Point distributions on the sphere: energy minimization, discrepancy, - PowerPoint PPT Presentation

Point distributions on the sphere: energy minimization, discrepancy, and more. Dmitriy Bilyk University of Minnesota ICERM Semester Program on “Point Configurations in Geometry, Physics and Computer Science” Workshop “Optimal and Random Point Configurations” Providence, RI February 27, 2018 Dmitriy Bilyk Points on the sphere

Discrepancy U : a set with a natural probability measure µ (e.g., [0 , 1] d , T d , S d , etc.) Dmitriy Bilyk Points on the sphere

Discrepancy U : a set with a natural probability measure µ (e.g., [0 , 1] d , T d , S d , etc.) A - a collection of subsets of U (“test sets”, e.g., balls, cubes, convex sets, spherical caps) Dmitriy Bilyk Points on the sphere

Discrepancy U : a set with a natural probability measure µ (e.g., [0 , 1] d , T d , S d , etc.) A - a collection of subsets of U (“test sets”, e.g., balls, cubes, convex sets, spherical caps) Let Z = { z 1 , . . . , z N } ⊂ U be an N -point set. Dmitriy Bilyk Points on the sphere

Discrepancy U : a set with a natural probability measure µ (e.g., [0 , 1] d , T d , S d , etc.) A - a collection of subsets of U (“test sets”, e.g., balls, cubes, convex sets, spherical caps) Let Z = { z 1 , . . . , z N } ⊂ U be an N -point set. Discrepancy of Z with respect to A : � � � � #( Z ∩ A ) � � D A ( Z ) = sup − µ ( A ) � . � N A ∈A Dmitriy Bilyk Points on the sphere

Discrepancy U : a set with a natural probability measure µ (e.g., [0 , 1] d , T d , S d , etc.) A - a collection of subsets of U (“test sets”, e.g., balls, cubes, convex sets, spherical caps) Let Z = { z 1 , . . . , z N } ⊂ U be an N -point set. 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 Points on the sphere

Discrepancy U : a set with a natural probability measure µ (e.g., [0 , 1] d , T d , S d , etc.) A - a collection of subsets of U (“test sets”, e.g., balls, cubes, convex sets, spherical caps) Let Z = { z 1 , . . . , z N } ⊂ U be an N -point set. 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 Points on the sphere

Disk (ball) disrepancy on T d Theorem (Montgomery; Beck; 80’s) For any N -point set Z = { z 1 , . . . , z N } ⊂ T 2 ≃ [0 , 1) 2 there exists a disk D ⊂ T 2 of radius 1 / 4 or 1 / 2 such that � � � � # { 1 ≤ i ≤ N : z i ∈ D } � � � � N − 3 4 . − | D | � N Dmitriy Bilyk Points on the sphere

Disk (ball) disrepancy on T d Theorem (Montgomery; Beck; 80’s) For any N -point set Z = { z 1 , . . . , z N } ⊂ T 2 ≃ [0 , 1) 2 there exists a disk D ⊂ T 2 of radius 1 / 4 or 1 / 2 such that � � � � # { 1 ≤ i ≤ N : z i ∈ D } � � � � N − 3 4 . − | D | � N Higher-dimensional version for Z ⊂ T d holds with N − 1 2 − 1 2 d . Dmitriy Bilyk Points on the sphere

Disk (ball) disrepancy on T d Theorem (Montgomery; Beck; 80’s) For any N -point set Z = { z 1 , . . . , z N } ⊂ T 2 ≃ [0 , 1) 2 there exists a disk D ⊂ T 2 of radius 1 / 4 or 1 / 2 such that � � � � # { 1 ≤ i ≤ N : z i ∈ D } � � � � N − 3 4 . − | D | � N Higher-dimensional version for Z ⊂ T d holds with N − 1 2 − 1 2 d . Is one radius enough? Still an open question! Dmitriy Bilyk Points on the sphere

Disk (ball) disrepancy on T d Theorem (Montgomery; Beck; 80’s) For any N -point set Z = { z 1 , . . . , z N } ⊂ T 2 ≃ [0 , 1) 2 there exists a disk D ⊂ T 2 of radius 1 / 4 or 1 / 2 such that � � � � # { 1 ≤ i ≤ N : z i ∈ D } � � � � N − 3 4 . − | D | � N Higher-dimensional version for Z ⊂ T d holds with N − 1 2 − 1 2 d . Is one radius enough? Still an open question! Sharp up to logarithms: jittered sampling. Dmitriy Bilyk Points on the sphere

Disk (ball) disrepancy on T d Theorem (Montgomery; Beck; 80’s) For any N -point set Z = { z 1 , . . . , z N } ⊂ T 2 ≃ [0 , 1) 2 there exists a disk D ⊂ T 2 of radius 1 / 4 or 1 / 2 such that � � � � # { 1 ≤ i ≤ N : z i ∈ D } � � � � N − 3 4 . − | D | � N Higher-dimensional version for Z ⊂ T d holds with N − 1 2 − 1 2 d . Is one radius enough? Still an open question! Sharp up to logarithms: jittered sampling. Sharp in L 2 sense: lattice. Dmitriy Bilyk Points on the sphere

L 2 discrepancy: lattice vs jittered sampling Denote � � � 2 � � # { Z ∩ B r ( x ) } D 2 � � L 2 ( Z ) = − | B r | dx � � N T d Dmitriy Bilyk Points on the sphere

L 2 discrepancy: lattice vs jittered sampling Denote � � � 2 � � # { Z ∩ B r ( x ) } D 2 � � L 2 ( Z ) = − | B r | dx � � N T d �� r 1 � � M , . . . , r d ⊂ T d Let L M = : r i = 0 , 1 , . . . , M − 1 M Dmitriy Bilyk Points on the sphere

L 2 discrepancy: lattice vs jittered sampling Denote � � � 2 � � # { Z ∩ B r ( x ) } D 2 � � L 2 ( Z ) = − | B r | dx � � N T d �� r 1 � � M , . . . , r d ⊂ T d and set Let L M = : r i = 0 , 1 , . . . , M − 1 M � � � � L jittered D lattice ( M ) = D L 2 L M and D jittered ( M ) = E D L 2 . M Dmitriy Bilyk Points on the sphere

L 2 discrepancy: lattice vs jittered sampling Denote � � � 2 � � # { Z ∩ B r ( x ) } D 2 � � L 2 ( Z ) = − | B r | dx � � N T d �� r 1 � � M , . . . , r d ⊂ T d and set Let L M = : r i = 0 , 1 , . . . , M − 1 M � � � � L jittered D lattice ( M ) = D L 2 L M and D jittered ( M ) = E D L 2 . M Theorem (Chen, Travaglini, ’08) For d = 1 or 2 , D lattice ( M ) < D jittered ( M ) . Dmitriy Bilyk Points on the sphere

L 2 discrepancy: lattice vs jittered sampling Denote � � � 2 � � # { Z ∩ B r ( x ) } D 2 � � L 2 ( Z ) = − | B r | dx � � N T d �� r 1 � � M , . . . , r d ⊂ T d and set Let L M = : r i = 0 , 1 , . . . , M − 1 M � � � � L jittered D lattice ( M ) = D L 2 L M and D jittered ( M ) = E D L 2 . M Theorem (Chen, Travaglini, ’08) For d = 1 or 2 , D lattice ( M ) < D jittered ( M ) . For large d , D lattice ( M ) > D jittered ( M ) . Dmitriy Bilyk Points on the sphere

L 2 discrepancy: lattice vs jittered sampling Denote � � � 2 � � # { Z ∩ B r ( x ) } D 2 � � L 2 ( Z ) = − | B r | dx � � N T d �� r 1 � � M , . . . , r d ⊂ T d and set Let L M = : r i = 0 , 1 , . . . , M − 1 M � � � � L jittered D lattice ( M ) = D L 2 L M and D jittered ( M ) = E D L 2 . M Theorem (Chen, Travaglini, ’08) For d = 1 or 2 , D lattice ( M ) < D jittered ( M ) . For large d , D lattice ( M ) > D jittered ( M ) . unless d ≡ 1 mod 4 and d > 1 . Dmitriy Bilyk Points on the sphere

Montgomery’s lower bound Let Z = { z 1 , . . . , z N } ⊂ T 2 . Then # { Z ∩ B r ( x ) } − | B r | = ( 1 B r ∗ D Z ) ( x ) , N N � where D Z = 1 δ z i − λ 2 (discrepancy measure). N i =1 � 1 B r ( n ) | 2 · | � D 2 | � D Z ( n ) | 2 L 2 ( Z ) = n ∈ Z 2 � r 1 B r ( n ) = | n | J 1 (2 π | n | r ) - Bessel function of the first kind � 2 πt cos( t − 3 π/ 4) + O ( t − 3 / 2 ) J 1 ( t ) = Dmitriy Bilyk Points on the sphere

Montgomery’s lower bound Let Z = { z 1 , . . . , z N } ⊂ T 2 . Then # { Z ∩ B r ( x ) } − | B r | = ( 1 B r ∗ D Z ) ( x ) , N N � where D Z = 1 δ z i − λ 2 (discrepancy measure). N i =1 � 1 B r ( n ) | 2 · | � D 2 | � D Z ( n ) | 2 L 2 ( Z ) = n ∈ Z 2 � r 1 B r ( n ) = | n | J 1 (2 π | n | r ) - Bessel function of the first kind � 2 πt cos( t − 3 π/ 4) + O ( t − 3 / 2 ) J 1 ( t ) = Dmitriy Bilyk Points on the sphere

Montgomery’s lower bound Let Z = { z 1 , . . . , z N } ⊂ T 2 . Then # { Z ∩ B r ( x ) } − | B r | = ( 1 B r ∗ D Z ) ( x ) , N N � where D Z = 1 δ z i − λ 2 (discrepancy measure). N i =1 � 1 B r ( n ) | 2 · | � D 2 | � D Z ( n ) | 2 L 2 ( Z ) = n ∈ Z 2 � r 1 B r ( n ) = | n | J 1 (2 π | n | r ) - Bessel function of the first kind � 2 πt cos( t − 3 π/ 4) + O ( t − 3 / 2 ) J 1 ( t ) = Dmitriy Bilyk Points on the sphere

Montgomery’s lower bound Let Z = { z 1 , . . . , z N } ⊂ T 2 . Then # { Z ∩ B r ( x ) } − | B r | = ( 1 B r ∗ D Z ) ( x ) , N N � where D Z = 1 δ z i − λ 2 (discrepancy measure). N i =1 � 1 B r ( n ) | 2 · | � D 2 | � D Z ( n ) | 2 L 2 ( Z ) = n ∈ Z 2 � r 1 B r ( n ) = | n | J 1 (2 π | n | r ) - Bessel function of the first kind � 2 πt cos( t − 3 π/ 4) + O ( t − 3 / 2 ) J 1 ( t ) = Dmitriy Bilyk Points on the sphere

Montgomery’s lower bound Let Z = { z 1 , . . . , z N } ⊂ T 2 . Then # { Z ∩ B r ( x ) } − | B r | = ( 1 B r ∗ D Z ) ( x ) , N N � where D Z = 1 δ z i − λ 2 (discrepancy measure). N i =1 � 1 B r ( n ) | 2 · | � D 2 | � D Z ( n ) | 2 L 2 ( Z ) = n ∈ Z 2 � r 1 B r ( n ) = | n | J 1 (2 π | n | r ) - Bessel function of the first kind � 2 πt cos( t − 3 π/ 4) + O ( t − 3 / 2 ) J 1 ( t ) = Dmitriy Bilyk Points on the sphere