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 ⊂ R p ◮ 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). Ω N ∋ ( x 1 , . . . , x N ) ◮ 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). � Ω N ∋ ( x 1 , . . . , x N ) �→ g ( x i , x j ) i � j ◮ 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). � � Ω N ∋ ( x 1 , . . . , x N ) �→ g ( x i , x j ) + q ( x i ) i � j i ◮ 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). � � Ω N ∋ ( x 1 , . . . , x N ) �→ g ( x i , x j ) + τ ( N ) q ( x i ) i � j i ◮ 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 ) : ( x 1 , . . . , x N ) �→ g ( x i , x j ) + τ ( N ) q ( x i ) , ( ⋆ ) i � j i ◮ 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 � i � j � x i − x j � − 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 � i � j � x i − x j � − 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 � i � j � x i − x j � − 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 ⊂ R p 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 �→ g ( x , y ) d µ ( x ) d µ ( y ) Ω × Ω ◮ 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 �→ g ( x , y ) d µ ( x ) d µ ( y ) + q ( x ) d µ ( x ) Ω × Ω Ω ◮ 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 �→ g ( x , y ) d µ ( x ) d µ ( y ) + q ( x ) d µ ( x ) ( ⋆⋆ ) Ω × Ω Ω ◮ 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 ( ω N ; g , q ) � g ( x i , x j ) + τ ( N ) q ( x i ) , ( ⋆ ) i � j i ∬ ∫ I ( µ ; g , q ) � g ( x , y ) d µ ( x ) d µ ( y ) + q ( x ) d µ ( x ) ( ⋆⋆ ) Ω × Ω Ω 6
Discrete vs. continuous � � E ( ω N ; g , q ) � g ( x i , x j ) + τ ( N ) q ( x i ) , ( ⋆ ) i � j i ∬ ∫ I ( µ ; g , q ) � g ( x , y ) d µ ( x ) d µ ( y ) + q ( x ) d µ ( x ) ( ⋆⋆ ) Ω × Ω Ω Theorem (Frostman, Fekete, Choquet, etc.) For any sequence of minimizers ˆ ω N , N ≥ 2 , of ( ⋆ ), associate the normalized counting measures N � x N ) ←→ 1 ω N : � (ˆ ˆ x 1 , . . . , ˆ δ ˆ x i . N i � 1 Then any weak ∗ limit ˆ µ is a minimizer of ( ⋆⋆ ). Ditto for renormalized values of minima: E ( ˆ ω N ; g , q ) −→ I ( ˆ µ ; g , q ) , N → ∞ . N 2 6
Discrete vs. continuous � � E ( ω N ; g , q ) � g ( x i , x j ) + τ ( N ) q ( x i ) , ( ⋆ ) i � j i ∬ ∫ I ( µ ; g , q ) � g ( x , y ) d µ ( x ) d µ ( y ) + q ( x ) d µ ( x ) ( ⋆⋆ ) Ω × Ω Ω Theorem (Frostman, Fekete, Choquet, etc.) For any sequence of minimizers ˆ ω N , N ≥ 2 , of ( ⋆ ), associate the normalized counting measures N � x N ) ←→ 1 ω N : � (ˆ ˆ x 1 , . . . , ˆ δ ˆ x i . N i � 1 Then any weak ∗ limit ˆ µ is a minimizer of ( ⋆⋆ ). Ditto for renormalized values of minima: E ( ˆ ω N ; g , q ) −→ I ( ˆ µ ; g , q ) , N → ∞ . N 2 ◮ This convergence allows to analyze the asymptotics of ( ⋆ ), and to compute ( ⋆⋆ ). 6
Riesz kernel ◮ Ω is compact. Hausdorff dimension dim H Ω � d . RK is given by g s ( x , y ) : � � x − y � − s for s > 0. 7
Riesz kernel ◮ Ω is compact. Hausdorff dimension dim H Ω � d . RK is given by g s ( x , y ) : � � x − y � − s for s > 0. ◮ positive definite ◮ harmonic when s � d − 2 7
Riesz kernel ◮ Ω is compact. Hausdorff dimension dim H Ω � d . RK is given by g s ( x , y ) : � � x − y � − s for s > 0. ◮ positive definite ◮ harmonic when s � d − 2 ◮ scale-invariant. Thus possible to compute the scaling factor: N , s < d , τ ( N ) � τ s , d ( N ) : � N log N , s � d , N s / d , s > d . 7
Riesz kernel ◮ Ω is compact. Hausdorff dimension dim H Ω � d . RK is given by g s ( x , y ) : � � x − y � − s for s > 0. ◮ positive definite ◮ harmonic when s � d − 2 ◮ scale-invariant. Thus possible to compute the scaling factor: N , s < d , τ ( N ) � τ s , d ( N ) : � N log N , s � d , N s / 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 ) g s ( 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 ) g s ( x , y ) ◮ When s ≥ d , energies of continuous w.r.t. H d measures are not defined. E ( ˆ ω N ; g s , q ) −→ ? , N → ∞ . N 1 + s / d ( 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 { F n } functionals on X , element x ∈ X fixed. We say that the Γ -convergence at x holds, Γ - lim n →∞ F n ( x ) � F ( x ) if 1. for every sequence { x n } ⊂ X such that lim n →∞ x n � x , there holds lim inf n →∞ F n ( x n ) ≥ F ( x ) ; 9
Γ -convergence in definitions [ De Giorgi - Franzoni ’75 ] X –compact metric space; F and { F n } functionals on X , element x ∈ X fixed. We say that the Γ -convergence at x holds, Γ - lim n →∞ F n ( x ) � F ( x ) if 1. for every sequence { x n } ⊂ X such that lim n →∞ x n � x , there holds lim inf n →∞ F n ( x n ) ≥ F ( x ) ; 2. there exists a sequence for which lim n →∞ x n � x and lim n →∞ F n ( x n ) � F ( x ) . If the convergence holds at every point x ∈ X , we say that the functionals have the Γ -limit: Γ - lim n →∞ F n � F . 9
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