-convergence of hypersingular Riesz energy functionals Alex - - PowerPoint PPT Presentation

Γ-convergence of hypersingular Riesz energy functionals

Alex Vlasiuk

Florida State University

Multivariate Algorithms and their Foundations in Number Theory November 2018

Based on joint work with Douglas Hardin and Edward Saff

Discrete energy problem

Diverse physical phenomena can be modeled as N particles (bodies) with given pairwise interactions: gravitational, electrostatic, transport (with additional constraints). Ω ⊂ Rp

◮ Ω is compact.

3

Discrete energy problem

Diverse physical phenomena can be modeled as N particles (bodies) with given pairwise interactions: gravitational, electrostatic, transport (with additional constraints). ΩN ∋ (x1, . . . , xN)

◮ Ω is compact.

3

Discrete energy problem

Diverse physical phenomena can be modeled as N particles (bodies) with given pairwise interactions: gravitational, electrostatic, transport (with additional constraints). ΩN ∋ (x1, . . . , xN) →

• ij

g(xi, xj)

◮ Ω is compact. ◮ g(x, y) stands for pairwise interactions. Lower semicontinuous; can be infinite

when x y.

3

Discrete energy problem

Diverse physical phenomena can be modeled as N particles (bodies) with given pairwise interactions: gravitational, electrostatic, transport (with additional constraints). ΩN ∋ (x1, . . . , xN) →

• ij

g(xi, xj) +

• i

q(xi)

◮ Ω is compact. ◮ g(x, y) stands for pairwise interactions. Lower semicontinuous; can be infinite

when x y.

◮ q can be confining potential; introduces additional data; l.s.c.

3

Discrete energy problem

Diverse physical phenomena can be modeled as N particles (bodies) with given pairwise interactions: gravitational, electrostatic, transport (with additional constraints). ΩN ∋ (x1, . . . , xN) →

• ij

g(xi, xj) + τ(N)

• i

q(xi)

◮ Ω is compact. ◮ g(x, y) stands for pairwise interactions. Lower semicontinuous; can be infinite

when x y.

◮ q can be confining potential; introduces additional data; l.s.c. ◮ the way τ(N) grows depends on g. Integrable g ⇒ τ(N) N.

3

Discrete energy problem

Diverse physical phenomena can be modeled as N particles (bodies) with given pairwise interactions: gravitational, electrostatic, transport (with additional constraints). E(· ; g, q) : (x1, . . . , xN) →

• ij

g(xi, xj) + τ(N)

• i

q(xi), (⋆)

◮ Ω is compact. ◮ g(x, y) stands for pairwise interactions. Lower semicontinuous; can be infinite

when x y.

◮ q can be confining potential; introduces additional data; l.s.c. ◮ the way τ(N) grows depends on g. Integrable g ⇒ τ(N) N.

3

An example

◮ (Uniform) random points exhibit clustering

4

An example

◮ (Uniform) random points exhibit clustering ◮ Detecting and removing it by minimizing

ij xi − xj−s for a fixed s > 2

Delaunay triangulations of: left, 500 uniformly random nodes in [0, 1]2; right, output of the Riesz gradient flow for the same number of points.

4

An example

◮ (Uniform) random points exhibit clustering ◮ Detecting and removing it by minimizing

ij xi − xj−s for a fixed s > 2

Delaunay triangulations of: left, 500 uniformly random nodes in [0, 1]2; right, output of the Riesz gradient flow for the same number of points.

4

An example

◮ (Uniform) random points exhibit clustering ◮ Detecting and removing it by minimizing

ij xi − xj−s for a fixed s > 2

Delaunay triangulations of: left, 500 uniformly random nodes in [0, 1]2; right, output of the Riesz gradient flow for the same number of points.

4

Continuous problem

The analog for integrals w.r.t. continuous measure instead of summations over discrete points. Ω ⊂ Rp

5

Continuous problem

The analog for integrals w.r.t. continuous measure instead of summations over discrete points. P(Ω) ∋ µ

5

Continuous problem

The analog for integrals w.r.t. continuous measure instead of summations over discrete points. P(Ω) ∋ µ →

∬

Ω×Ω

g(x, y)dµ(x)dµ(y)

◮ g(x, y) denotes pairwise interaction.

5

Continuous problem

The analog for integrals w.r.t. continuous measure instead of summations over discrete points. P(Ω) ∋ µ →

∬

Ω×Ω

g(x, y)dµ(x)dµ(y) +

∫

Ω

q(x)dµ(x)

◮ g(x, y) denotes pairwise interaction. ◮ q can be confining potential; introduces additional data; l.s.c.

5

Continuous problem

The analog for integrals w.r.t. continuous measure instead of summations over discrete points. I(· ; g, q) : µ →

∬

Ω×Ω

g(x, y)dµ(x)dµ(y) +

∫

Ω

q(x)dµ(x) (⋆⋆)

◮ g(x, y) denotes pairwise interaction. ◮ q can be confining potential; introduces additional data; l.s.c. ◮ a positive definite g(x, y) corresponds to a scalar product on P(Ω). ◮ with harmonic Riesz kernel g(x, y), (⋆⋆) is equivalent to the obstacle problem. ◮ potential-theoretic tools, balayage.

5

Discrete vs. continuous

E(ωN; g, q)

• ij

g(xi, xj) + τ(N)

• i

q(xi), (⋆) I(µ; g, q)

∬

Ω×Ω

g(x, y)dµ(x)dµ(y) +

∫

Ω

q(x)dµ(x) (⋆⋆)

6

Discrete vs. continuous

E(ωN; g, q)

• ij

g(xi, xj) + τ(N)

• i

q(xi), (⋆) I(µ; g, q)

∬

Ω×Ω

g(x, y)dµ(x)dµ(y) +

∫

Ω

q(x)dµ(x) (⋆⋆)

Theorem (Frostman, Fekete, Choquet, etc.)

For any sequence of minimizers ˆ ωN, N ≥ 2, of (⋆), associate the normalized counting measures ˆ ωN : (ˆ x1, . . . , ˆ xN) ←→ 1 N

N

• i1

δˆ

xi.

Then any weak∗ limit ˆ µ is a minimizer of (⋆⋆). Ditto for renormalized values of minima: E( ˆ ωN; g, q) N2 −→ I( ˆ µ; g, q), N → ∞.

6

Discrete vs. continuous

E(ωN; g, q)

• ij

g(xi, xj) + τ(N)

• i

q(xi), (⋆) I(µ; g, q)

∬

Ω×Ω

g(x, y)dµ(x)dµ(y) +

∫

Ω

q(x)dµ(x) (⋆⋆)

Theorem (Frostman, Fekete, Choquet, etc.)

For any sequence of minimizers ˆ ωN, N ≥ 2, of (⋆), associate the normalized counting measures ˆ ωN : (ˆ x1, . . . , ˆ xN) ←→ 1 N

N

• i1

δˆ

xi.

Then any weak∗ limit ˆ µ is a minimizer of (⋆⋆). Ditto for renormalized values of minima: E( ˆ ωN; g, q) N2 −→ I( ˆ µ; g, q), N → ∞.

◮ This convergence allows to analyze the asymptotics of (⋆), and to compute (⋆⋆).

6

Riesz kernel

◮ Ω is compact. Hausdorff dimension dimH Ω d.

RK is given by gs(x, y) : x − y−s for s > 0.

7

Riesz kernel

◮ Ω is compact. Hausdorff dimension dimH Ω d.

RK is given by gs(x, y) : x − y−s for s > 0.

◮ positive definite ◮ harmonic when s d − 2

7

Riesz kernel

◮ Ω is compact. Hausdorff dimension dimH Ω d.

RK is given by gs(x, y) : x − y−s for s > 0.

◮ positive definite ◮ harmonic when s d − 2 ◮ scale-invariant. Thus possible to compute the scaling factor:

τ(N) τs,d(N) :

        

N, s < d, N log N, s d, Ns/d, s > d.

7

Riesz kernel

◮ Ω is compact. Hausdorff dimension dimH Ω d.

RK is given by gs(x, y) : x − y−s for s > 0.

◮ positive definite ◮ harmonic when s d − 2 ◮ scale-invariant. Thus possible to compute the scaling factor:

τ(N) τs,d(N) :

        

N, s < d, N log N, s d, Ns/d, s > d.

7

Our kernel

◮ For s ≥ d, add a multiplicative weight κ(x, y) : Ω × Ω → R+, continuous at

diag (Ω × Ω): g(x, y) κ(x, y)gs(x, y)

8

Our kernel

◮ For s ≥ d, add a multiplicative weight κ(x, y) : Ω × Ω → R+, continuous at

diag (Ω × Ω): g(x, y) κ(x, y)gs(x, y)

◮ When s ≥ d, energies of continuous w.r.t. H

d measures are not defined.

E( ˆ ωN; gs, q) N1+s/d −→ ?, N → ∞. (H

d is normalized as the d-dimensional Lebesgue measure.)

8

Γ-convergence

in definitions

[De Giorgi - Franzoni ’75] X–compact metric space; F and {Fn} functionals on X, element x ∈ X fixed. We say that the Γ-convergence at x holds, Γ- lim

n→∞ Fn(x) F(x)

if

• 1. for every sequence {xn} ⊂ X such that limn→∞ xn x, there holds

lim infn→∞ Fn(xn) ≥ F(x);

9

Γ-convergence

in definitions

[De Giorgi - Franzoni ’75] X–compact metric space; F and {Fn} functionals on X, element x ∈ X fixed. We say that the Γ-convergence at x holds, Γ- lim

n→∞ Fn(x) F(x)

if

• 1. for every sequence {xn} ⊂ X such that limn→∞ xn x, there holds

lim infn→∞ Fn(xn) ≥ F(x);

• 2. there exists a sequence for which limn→∞ xn x and limn→∞ Fn(xn) F(x).

If the convergence holds at every point x ∈ X, we say that the functionals have the Γ-limit: Γ- lim

n→∞ Fn F.

9

Γ-convergence

in definitions

[De Giorgi - Franzoni ’75] X–compact metric space; F and {Fn} functionals on X, element x ∈ X fixed. We say that the Γ-convergence at x holds, Γ- lim

n→∞ Fn(x) F(x)

if

• 1. for every sequence {xn} ⊂ X such that limn→∞ xn x, there holds

lim infn→∞ Fn(xn) ≥ F(x);

• 2. there exists a sequence for which limn→∞ xn x and limn→∞ Fn(xn) F(x).

If the convergence holds at every point x ∈ X, we say that the functionals have the Γ-limit: Γ- lim

n→∞ Fn F.

Note:

◮ for the Γ-limit at x, none of Fn(x) has to be finite ◮ if the functionals Γ-converge, minimizers converge to minimizers ◮ sequence in 2 is called the recovery sequence

9

Γ-convergence

in pictures

Example Γ- lim n→∞(x2 + sin(nx)) x2 − 1

2 1 1 2 3 4 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 x2

x2 + sin(10x)

10

Γ-convergence

in pictures

Example Γ- lim n→∞(x2 + sin(nx)) x2 − 1

2 1 1 2 3 4 0.0 2.5 5.0 7.5 10.0 12.5 15.0 x2

x2 + sin(20x)

10

Γ-convergence

in pictures

Example Γ- lim n→∞(x2 + sin(nx)) x2 − 1

2 1 1 2 3 4 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 x2

x2 + sin(100x)

10

Γ-convergence

in pictures

Example Γ- lim n→∞(x2 + sin(nx)) x2 − 1

2 1 1 2 3 4 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 x2

x2 + sin(1000x)

10

Γ-convergence

in pictures

Example Γ- lim n→∞(x2 + sin(nx)) x2 − 1

2 1 1 2 3 4 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 x2 x2 1

x2 + sin(1000x)

10

Γ-convergence for s < d

◮ in the case of arbitrary lower semicontinuous kernel g, the expression ∬

Ω×Ω

g(x, y)dµ(x)dν(y) is lower semicontinuous on P(Ω) × P(Ω) as a function of (µ, ν).

11

Γ-convergence for s < d

◮ in the case of arbitrary lower semicontinuous kernel g, the expression ∬

Ω×Ω

g(x, y)dµ(x)dν(y) is lower semicontinuous on P(Ω) × P(Ω) as a function of (µ, ν).

◮

1 N2 EN(µ; g, q) :

• 1

N2 E({x : x ∈ supp µ}; g, q)

if µ ∈ P

N(Ω),

+∞,

• therwise.

11

Γ-convergence for s < d

◮ in the case of arbitrary lower semicontinuous kernel g, the expression ∬

Ω×Ω

g(x, y)dµ(x)dν(y) is lower semicontinuous on P(Ω) × P(Ω) as a function of (µ, ν).

◮

1 N2 EN(µ; g, q) :

• 1

N2 E({x : x ∈ supp µ}; g, q)

if µ ∈ P

N(Ω),

+∞,

• therwise.

◮ the theorem above can be restated as

Theorem (Frostman, Fekete, Choquet, etc.)

Suppose g is integrable, l.s.c., positive definite, and let ˆ µ be its (unique) minimizer; let further q be continuous. Then Γ- lim

N→∞

1 N2 EN(·; g, q) ( ˆ µ) I(· ; g, q) ( ˆ µ).

11

Γ-convergence for s > d

◮

1 N1+s/d EN(µ; g, q) :

• 1

N1+s/d E({x : x ∈ supp µ}; g, q)

if µ ∈ P

N(Ω),

+∞,

• therwise.

12

Γ-convergence for s > d

◮

1 N1+s/d EN(µ; g, q) :

• 1

N1+s/d E({x : x ∈ supp µ}; g, q)

if µ ∈ P

N(Ω),

+∞,

• therwise.

◮ writing κ κ(x, x) for brevity, let

S(µ; gs, κ, q) :

• Cs,d

∫

Ω κϕ1+s/d dH d +

∫

Ω qϕ dH d,

µ ϕ dH

d,

+∞,

• therwise.

12

Γ-convergence for s > d

Theorem

Assume Ω is d-rectifiable and d-regular. If κ, q are continuous on diag (Ω × Ω) and Ω respectively, kernel gs is the hypersingular Riesz kernel with s > d dimH Ω, then Γ- lim

N→∞

1 N1+s/d EN(· ; gs, κ, q) S(· ; gs, κ, q),

• n P(Ω) equipped with the weak∗ topology.

◮ as above, recovery sequences are given by appropriately chosen minimizers.

13

Γ-convergence for s > d

Theorem

Assume Ω is d-rectifiable and d-regular. If κ, q are continuous on diag (Ω × Ω) and Ω respectively, kernel gs is the hypersingular Riesz kernel with s > d dimH Ω, then Γ- lim

N→∞

1 N1+s/d EN(· ; gs, κ, q) S(· ; gs, κ, q),

• n P(Ω) equipped with the weak∗ topology.

◮ as above, recovery sequences are given by appropriately chosen minimizers. ◮ the minimizer of

Cs,d

∫

Ω

κϕ1+s/d dH

d +

∫

Ω

qϕ dH

d

is ˆ ϕ(x)

• L1 − q(x)

Cs,d(1 + s/d) κ(x, x)

d/s

+

13

Γ-convergence for s > d

Theorem

Assume Ω is d-rectifiable and d-regular. If κ, q are continuous on diag (Ω × Ω) and Ω respectively, kernel gs is the hypersingular Riesz kernel with s > d dimH Ω, then Γ- lim

N→∞

1 N1+s/d EN(· ; gs, κ, q) S(· ; gs, κ, q),

• n P(Ω) equipped with the weak∗ topology.

◮ as above, recovery sequences are given by appropriately chosen minimizers. ◮ the minimizer of

Cs,d

∫

Ω

κϕ1+s/d dH

d +

∫

Ω

qϕ dH

d

is ˆ ϕ(x)

• L1 − q(x)

Cs,d(1 + s/d) κ(x, x)

d/s

+

◮ here Γ-convergence of functionals, previously merely Γ-limit at ˆ

ϕ

13

Family of energies

14

Family of energies

E(ωN; gs, κ, q) :

• ij

κ(xi, xj)gs(xi, xj) + Ns/d

i

q(xi),

14

Family of energies

E(ωN; gs, κ, q) :

• ij

κ(xi, xj)gs(xi, xj) + Ns/d

i

q(xi),

◮ flexibility in choosing the functional

minimizers converge to

• L1−q(x)

Cs,d(1+s/d) κ(x,x)

d/s

+

dH

d for a normalizing L1

14

Family of energies

E(ωN; gs, κ, q) :

• ij

κ(xi, xj)gs(xi, xj) + Ns/d

i

q(xi),

◮ flexibility in choosing the functional

minimizers converge to

• L1−q(x)

Cs,d(1+s/d) κ(x,x)

d/s

+

dH

d for a normalizing L1

◮ the cost of evaluation of the density defines the costs of κ and q

14

Family of energies

E(ωN; gs, κ, q) :

• ij

κ(xi, xj)gs(xi, xj) + Ns/d

i

q(xi),

◮ flexibility in choosing the functional

minimizers converge to

• L1−q(x)

Cs,d(1+s/d) κ(x,x)

d/s

+

dH

d for a normalizing L1

◮ the cost of evaluation of the density defines the costs of κ and q ◮ local forces in κ vs global in q

14

Family of energies

E(ωN; gs, κ, q) :

• ij

κ(xi, xj)gs(xi, xj) + Ns/d

i

q(xi),

◮ flexibility in choosing the functional

minimizers converge to

• L1−q(x)

Cs,d(1+s/d) κ(x,x)

d/s

+

dH

d for a normalizing L1

◮ the cost of evaluation of the density defines the costs of κ and q ◮ local forces in κ vs global in q ◮ truncating weight to lower computational cost

14

Example: implicit surface 1

15

Example: implicit surface 1

◮ Handle: 3

k1

• (x2

k − 4)2 + 3x2 k+1x2 k+2 + 2xkxk+1xk+2 − 10x2 k

• 15

Example: implicit surface 1

◮ Handle: 3

k1

• (x2

k − 4)2 + 3x2 k+1x2 k+2 + 2xkxk+1xk+2 − 10x2 k

• ◮ 20K points distributed according to x2
• 3. Left: color-coded density, blue/orange is

lower/higher. Right: surface Voronoi diagram.

15

Example: implicit surface 2

16

Example: implicit surface 2

◮ Chmutov-Banchoff-type surface: x2(x2 − 5) + y2(y2 − 5) + z2(z2 − 5) + 11 0

16

Example: implicit surface 2

◮ Chmutov-Banchoff-type surface: x2(x2 − 5) + y2(y2 − 5) + z2(z2 − 5) + 11 0 ◮ 40K points distributed according to the absolute value of the Gaussian

• curvature. Left: color-coded Gaussian curvature, blue/orange is lower/higher. Right:

surface Voronoi diagram.

16

Local properties of minimizers

17

Local properties of minimizers

◮ Optimal order of minimal pairwise distances, N−1/d, for s > d.

17

Local properties of minimizers

◮ Optimal order of minimal pairwise distances, N−1/d, for s > d. ◮ Under mild smoothness assumptions minimizers have the optimal covering

radius of order N−1/d on the sublevel sets {x ∈ Ω : q(x) ≤ u}, for all u < L1.

17

Thank you!

Thank you!

◮ A. Braides. Local Minimization, Variational Evolution and Γ-Convergence. Vol. 2094, Springer International Publishing, 2014. DOI:10.1007/978-3-319-01982-6. ◮ D. P. Hardin, E. B. Saff and O. V. Generating point configurations via hypersingular Riesz energy with an external field, SIAM J. Math. Anal., 49(1), 646-673, 2017 ◮ S. V. Borodachov, D. P. Hardin, and E. B. Saff. Low Complexity Methods For Discretizing Manifolds Via Riesz Energy Minimization. Found. Comput. Math., 14, 2014