Discrepancy and energy optimization on the sphere Dmitriy Bilyk University of Minnesota “Optimal point configurations and orthogonal polynomials” CIEM, Castro Urdiales, Cantabria, Spain April 2017 April 19, 2017 Dmitriy Bilyk Discrepancy and energy optimization
Good point distributions Lattices Energy minimization Monte-Carlo Other random point processes (jittered sampling, determinantal point process) Covering/packing problems Low-discrepancy sets Cubature formulas, numerical integration Uniform tessellation, almost isometric embeddings Dmitriy Bilyk Discrepancy and energy optimization
Discrepancy U : a set with a natural probability measure µ (e.g., [0 , 1] d , S d , R d , etc.) Dmitriy Bilyk Discrepancy and energy optimization
Discrepancy U : a set with a natural probability measure µ (e.g., [0 , 1] d , S d , R d , etc.) A - a collection of subsets of U (“test sets”, e.g., balls, cubes, convex sets, spherical caps) Dmitriy Bilyk Discrepancy and energy optimization
Discrepancy U : a set with a natural probability measure µ (e.g., [0 , 1] d , S d , R d , etc.) A - a collection of subsets of U (“test sets”, e.g., balls, cubes, convex sets, spherical caps) Choose an N -point set in Z ⊂ U Discrepancy of Z with respect to A : � � � � #( Z ∩ A ) � � D A ( Z ) = sup − µ ( A ) � . � N A ∈A Dmitriy Bilyk Discrepancy and energy optimization
Discrepancy U : a set with a natural probability measure µ (e.g., [0 , 1] d , S d , R d , etc.) A - a collection of subsets of U (“test sets”, e.g., balls, cubes, convex sets, spherical caps) Choose an N -point set in Z ⊂ U Discrepancy of Z with respect to A : � � � � #( Z ∩ A ) � � D A ( Z ) = sup − µ ( A ) � . � N A ∈A Optimal discrepancy wrt A : D N ( A ) = # Z = N D A ( Z ) . inf Dmitriy Bilyk Discrepancy and energy optimization
Discrepancy U : a set with a natural probability measure µ (e.g., [0 , 1] d , S d , R d , etc.) A - a collection of subsets of U (“test sets”, e.g., balls, cubes, convex sets, spherical caps) Choose an N -point set in Z ⊂ U Discrepancy of Z with respect to A : � � � � #( Z ∩ A ) � � D A ( Z ) = sup − µ ( A ) � . � N A ∈A Optimal discrepancy wrt A : D N ( A ) = # Z = N D A ( Z ) . inf sup → L 2 -average: L 2 discrepancy. Dmitriy Bilyk Discrepancy and energy optimization
Spherical cap discrepancy For x ∈ S d , t ∈ [ − 1 , 1] define spherical caps: C ( x, t ) = { y ∈ S d : � x, y � ≥ t } . For a finite set Z = { z 1 , z 2 , ..., z N } ⊂ S d define � � � � � � � � # Z ∩ C ( x, t ) � � D cap ( Z ) = sup − σ C ( x, t ) � . � � � N x ∈ S d ,t ∈ [ − 1 , 1] Theorem (Beck, ’84) There exists constants c d , C d > 0 such that 2 d � c d N − 1 2 − 1 # Z = N D cap ( Z ) ≤ C d N − 1 2 − 1 2 d ≤ inf log N. Dmitriy Bilyk Discrepancy and energy optimization
Spherical caps: L 2 Stolarsky Principle Define the spherical cap L 2 discrepancy � � 1 � � � � 1 2 � � 2 # Z ∩ C ( x, t ) � � � � D cap,L 2 ( Z ) = − σ C ( x, t ) dt dσ ( x ) � � . � N � S d − 1 Dmitriy Bilyk Discrepancy and energy optimization
Spherical caps: L 2 Stolarsky Principle Define the spherical cap L 2 discrepancy � � 1 � � � � 1 2 � � 2 # Z ∩ C ( x, t ) � � � � D cap,L 2 ( Z ) = − σ C ( x, t ) dt dσ ( x ) � � . � � N S d − 1 Theorem (Stolarsky invariance principle) For any finite set Z = { z 1 , ..., z N } ⊂ S d N � � 2 � 1 � z i − z j � + c d D L 2 ,cap = const N 2 i,j =1 � � = � x − y � dσ ( x ) dσ ( y ) . S d S d Dmitriy Bilyk Discrepancy and energy optimization
Spherical caps: L 2 Stolarsky Principle Define the spherical cap L 2 discrepancy � � 1 � � � � 1 2 � � 2 � � # Z ∩ C ( x, t ) � � D cap,L 2 ( Z ) = − σ C ( x, t ) dt dσ ( x ) . � � � � N S d − 1 Theorem (Stolarsky invariance principle) For any finite set Z = { z 1 , ..., z N } ⊂ S d � � 2 D cap,L 2 ( Z ) = c d � � N � � x − y � dσ ( x ) dσ ( y ) − 1 = � z i − z j � . N 2 i,j =1 S d S d Dmitriy Bilyk Discrepancy and energy optimization
Spherical caps: L 2 Stolarsky Principle Define the spherical cap L 2 discrepancy � � 1 � � � � 1 2 � � 2 � � # Z ∩ C ( x, t ) � � D cap,L 2 ( Z ) = − σ C ( x, t ) dt dσ ( x ) . � � � � N S d − 1 Theorem (Stolarsky invariance principle) For any finite set Z = { z 1 , ..., z N } ⊂ S d � � 2 D cap,L 2 ( Z ) = c d � � N � � x − y � dσ ( x ) dσ ( y ) − 1 = � z i − z j � . N 2 i,j =1 S d S d Stolarsky ’73, Brauchart, Dick ’12, DB, Dai, Matzke ’16. Dmitriy Bilyk Discrepancy and energy optimization
Spherical caps: L 2 Stolarsky Principle Define the spherical cap discrepancy of fixed height t : 1 / 2 � �� � N � 2 � � � 1 D ( t ) � � L 2 , cap ( Z ) := 1 C ( x,t ) ( z j ) − σ C ( x, t ) dσ ( x ) � � N S d j =1 Dmitriy Bilyk Discrepancy and energy optimization
Spherical caps: L 2 Stolarsky Principle Define the spherical cap discrepancy of fixed height t : 1 / 2 � �� � N � 2 � � � 1 D ( t ) � � L 2 , cap ( Z ) := 1 C ( x,t ) ( z j ) − σ C ( x, t ) dσ ( x ) � � N S d j =1 � � 2 N � � � � � �� 2 . 1 D ( t ) L 2 , cap ( Z ) = C ( z i , t ) ∩ C ( z j , t ) − C ( p, t ) σ σ N 2 i,j =1 Dmitriy Bilyk Discrepancy and energy optimization
Spherical caps: L 2 Stolarsky Principle Define the spherical cap discrepancy of fixed height t : 1 / 2 � �� � N � 2 � � � 1 D ( t ) � � L 2 , cap ( Z ) := 1 C ( x,t ) ( z j ) − σ C ( x, t ) dσ ( x ) � � N S d j =1 � � 2 N � � � � � �� 2 . 1 D ( t ) L 2 , cap ( Z ) = C ( z i , t ) ∩ C ( z j , t ) − C ( p, t ) σ σ N 2 i,j =1 Averaging over t ∈ [ − 1 , 1] Dmitriy Bilyk Discrepancy and energy optimization
Spherical caps: L 2 Stolarsky Principle Define the spherical cap discrepancy of fixed height t : 1 / 2 � �� � N � 2 � � � 1 D ( t ) � � L 2 , cap ( Z ) := 1 C ( x,t ) ( z j ) − σ C ( x, t ) dσ ( x ) � � N S d j =1 � � 2 N � � � � � �� 2 . 1 D ( t ) L 2 , cap ( Z ) = C ( z i , t ) ∩ C ( z j , t ) − C ( p, t ) σ σ N 2 i,j =1 Averaging over t ∈ [ − 1 , 1] � 1 � � C ( x, t ) ∩ C ( y, t ) = 1 − C d � x − y � σ dt − 1 Dmitriy Bilyk Discrepancy and energy optimization
Spherical caps: L 2 Stolarsky Principle Define the spherical cap discrepancy of fixed height t : 1 / 2 � �� � N � 2 � � � 1 D ( t ) � � L 2 , cap ( Z ) := 1 C ( x,t ) ( z j ) − σ C ( x, t ) dσ ( x ) � � N S d j =1 � � 2 N � � � � � �� 2 . 1 D ( t ) L 2 , cap ( Z ) = C ( z i , t ) ∩ C ( z j , t ) − C ( p, t ) σ σ N 2 i,j =1 Averaging over t ∈ [ − 1 , 1] � 1 � � C ( x, t ) ∩ C ( y, t ) = 1 − C d � x − y � σ dt − 1 � 1 � � � � �� 2 dt σ C ( p, t ) = 1 − C d � x − y � dσ ( x ) dσ ( y ) . − 1 S d S d Dmitriy Bilyk Discrepancy and energy optimization
Hemisphere discrepancy L 2 discrepancy for spherical cap discrepancy of fixed height t satisfies: � � 2 N � � � � � �� 2 . 1 D ( t ) L 2 , cap ( Z ) = σ C ( z i , t ) ∩ C ( z j , t ) − σ C ( p, t ) N 2 i,j =1 Dmitriy Bilyk Discrepancy and energy optimization
Hemisphere discrepancy L 2 discrepancy for spherical cap discrepancy of fixed height t satisfies: � � 2 N � � � � � �� 2 . 1 D ( t ) L 2 , cap ( Z ) = σ C ( z i , t ) ∩ C ( z j , t ) − σ C ( p, t ) N 2 i,j =1 Taking t = 0 (i.e. hemispheres): � � C ( z i , t ) ∩ C ( z j , t ) = 1 1 − d ( x, y ) , where d is the 2 normalized geodesic distance d ( x, y ) = 1 π cos − 1 ( x · y ). Dmitriy Bilyk Discrepancy and energy optimization
Hemisphere discrepancy L 2 discrepancy for spherical cap discrepancy of fixed height t satisfies: � � 2 N � � � � � �� 2 . 1 D ( t ) L 2 , cap ( Z ) = σ C ( z i , t ) ∩ C ( z j , t ) − σ C ( p, t ) N 2 i,j =1 Taking t = 0 (i.e. hemispheres): � � C ( z i , t ) ∩ C ( z j , t ) = 1 1 − d ( x, y ) , where d is the 2 normalized geodesic distance d ( x, y ) = 1 π cos − 1 ( x · y ). Theorem (Stolarsky for hemispheres, DB ’16, Skriganov ’16) [ D L 2 , hem ( Z )] 2 = [ D (0) L 2 , cap ( Z )] 2 � � N � = 1 d ( x, y ) dσ ( x ) dσ ( y ) − 1 . d ( z i , z j ) N 2 2 i,j =1 S d S d Dmitriy Bilyk Discrepancy and energy optimization
Hemisphere Stolarsky: simple corollaries N � [ D L 2 , hem ( Z )] 2 = 1 1 2 − 1 . d ( z i , z j ) N 2 2 i,j =1 Dmitriy Bilyk Discrepancy and energy optimization
Hemisphere Stolarsky: simple corollaries N � [ D L 2 , hem ( Z )] 2 = 1 1 2 − 1 . d ( z i , z j ) N 2 2 i,j =1 For any Z = { z 1 , ..., z N } ⊂ S d N � 1 d ( z i , z j ) ≤ 1 N 2 2 i,j =1 Dmitriy Bilyk Discrepancy and energy optimization
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