Uniform distribution: approximating continuous objects by discrete - PowerPoint PPT Presentation

Uniform distribution: approximating continuous objects by discrete ones Dmitriy Bilyk School of Mathematics, University of Minnesota Minneapolis, MN “Introduction to Research” seminar February 10, 2016 Dmitriy Bilyk Uniform distribution: discrete vs. continuous

Uniform distribution of sequences A sequence ( x n ) is uniformly distributed in [0 , 1] iff # { n ≤ N : x n ∈ I } for any interval I ⊂ [0 , 1] : lim = | I | N N →∞ Dmitriy Bilyk Uniform distribution: discrete vs. continuous

Uniform distribution of sequences A sequence ( x n ) is uniformly distributed in [0 , 1] iff # { n ≤ N : x n ∈ I } for any interval I ⊂ [0 , 1] : lim = | I | N N →∞ Equivalently, for all continuous f on [0 , 1]: � 1 � N 1 n =1 f ( x n ) − → 0 f ( x ) dx as N → ∞ . N Dmitriy Bilyk Uniform distribution: discrete vs. continuous

Uniform distribution of sequences A sequence ( x n ) is uniformly distributed in [0 , 1] iff # { n ≤ N : x n ∈ I } for any interval I ⊂ [0 , 1] : lim = | I | N N →∞ Equivalently, for all continuous f on [0 , 1]: � 1 � N 1 n =1 f ( x n ) − → 0 f ( x ) dx as N → ∞ . N Weyl Criterion (1916): ( x n ) is uniformly distributed in [0 , 1] iff for all k ∈ Z , k � = 0 : N 1 � e 2 πikx n = 0 lim N N →∞ n =1 Dmitriy Bilyk Uniform distribution: discrete vs. continuous

Uniform distribution of sequences A sequence ( x n ) is uniformly distributed in [0 , 1] iff # { n ≤ N : x n ∈ I } for any interval I ⊂ [0 , 1] : lim = | I | N N →∞ Equivalently, for all continuous f on [0 , 1]: � 1 � N 1 n =1 f ( x n ) − → 0 f ( x ) dx as N → ∞ . N Weyl Criterion (1916): ( x n ) is uniformly distributed in [0 , 1] iff for all k ∈ Z , k � = 0 : N 1 � e 2 πikx n = 0 lim N N →∞ n =1 The sequence { nθ } is uniformly distributed in [0 , 1] iff θ is irrational. Dmitriy Bilyk Uniform distribution: discrete vs. continuous

Uniform distribution of sequences A sequence ( x n ) is uniformly distributed in [0 , 1] iff # { n ≤ N : x n ∈ I } for any interval I ⊂ [0 , 1] : lim = | I | N N →∞ Equivalently, for all continuous f on [0 , 1]: � 1 � N 1 n =1 f ( x n ) − → 0 f ( x ) dx as N → ∞ . N Weyl Criterion (1916): ( x n ) is uniformly distributed in [0 , 1] iff for all k ∈ Z , k � = 0 : N 1 � e 2 πikx n = 0 lim N N →∞ n =1 The sequence { nθ } is uniformly distributed in [0 , 1] iff θ is irrational. For any subsequence ( n k ) of integers, the sequence { n k θ } is uniformly distributed for a.e. θ . Dmitriy Bilyk Uniform distribution: discrete vs. continuous

Discrepancy Discrepancy of a sequence For a sequence ω = ( ω n ) ∞ n =1 and an interval I ⊂ [0 , 1] consider the quantity ∆ N,I = ♯ { ω n : ω n ∈ I ; n ≤ N } − N | I | . Define � � D N = sup � ∆ N,I � . I ⊂ [0 , 1] Dmitriy Bilyk Uniform distribution: discrete vs. continuous

Discrepancy Discrepancy of a sequence For a sequence ω = ( ω n ) ∞ n =1 and an interval I ⊂ [0 , 1] consider the quantity ∆ N,I = ♯ { ω n : ω n ∈ I ; n ≤ N } − N | I | . Define � � D N = sup � ∆ N,I � . I ⊂ [0 , 1] A sequence ( ω n ) ∞ n =1 is u.d. in [0 , 1] if and only if D N lim = 0 . N N →∞ Dmitriy Bilyk Uniform distribution: discrete vs. continuous

Erd˝ os-Turan inequality Theorem (Erd˝ os-Turan) For any sequence ω ⊂ [0 , 1] we have � � m N � � D N ( ω ) � N 1 � � e 2 πihω n � � m + � � h � � n =1 h =1 for all natural numbers m . Dmitriy Bilyk Uniform distribution: discrete vs. continuous

Erd˝ os-Turan inequality Theorem (Erd˝ os-Turan) For any sequence ω ⊂ [0 , 1] we have � � m N � � D N ( ω ) � N 1 � � e 2 πihω n � � m + � � h � � n =1 h =1 for all natural numbers m . Folklore: misses optimal estimates by a logarithm. Dmitriy Bilyk Uniform distribution: discrete vs. continuous

Erd˝ os-Turan inequality Theorem (Erd˝ os-Turan) For any sequence ω ⊂ [0 , 1] we have � � m N � � D N ( ω ) � N 1 � � e 2 πihω n � � m + � � h � � n =1 h =1 for all natural numbers m . Folklore: misses optimal estimates by a logarithm. E.g., for a badly approximable irrational θ , Erd˝ os-Turan yields D N ( { nθ } ) � log 2 N , while in fact D N ( { nθ } ) � log N. Dmitriy Bilyk Uniform distribution: discrete vs. continuous

Erd˝ os-Turan inequality Theorem (Erd˝ os-Turan) For any sequence ω ⊂ [0 , 1] we have � � m N � � D N ( ω ) � N 1 � � e 2 πihω n � � m + � � h � � n =1 h =1 for all natural numbers m . Folklore: misses optimal estimates by a logarithm. E.g., for a badly approximable irrational θ , Erd˝ os-Turan yields D N ( { nθ } ) � log 2 N , while in fact D N ( { nθ } ) � log N. For ω = { nθ } sharper bounds can be obtained using continued fractions. Dmitriy Bilyk Uniform distribution: discrete vs. continuous

Irregularities of distribution Can discrepancy stay small? Consider a sequence ω = ( ω n ) ∞ n =1 ⊂ [0 , 1]. Dmitriy Bilyk Uniform distribution: discrete vs. continuous

Irregularities of distribution Can discrepancy stay small? Consider a sequence ω = ( ω n ) ∞ n =1 ⊂ [0 , 1]. van der Corput (1934): Can D N ( ω ) be bounded as N → ∞ ? Dmitriy Bilyk Uniform distribution: discrete vs. continuous

Irregularities of distribution Can discrepancy stay small? Consider a sequence ω = ( ω n ) ∞ n =1 ⊂ [0 , 1]. van der Corput (1934): Can D N ( ω ) be bounded as N → ∞ ? van Aardenne-Ehrenfest (1945): NO! Dmitriy Bilyk Uniform distribution: discrete vs. continuous

Irregularities of distribution Can discrepancy stay small? Consider a sequence ω = ( ω n ) ∞ n =1 ⊂ [0 , 1]. van der Corput (1934): Can D N ( ω ) be bounded as N → ∞ ? van Aardenne-Ehrenfest (1945): NO! Theorem (K. Roth, 1954) The following are equivalent: (i) For every ω = ( ω n ) ∞ n =1 ⊂ [0 , 1] , D N ( ω ) � f ( N ) for infinitely many values of N . (ii) For any distribution P N ⊂ [0 , 1] 2 of N points, � � � � sup � # P N ∩ R − N · | R | � � f ( N ) R − rectangle Dmitriy Bilyk Uniform distribution: discrete vs. continuous

Irregularities of Distribution: simplest example X – roll of a single die Dmitriy Bilyk Uniform distribution: discrete vs. continuous

Irregularities of Distribution: simplest example X – roll of a single die � � � ≥ 1 � � � X − E X 2 Dmitriy Bilyk Uniform distribution: discrete vs. continuous

Geometric Discrepancy P N – a set of N points in [0 , 1] d R –a geometric family (e.g. axis-parallel rectangles, all rectangles, polytopes, balls, convex sets etc.) Dmitriy Bilyk Uniform distribution: discrete vs. continuous

Geometric Discrepancy P N – a set of N points in [0 , 1] d R –a geometric family (e.g. axis-parallel rectangles, all rectangles, polytopes, balls, convex sets etc.) Discrepancy of P N with respect to R ∈ R D ( P N , R ) = ♯ {P N ∩ R } − N · vol ( R ) Dmitriy Bilyk Uniform distribution: discrete vs. continuous

Geometric Discrepancy P N – a set of N points in [0 , 1] d R –a geometric family (e.g. axis-parallel rectangles, all rectangles, polytopes, balls, convex sets etc.) Discrepancy of P N with respect to R ∈ R D ( P N , R ) = ♯ {P N ∩ R } − N · vol ( R ) Discrepancy of P N with respect to R D ( P N ) = sup | D ( P N , R ) | R ∈R Dmitriy Bilyk Uniform distribution: discrete vs. continuous

Geometric Discrepancy P N – a set of N points in [0 , 1] d R –a geometric family (e.g. axis-parallel rectangles, all rectangles, polytopes, balls, convex sets etc.) Discrepancy of P N with respect to R ∈ R D ( P N , R ) = ♯ {P N ∩ R } − N · vol ( R ) Discrepancy of P N with respect to R D ( P N ) = sup | D ( P N , R ) | R ∈R D ( N ) = inf P N D ( P N ) Dmitriy Bilyk Uniform distribution: discrete vs. continuous

Discrepancy function Consider a set P N ⊂ [0 , 1] d consisting of N points: Define the discrepancy function of the set P N as D N ( x ) = ♯ {P N ∩ [0 , x ) } − Nx 1 x 2 . . . x d Dmitriy Bilyk Uniform distribution: discrete vs. continuous

Numerical integration Koksma-Hlawka inequality: � � � � � [0 , 1] d f ( x ) dx − 1 � 1 � � � f ( p ) N V ( f ) · � D N � ∞ � � � N � � � p ∈P N Dmitriy Bilyk Uniform distribution: discrete vs. continuous

Numerical integration Koksma-Hlawka inequality: � � � � � [0 , 1] d f ( x ) dx − 1 � 1 � � � f ( p ) N V ( f ) · � D N � ∞ � � � N � � � p ∈P N V ( f ) is the Hardy-Krause variation of f Dmitriy Bilyk Uniform distribution: discrete vs. continuous

Numerical integration Koksma-Hlawka inequality: � � � � � [0 , 1] d f ( x ) dx − 1 � 1 � � � f ( p ) N � f x 1 ...x d � 1 · � D N � ∞ � � � N � � � p ∈P N V ( f ) is the Hardy-Krause variation of f � � � ∂ d f � � V ( f ) = � dx 1 . . . dx d � � � ∂x 1 ∂x 2 . . . ∂x d [0 , 1] d � 1 � 1 e.g., if f ( x 1 , ..., x d ) = x 1 ... x d φ ( y ) dy Dmitriy Bilyk Uniform distribution: discrete vs. continuous