3) K1 = λ1 α(G) ≤ |V | λ max

x∈V K(x, x).

Spectral decomposition: K − λ |V |11T 0 λ |V ||I|2 ≤

• x,y∈I

K(x, y)

Finding optimal K can be done by SDP

• Also works if V is compact measure space
• If V is not compact, then (3) makes trouble.
• Symmetries of G and harmonic analysis on V help to simplify the SDP
• K : V × V → R be a symmetric matrix such that

(1) K is positive semidefinite (2) K(x, y) ≤ 0 if {x, y} ∈ E Then,

Hoffman (1970) Delsarte (1973) Lov´ asz (1979)

G = (V, E) be finite graph, (3) K1 = λ1 α(G) ≤ |V | λ max

x∈V K(x, x).

packing problem vertex set harmonic analysis authors

spherical caps compact, non- abelian Delsarte, Goethals, Seidel (1977), Kabatiansky, Levenshtein (1978) sphere packing non-compact, abelian Cohn, Elkies (2003) body packing non-compact, non-abelian Oliveira, V. (2011)

Sn−1 Rn Rn SO(n)

Main theorem

∀N ∈ N ∀(x1, A1), . . . , (xN, AN) ∈ Rn SO(n):

• f((xi, Ai)(xj, Aj)−1)
• i,j 0

Let P be a packing with body K, and f ∈ L1(Rn SO(n)) be continuous such that (2) f(x, A) ≤ 0 if Ko ∩ (x + AKo) = ∅ (3) λ =

• RnSO(n) f(x, A)d(x, A) > 0.

Then, δ(P) ≤ f(0,I)

λ

· vol K

(1) f is of positive type:

compact Define K : (Rn SO(n))/L × (Rn SO(n))/L → R K((x, A), (y, B)) =

• v∈L

f((y − v, B)−1(x, A)) by and ”apply” previous theorem. P =

• v∈L

N

• i=1

v + xi + AiK for some lattice L ⊆ Rn and (xi, Ai) ∈ Rn SO(n) Approximate P by periodic packing P

• 3. Computational strategies

image credit: wikipedia

Fourier basis

Question: How to find a good function f?

Main technical step: Parametrize functions using its Fourier coefficients. Here: n = 2, other n only on demand. . . Fourier coefficients are Hilbert Schmidt operators ∀p ≥ 0 :

• frs(p)
• rs 0

Answer: Finding an optimal f is an ∞-dimensional SDP.

∞

∞

• r,s=−∞
• frs(p)is−re−i(sθ+(r−s)φ)Js−r(pa)pdp

f a cos φ a sin φ

• ,

cos θ − sin θ sin θ cos θ

• =

Now

Use tools from polynomial optimization to approximate optimal solution

• f this ∞-dimensional SDP.

Main trick

p ∈ R[x] is sum of squares (SOS) p(x) =

n

• i=1

(qi(x))2 iff p(x) =      1 x . . . xd     

T

Q      1 x . . . xd      for Q 0.

Case K = B(0, 1)

Laguerre polynomials Instead of

• frs(p)
• rs 0

use

• f00(p) ≥ 0
• f00(p) =

d

• k=0

f2kp2ke−p2 with f(a) = ∞

• f00(p)J0(pa)pdp =

d

• k=0

f2k k! 2 Lk(a2/4)e−a2 then

Case K = B(0, 1); reproving Cohn-Elkies

Conditions (1) : ∀p ∈ R : f00(p) ≥ 0 (2) : ∀a ∈ R≥2 : f(a) ≤ 0 are equivalent to SOS conditions.

(Differs from numerical scheme of Cohn-Elkies and gives global optima)

with d = 40 and high accuracy SDP solver and numerical help from Hans Mittelmann

0.28867493155 . . . 0.18615073311 . . . 0.13125261809 . . . 0.09973444216 . . . 0.08082433040 . . . 0.06930754913 . . . 0.06250430042 . . . 0.05898945087 . . .

Numerical results

Case K = C5

∀t ∈ R Restrict to ”polynomial” Fourier coefficients

• frs(t) =

d

• k=0

frs,2kt2ke−t2 P(t) =

• frs(t)
• −N≤r,s≤N 0

Then is a univariate PSD matrix inequality. p(t, y) = yTP(t)y ∈ R[t, y−N, . . . , yN] is a sum of squares This is equivalent to: Multivariate polynomial

Step 1: Making things of positive type

f(a, φ, θ) =

N

• r,s=−N

d

• k=0

frs,2kis−re−i(sθ+(r−s)φ) ∞ t2k+1e−t2Js−r(at)dt

= Γ((2k + 2 + s − r)/2)(a/2)s−re−a2/4 2Γ(s − r + 1)

1F1

(s − r − 2k − 2)/2 + 1 s − r + 1 ; a2 4

• if 2k + 2 + s − r > 0

AAR, page 222 (4.11.24)

If s − r even and (s − r − 2k − 2)/2 + 1 ≤ 0, then polynomial in a:

Lα

n(a) = (α + 1)n

n!

1F1

−n α ; a

• Laguerre polynomial, orthogonal on [0, ∞) wrt aαe−adx

So f is a polynomial in a and trigonometric in φ, θ.

linear combination of ”monomials”: ak sin rφ cos tθ

Step 2: Evaluating the function

Fix A ∈ SO(n). x ∈ Rn with Ko ∩ x + AKo = ∅ is Minkowski difference Ko − AKo

(2) f(x, A) ≤ 0 if Ko ∩ (x + AKo) = ∅

If K is a polytope, this is a linear condition in x.

Ko − AKo is an open 10-gon

Step 3: Dealing with the geometry

(If K is the ball, then shape is a round cylinder.)

x ∈ R2, θ ∈ [−2π/10, 2π/10] with x ∈ Ko − A(θ)Ko

gi(a, φ, θ) < 0 ”facet” defining ”polynomial” f ≤ 0 on the ten ”semialgebraic” sets {(a, φ, θ) : gi(a, φ, θ) ≥ 0} Can be relaxed as SOS condition.

Numerical results

image credit: WWW

Conclusions

We are developing the first step of an algorithmic solution for a large class of packing problems

• Complexity of body K is reflected in the complexity
• f the computation
• Numerical calculations are challenging

but seem to be in reach (in dimensions 2, 3)