Point sets of minimal energy Peter Grabner Institut fr Analysis und - - PowerPoint PPT Presentation

Point sets of minimal energy

Peter Grabner

Institut für Analysis und Computational Number Theory Graz University of Technology

October 16, 2013

Peter Grabner Point sets of minimal energy

Self-organisation by local interaction

Self-organisation by local interaction is a far-reaching principle

• ccurring in Biology, Chemistry, Physics. . . . It explains inter alia

Peter Grabner Point sets of minimal energy

Self-organisation by local interaction

Self-organisation by local interaction is a far-reaching principle

• ccurring in Biology, Chemistry, Physics. . . . It explains inter alia

the distribution of electrons on a surface (Thomson’s problem) the distribution of pollen grains (Tammes’ problem) the structure of polymers structure of ground states of particle systems viral morphology the arrangement of colloidal particles the structure of fullerenes

Peter Grabner Point sets of minimal energy

Self-organisation by local interaction

Self-organisation by local interaction is a far-reaching principle

• ccurring in Biology, Chemistry, Physics. . . . It explains inter alia

the distribution of electrons on a surface (Thomson’s problem) the distribution of pollen grains (Tammes’ problem) the structure of polymers structure of ground states of particle systems viral morphology the arrangement of colloidal particles the structure of fullerenes Idea: use this principle for generating Quasi-Monte Carlo point sets

• n manifolds, especially spheres.

Peter Grabner Point sets of minimal energy

“Even” point distributions on the sphere

The question of how to distribute N points “evenly” on the sphere has many important applications, such as

Peter Grabner Point sets of minimal energy

“Even” point distributions on the sphere

The question of how to distribute N points “evenly” on the sphere has many important applications, such as sampling functions on spheres

Peter Grabner Point sets of minimal energy

“Even” point distributions on the sphere

The question of how to distribute N points “evenly” on the sphere has many important applications, such as sampling functions on spheres integrating functions over spherical domains

Peter Grabner Point sets of minimal energy

“Even” point distributions on the sphere

The question of how to distribute N points “evenly” on the sphere has many important applications, such as sampling functions on spheres integrating functions over spherical domains solving PDEs by discretisation

Peter Grabner Point sets of minimal energy

“Even” point distributions on the sphere

The question of how to distribute N points “evenly” on the sphere has many important applications, such as sampling functions on spheres integrating functions over spherical domains solving PDEs by discretisation sampling spacial directions

Peter Grabner Point sets of minimal energy

Pollen grains

Peter Grabner Point sets of minimal energy

Potential function of a point distribution

Peter Grabner Point sets of minimal energy

Quantify evenness

For every point set XN = {x1, . . . , xN} of distinct points, we assign several qualitative measures that describe aspects of even distribution.

Peter Grabner Point sets of minimal energy

Quantify evenness

For every point set XN = {x1, . . . , xN} 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.

Peter Grabner Point sets of minimal energy

Combinatorial measures

discrepancy DN(XN) = sup

C

• 1

N

N

• n=1

χC(xn) − σ(C)

• Peter Grabner

Point sets of minimal energy

Combinatorial measures

discrepancy DN(XN) = sup

C

• 1

N

N

• n=1

χC(xn) − σ(C)

• dispersion

δN(XN) = sup

x∈Sd min k |x − xk|

Peter Grabner Point sets of minimal energy

Combinatorial measures

discrepancy DN(XN) = sup

C

• 1

N

N

• n=1

χC(xn) − σ(C)

• dispersion

δN(XN) = sup

x∈Sd min k |x − xk|

separation ∆N(XN) = min

i=j |xi − xj|

Peter Grabner Point sets of minimal energy

Analytic measures

error in numerical integration IN(f , XN) =

• N
• n=1

f (xn) −

• Sd f (x) dσd(x)
• Peter Grabner

Point sets of minimal energy

Analytic measures

error in numerical integration IN(f , XN) =

• N
• n=1

f (xn) −

• Sd f (x) dσd(x)
• Worst-case error for integration in a normed space H:

IN(XN, H) = sup

f ∈H f =1

IN(f , XN)),

Peter Grabner Point sets of minimal energy

Analytic measures

error in numerical integration IN(f , XN) =

• N
• n=1

f (xn) −

• Sd f (x) dσd(x)
• Worst-case error for integration in a normed space H:

IN(XN, H) = sup

f ∈H f =1

IN(f , XN)), (generalised) energy: Eg(XN) =

N

• i,j=1

i=j

g(xi, xj) =

N

• i,j=1

i=j

˜ g(xi − xj), where g denotes a positive definite function.

Peter Grabner Point sets of minimal energy

Other concepts

designs: 1 N

N

• n=1

f (xn) =

• Sd f (x) dσ(x)

for all polynomials of degree ≤ t.

Peter Grabner Point sets of minimal energy

Other concepts

designs: 1 N

N

• n=1

f (xn) =

• Sd f (x) dσ(x)

for all polynomials of degree ≤ t. L2-discrepancy: π

• Sd
• 1

N

N

• n=1

χC(x,t)(xn) − σd(C(x, t))

• 2

dσd(x) dt

Peter Grabner Point sets of minimal energy

Discrepancy

Discrepancy is the most classical measure for the difference of two distributions DN(XN) = sup

C

• 1

N

N

• n=1

χC(xn) − σ(C)

• .

Peter Grabner Point sets of minimal energy

Discrepancy

Discrepancy is the most classical measure for the difference of two distributions DN(XN) = sup

C

• 1

N

N

• n=1

χC(xn) − σ(C)

• .

It is rather difficult to compute explicitly, even for moderate values

• f N.

Peter Grabner Point sets of minimal energy

Estimates for discrepancy

Thus estimates for DN(XN) are of interest (PG 1991, X.-J. Li &

• J. Vaaler, 1999), Erdős-Turán type inequality

DC

N (XN) ≤ Cd

  1 M +

M

• k=1

1 k

Z(d,k)

• j=1
• 1

N

N

• n=1

Yk,j(xn)

• 



Peter Grabner Point sets of minimal energy

Estimates for discrepancy

Thus estimates for DN(XN) are of interest (PG 1991, X.-J. Li &

• J. Vaaler, 1999), Erdős-Turán type inequality

DC

N (XN) ≤ Cd

  1 M +

M

• k=1

1 k

Z(d,k)

• j=1
• 1

N

N

• n=1

Yk,j(xn)

• 



• F. J. Narcowich, X. Sun, J. D. Ward, and Z. Wu (2010),

LeVeque-type inequality: D(XN) ≤ B(d)  

∞

• ℓ=0

ℓ−(d+1)

Z(d,ℓ)

• m=1
• 1

N

N

• n=1

Yℓ,m(xn) 2 

1 d+2

.

Peter Grabner Point sets of minimal energy

Irregularities

On the other hand the theory of irregularities of distributions developed by K. F. Roth, W. Schmidt, J. Beck, W. Chen, . . . gives a lower bound DN(XN) ≥ CN− 1

2− 1 2d . Peter Grabner Point sets of minimal energy

Irregularities

On the other hand the theory of irregularities of distributions developed by K. F. Roth, W. Schmidt, J. Beck, W. Chen, . . . gives a lower bound DN(XN) ≥ CN− 1

2− 1 2d .

This is essentially best possible. Namely, for every N there exists a point set XN such that DN(XN) ≤ CN− 1

2 − 1 2d log N.

The construction of this point set is probabilistic.

Peter Grabner Point sets of minimal energy

Energy

Depending on the behaviour of the positive definite function g at 1, the corresponding energy-functionals are categorised

Peter Grabner Point sets of minimal energy

Energy

Depending on the behaviour of the positive definite function g at 1, the corresponding energy-functionals are categorised if g(1) exists, the energy functional Eg(XN) =

N

• i,j=1

g(xi, xj) is called non-singular

Peter Grabner Point sets of minimal energy

Energy

Depending on the behaviour of the positive definite function g at 1, the corresponding energy-functionals are categorised if g(1) exists, the energy functional Eg(XN) =

N

• i,j=1

g(xi, xj) is called non-singular if g(1 − t) = O(t−s/2) for 0 ≤ s < d the energy functional Eg(XN) =

N

• i,j=1

i=j

g(xi, xj) is called singular.

Peter Grabner Point sets of minimal energy

Energy

Depending on the behaviour of the positive definite function g at 1, the corresponding energy-functionals are categorised if g(1) exists, the energy functional Eg(XN) =

N

• i,j=1

g(xi, xj) is called non-singular if g(1 − t) = O(t−s/2) for 0 ≤ s < d the energy functional Eg(XN) =

N

• i,j=1

i=j

g(xi, xj) is called singular. if g(1 − t) = Ω(t−s/2) for s ≥ d the energy functional is called super-singular.

Peter Grabner Point sets of minimal energy

Energy

For s > 0 consider the Riesz-energy functional of XN Es(XN) =

N

• i,j=1

i=j

xi − xj−s and find its minimal value and the corresponding point sets for given N.

Peter Grabner Point sets of minimal energy

Minimal energy point configuration provided by J. Brauchart

Peter Grabner Point sets of minimal energy

Energy continued

For 0 < s < d classical potential theory asserts that the discrete distribution measures of minimal energy point sets 1 N

N

• n=1

δxn weakly tend to the unique minimiser of the continuous energy functional E(µ) =

• Sd
• Sd x − y−s dµ(x) dµ(y),

the surface measure σ.

Peter Grabner Point sets of minimal energy

Energy continued

This gives lim

N→∞

1 N

N

• n=1

f (xn) =

• S2 f (x) dσ(x)

Peter Grabner Point sets of minimal energy

Energy continued

This gives lim

N→∞

1 N

N

• n=1

f (xn) =

• S2 f (x) dσ(x)

and lim

N→∞ DC N (XN) = 0.

Peter Grabner Point sets of minimal energy

Energy continued

This gives lim

N→∞

1 N

N

• n=1

f (xn) =

• S2 f (x) dσ(x)

and lim

N→∞ DC N (XN) = 0.

Minimal energy points give good behaviour for numerical integration and discrepancy.

Peter Grabner Point sets of minimal energy

Energy continued

Furthermore, for d − 2 ≤ s < d the point sets XN of minimal energy are well separated: min

i=j xi − xj ≥ C

N

1 d Peter Grabner Point sets of minimal energy

Energy continued

Furthermore, for d − 2 ≤ s < d the point sets XN of minimal energy are well separated: min

i=j xi − xj ≥ C

N

1 d

For 0 < s < d the minimal value of the energy satisfies (G. Wagner) CsN2 − c(1)

s

N1+ s

d ≤ min

XN

Es(XN) ≤ CsN2 − c(2)

s

N1+ s

d

for positive constants c(1)

s

and c(2)

s

Peter Grabner Point sets of minimal energy

Energy continued

Furthermore, for d − 2 ≤ s < d the point sets XN of minimal energy are well separated: min

i=j xi − xj ≥ C

N

1 d

For 0 < s < d the minimal value of the energy satisfies (G. Wagner) CsN2 − c(1)

s

N1+ s

d ≤ min

XN

Es(XN) ≤ CsN2 − c(2)

s

N1+ s

d

for positive constants c(1)

s

and c(2)

s

with Cs =

• S2
• S2 x − y−s dσ(x) dσ(y).

Peter Grabner Point sets of minimal energy

Energy continued

It is conjectured that lim

N→∞

minXN Es(XN) − CsN2 N1+ s

d

• exists. The conjectured value of the limit is given in terms of the

zeta-function of a lattice.

Peter Grabner Point sets of minimal energy

Energy continued

It is conjectured that lim

N→∞

minXN Es(XN) − CsN2 N1+ s

d

• exists. The conjectured value of the limit is given in terms of the

zeta-function of a lattice. This conjecture reflects the fact that on S2 locally the numerical examples of minimal energy configurations show a hexagonal structure.

Peter Grabner Point sets of minimal energy

Energy continued

It is conjectured that lim

N→∞

minXN Es(XN) − CsN2 N1+ s

d

• exists. The conjectured value of the limit is given in terms of the

zeta-function of a lattice. This conjecture reflects the fact that on S2 locally the numerical examples of minimal energy configurations show a hexagonal structure. Furthermore, in the plane the hexagonal lattice minimises the corresponding energy functional.

Peter Grabner Point sets of minimal energy

Minimal energy point sets

Figure: Mesh function and Voronoi cells for minimal energy point sets

Peter Grabner Point sets of minimal energy

Energy and numerical integration

Let 0 < s < d. For f with |f (x) − f (y)| ≤ C|x − y| (PG &

• S. Damelin)

IN(f , XN) ≤ 24C 1 m + max

1≤k≤m

2k + 1 ωdak(δ) 1

2

f 2× 1 N2 Es(XN) + 1 N δ−s − a0(δ) 1

2

, where 1 (1 + δ − cos(t))s =

∞

• k=0

ak(δ)Pk(cos(t)).

Peter Grabner Point sets of minimal energy

Super-singular energy

For s ≥ d potential theoretic methods are no more applicable, since the energy integral

• Sd
• Sd x − y−s dµ(x) dµ(y)

diverges for all measures µ.

Peter Grabner Point sets of minimal energy

Super-singular energy

For s ≥ d potential theoretic methods are no more applicable, since the energy integral

• Sd
• Sd x − y−s dµ(x) dµ(y)

diverges for all measures µ. The case s = d could still be managed by taking the limit s → d−; this was used by M. Götz and E. Saff to obtain asymptotic uniform distribution for N → ∞.

Peter Grabner Point sets of minimal energy

Super-singular energy and discrepancy

For the discrepancy of a point set minimising the energy for s = d an estimate of the form DC

N (XN) = O

• log log N

log N

• could be given (PG & S. Damelin).

Peter Grabner Point sets of minimal energy

Super-singular energy continued

The case s > d was open for a longer period of time. In 2005

• D. Hardin and E. Saff could show that for s > d the point sets of

minimal energy are asymptotically uniformly distributed.

Peter Grabner Point sets of minimal energy

Super-singular energy continued

The case s > d was open for a longer period of time. In 2005

• D. Hardin and E. Saff could show that for s > d the point sets of

minimal energy are asymptotically uniformly distributed. They could even prove more: For any rectifiable manifold M the minimal energy point sets are asymptotically uniformly distributed with respect to the normalised Hausdorff measure on M.

Peter Grabner Point sets of minimal energy

Super-singular energy continued

The case s > d was open for a longer period of time. In 2005

• D. Hardin and E. Saff could show that for s > d the point sets of

minimal energy are asymptotically uniformly distributed. They could even prove more: For any rectifiable manifold M the minimal energy point sets are asymptotically uniformly distributed with respect to the normalised Hausdorff measure on M. There exists a constant C(s, d) such that lim

N→∞

Es(XN) N1+ s

d

= C(s, d) H(M)

s d

The constant C(s, d) is again conjectured to be expressible in terms of the zeta-function of a lattice.

Peter Grabner Point sets of minimal energy

Super-singular energy continued

As opposed to the case s < d the energy for s > d mostly depends

• n local (short range) interactions between the points. This

provides a heuristic explanation of why the approaches for s < d are not applicable for the singular case.

Peter Grabner Point sets of minimal energy

Super-singular energy continued

As opposed to the case s < d the energy for s > d mostly depends

• n local (short range) interactions between the points. This

provides a heuristic explanation of why the approaches for s < d are not applicable for the singular case. On the other hand the dominance of the short range interactions allows to show separation results for minimal energy point sets by an easy argument δN(XN) = min

i=j xi − xj ≥ C

N

1 d

.

Peter Grabner Point sets of minimal energy

Super-singular energy continued

For s → ∞ the problem of minimising the s-energy becomes the problem of maximising the minimal distance δN(XN), the problem

• f best packing.

Peter Grabner Point sets of minimal energy

Super-singular energy continued

For s → ∞ the problem of minimising the s-energy becomes the problem of maximising the minimal distance δN(XN), the problem

• f best packing.

In the plane the problem of best packing is solved by the hexagonal lattice, which supports the conjecture about the constant C(s, 2).

Peter Grabner Point sets of minimal energy

Non-singular energy

For every positive definite zonal function g(t) =

∞

• k=0

Z(d, k)akPk(t) the corresponding energy functional of a set XN is defined by Eg(XN) =

N

• i,j=1

g(xi, xj).

Peter Grabner Point sets of minimal energy

Non-singular energy continued

As in the potential-theoretic case minimal energy configurations for N → ∞ tend to the unique minimiser of the energy integral Ig(µ) =

• Sd×Sd g(x, y) dµ(x) dµ(y)

amongst all Borel probability measures; the normalised surface measure σd on Sd

Peter Grabner Point sets of minimal energy

Non-singular energy continued

As in the potential-theoretic case minimal energy configurations for N → ∞ tend to the unique minimiser of the energy integral Ig(µ) =

• Sd×Sd g(x, y) dµ(x) dµ(y)

amongst all Borel probability measures; the normalised surface measure σd on Sd in the sense 1 N

N

• i=1

δxi ⇀ σd.

Peter Grabner Point sets of minimal energy

Example

For a reproducing kernel Hilbert space H with kernel K take g = K. Then the worst case integration error satisfies wce(XN, H)2 = Eg(XN).

Peter Grabner Point sets of minimal energy

Example

For a reproducing kernel Hilbert space H with kernel K take g = K. Then the worst case integration error satisfies wce(XN, H)2 = Eg(XN). Minimising the worst case integration error gives asymptotically uniformly distributed point sets.

Peter Grabner Point sets of minimal energy

Interpretation of L2-Discrepancy

“Random Functions” If g(1) exists we can define a stochastic process by the covariance matrix (g(xi, xj))i,j . Then the mean square of the integration error (with respect to the measure defined by the process) satisfies

• C(Sd)
• 1

N

N

• n=1

y(xn) −

• Sd y(x) dσ(x)

2 dλ(y) = 1 N2 Eg(XN) − a0.

Peter Grabner Point sets of minimal energy

Example

For instance, the process can be chosen in a way that

1 N2 Eg(XN) − a0 equals the usual L2-discrepancy

π

• Sd
• 1

N

N

• n=1

χB(x,r)(xn) − σ(B(x, r)) 2 dσ(x dr. The process can be given explicitely by Y (x) =

∞

• n=0

Z(d,n)

• k=1

An,kYn,k(x), where the An,k’s are independent normal random variables with mean 0 and variance

an Z(d,n).

Peter Grabner Point sets of minimal energy