Optimal order digital nets and sequences Takashi Goda University of Tokyo Joint work with Kosuke Suzuki and Takehito Yoshiki RICAM Discrepancy Workshop, November 2018 Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 1 / 36
This talk is NOT directly about discrepancy... BUT about how the idea from discrepancy problem can be used to obtain some results in numerical integration. Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 2 / 36
This talk is NOT directly about discrepancy... BUT about how the idea from discrepancy problem can be used to obtain some results in numerical integration. The following is a list of our papers which this talk is based on. ▶ TG, K. Suzuki, T. Yoshiki: Optimal order quadrature error bounds for infinite-dimensional higher-order digital sequences, Found. Comput. Math. 18 (2018) 433–458. ▶ TG, K. Suzuki, T. Yoshiki: Optimal order quasi-Monte Carlo integration in weighted Sobolev spaces of arbitrary smoothness, IMA J. Numer. Anal. 37 (2017), 505–518. ▶ TG, K. Suzuki, T. Yoshiki: An explicit construction of optimal order quasi-Monte Carlo rules for smooth integrands, SIAM J. Numer. Anal. 54 (2016), 2664–2683. Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 2 / 36
L p -discrepancy Definition For an N -element point set P ⊂ [0 , 1] s , the L p -discrepancy is defined by � � p ∫ � 1 � ∑ ( L p ( P )) p := � � 1 x ∈ [ 0 , y ) − λ ([ 0 , y )) d y , � � N [0 , 1] s � � x ∈ P where λ denotes the Lebesgue measure in R s . Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 3 / 36
L p -discrepancy Definition For an N -element point set P ⊂ [0 , 1] s , the L p -discrepancy is defined by � � p ∫ � 1 � ∑ ( L p ( P )) p := � � 1 x ∈ [ 0 , y ) − λ ([ 0 , y )) d y , � � N [0 , 1] s � � x ∈ P where λ denotes the Lebesgue measure in R s . In case of p = 2, s 1 − x 2 s ( L 2 ( P )) 2 = 1 3 s − 2 + 1 ∑ ∏ ∑ ∏ j min(1 − x j , 1 − y j ) . N 2 N 2 x ∈ P j =1 x , y ∈ P j =1 Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 3 / 36
Quasi-Monte Carlo (QMC) integration Problem Approximate/Estimate ∫ I ( f ) := [0 , 1] s f ( x ) d x , where s ∈ N and f : [0 , 1] s → R is integrable. Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 4 / 36
Quasi-Monte Carlo (QMC) integration Problem Approximate/Estimate ∫ I ( f ) := [0 , 1] s f ( x ) d x , where s ∈ N and f : [0 , 1] s → R is integrable. Choose an N -element point set P ⊂ [0 , 1] s . Approximate I ( f ) by Q P ( f ) := 1 ∑ f ( x ) . N x ∈ P For an infinite sequence of points S = { x n | n ≥ 0 } , the first N elements of S are used as P . Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 4 / 36
Worst-case error Definition For a function space V with norm ∥ · ∥ V , e wor ( V , P ) := sup | Q P ( f ) − I ( f ) | . ∥ f ∥ V ≤ 1 Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 5 / 36
Worst-case error Definition For a function space V with norm ∥ · ∥ V , e wor ( V , P ) := sup | Q P ( f ) − I ( f ) | . ∥ f ∥ V ≤ 1 We want to construct a good P depending on V . When V is an RKHS with kernel K , ∫ ∫ ( e wor ( V , P )) 2 = [0 , 1] s K ( x , y ) d x d y [0 , 1] s ∫ − 2 [0 , 1] s K ( x , y ) d y + 1 ∑ ∑ K ( x , y ) . N 2 N x ∈ P x , y ∈ P Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 5 / 36
L 2 -discrepancy (again) Let us consider an RKHS with s ∏ K ∗ 1 , s ( x , y ) = min(1 − x j , 1 − y j ) . j =1 The L 2 -discrepancy of P is same as the worst-case error in this RKHS. Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 6 / 36
L 2 -discrepancy (again) Let us consider an RKHS with s ∏ K ∗ 1 , s ( x , y ) = min(1 − x j , 1 − y j ) . j =1 The L 2 -discrepancy of P is same as the worst-case error in this RKHS. This RKHS is s ⊗ V = H ∗ H ∗ 1 , s = 1 j =1 where { } f : [0 , 1] → R | f (1) = 0 , f (1) ∈ L 2 H ∗ 1 = . Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 6 / 36
Optimal order QMC for L 2 -discrepancy QMC point sets, which achieve the optimal order worst-case error in this RKHS, also achieve the optimal order L 2 -discrepancy: L 2 ( P ) ≍ (log N ) ( s − 1) / 2 . N Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 7 / 36
Optimal order QMC for L 2 -discrepancy QMC point sets, which achieve the optimal order worst-case error in this RKHS, also achieve the optimal order L 2 -discrepancy: L 2 ( P ) ≍ (log N ) ( s − 1) / 2 . N For s = 1 and s = 2, there are many explicit constructions of such point sets. But for s ≥ 3, we only know the results from 1 Chen and Skriganov (2002), Skriganov (2006) 2 Dick and Pillichshammer (2014), Dick, Hinrichs, Markhasin and Pillichshammer (2017): higher order digital nets/sequences 3 Levin (2018): Halton sequences Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 7 / 36
Sobolev spaces of our interest In this work we consider functions of higher smoothness: s ⊗ V = H α, s = H α , j =1 where α ≥ 2 and { } f : [0 , 1] → R | f ( r ) : abs. conti. for 0 ≤ r ≤ α − 1 , f ( α ) ∈ L 2 H α = . Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 8 / 36
Sobolev spaces of our interest In this work we consider functions of higher smoothness: s ⊗ V = H α, s = H α , j =1 where α ≥ 2 and { } f : [0 , 1] → R | f ( r ) : abs. conti. for 0 ≤ r ≤ α − 1 , f ( α ) ∈ L 2 H α = . H α, s coincides with an RKHS with kernel [ α ] s B r ( x j ) B r ( y j ) + ( − 1) α +1 B 2 α ( | x j − y j | ) ∏ ∑ K α, s ( x , y ) = , ( r !) 2 (2 α )! r =0 j =1 where B r denotes the Bernoulli poly. of degree r (Wahba, 1990). Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 8 / 36
Higher order convergence Known results (Dick, 2008; Baldeaux and Dick, 2009) Higher order digital nets/sequences achieve e wor ( H α, s , P ) ≪ (log N ) c ( α, s ) . N α c ( α, s ) = α s is obtained for order α digital nets. The best possible exponent of log N term is ( s − 1) / 2 (for linear quadrature algorithm). Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 9 / 36
Higher order convergence Known results (Dick, 2008; Baldeaux and Dick, 2009) Higher order digital nets/sequences achieve e wor ( H α, s , P ) ≪ (log N ) c ( α, s ) . N α c ( α, s ) = α s is obtained for order α digital nets. The best possible exponent of log N term is ( s − 1) / 2 (for linear quadrature algorithm). Aim of this talk Prove that higher order digital nets/sequences achieve the optimal order of convergence e wor ( H α, s , P ) ≪ (log N ) ( s − 1) / 2 . N α Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 9 / 36
Main result Theorem (G., Suzuki and Yoshiki, 2018) For α ≥ 2 , order 2 α + 1 digital nets/sequences in prime base b achieve e wor ( H α, s , P ) ≍ (log N ) ( s − 1) / 2 N α for N = b m with m ∈ N . Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 10 / 36
In the rest of this talk I focus on 1 nets, and 2 an upper bound. (Remark: A lower bound follows from the “bumping function” argument by Bakhvalov (1959).) Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 11 / 36
In the rest of this talk I focus on 1 nets, and 2 an upper bound. (Remark: A lower bound follows from the “bumping function” argument by Bakhvalov (1959).) I want to 1 introduce higher order digital nets, and 2 show a sketch of our proof for the main result by highlighting an analogy to the proof by Dick and Pillichshammer (2014), who proved the optimal order L 2 -discrepancy bound for order 3 digital nets. Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 11 / 36
Digital nets Definition (Niederreiter, 1992) For prime b , F b denotes the b -element field. Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 12 / 36
Digital nets Definition (Niederreiter, 1992) For prime b , F b denotes the b -element field. For s , m , n ∈ N , let C 1 , . . . , C s ∈ F n × m . b Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 12 / 36
Digital nets Definition (Niederreiter, 1992) For prime b , F b denotes the b -element field. For s , m , n ∈ N , let C 1 , . . . , C s ∈ F n × m . b For 0 ≤ h < b m , write h = η 0 + η 1 b + · · · + η m − 1 b m − 1 . Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 12 / 36
Digital nets Definition (Niederreiter, 1992) For prime b , F b denotes the b -element field. For s , m , n ∈ N , let C 1 , . . . , C s ∈ F n × m . b For 0 ≤ h < b m , write h = η 0 + η 1 b + · · · + η m − 1 b m − 1 . For 1 ≤ j ≤ s , let x h , j = ξ 1 , h , j + · · · + ξ n , h , j ∈ [0 , 1] , b n b where ( ξ 1 , h , j , . . . , ξ n , h , j ) ⊤ = C j · ( η 0 , . . . , η m − 1 ) ⊤ ∈ F n b . Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 12 / 36
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