Optimal order digital nets and sequences Takashi Goda University of - - PowerPoint PPT Presentation
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
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
NOT directly about discrepancy... BUT about how the idea from discrepancy problem can be used to
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 2 / 36
NOT directly about discrepancy... BUT about how the idea from discrepancy problem can be used to
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
Definition
For an N-element point set P ⊂ [0, 1]s, the Lp-discrepancy is defined by (Lp(P))p := ∫
[0,1]s
N ∑
x∈P
1x∈[0,y) − λ([0, y))
dy, where λ denotes the Lebesgue measure in Rs.
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 3 / 36
Definition
For an N-element point set P ⊂ [0, 1]s, the Lp-discrepancy is defined by (Lp(P))p := ∫
[0,1]s
N ∑
x∈P
1x∈[0,y) − λ([0, y))
dy, where λ denotes the Lebesgue measure in Rs. In case of p = 2, (L2(P))2 = 1 3s − 2 N ∑
x∈P s
∏
j=1
1 − x2
j
2 + 1 N2 ∑
x,y∈P s
∏
j=1
min(1 − xj, 1 − yj).
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 3 / 36
Problem
Approximate/Estimate I(f ) := ∫
[0,1]s f (x) dx,
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
Problem
Approximate/Estimate I(f ) := ∫
[0,1]s f (x) dx,
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 QP(f ) := 1 N ∑
x∈P
f (x). For an infinite sequence of points S = {xn | 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
Definition
For a function space V with norm ∥ · ∥V , ewor(V , P) := sup
∥f ∥V ≤1
|QP(f ) − I(f )|.
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 5 / 36
Definition
For a function space V with norm ∥ · ∥V , ewor(V , P) := sup
∥f ∥V ≤1
|QP(f ) − I(f )|. We want to construct a good P depending on V . When V is an RKHS with kernel K, (ewor(V , P))2 = ∫
[0,1]s
∫
[0,1]s K(x, y) dx dy
− 2 N ∑
x∈P
∫
[0,1]s K(x, y) dy + 1
N2 ∑
x,y∈P
K(x, y).
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 5 / 36
Let us consider an RKHS with K ∗
1,s(x, y) = s
∏
j=1
min(1 − xj, 1 − yj). The L2-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
Let us consider an RKHS with K ∗
1,s(x, y) = s
∏
j=1
min(1 − xj, 1 − yj). The L2-discrepancy of P is same as the worst-case error in this RKHS. This RKHS is V = H∗
1,s = s
⊗
j=1
H∗
1
where H∗
1 =
{ f : [0, 1] → R | f (1) = 0, f (1) ∈ L2 } .
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 6 / 36
QMC point sets, which achieve the optimal order worst-case error in this RKHS, also achieve the optimal order L2-discrepancy: L2(P) ≍ (log N)(s−1)/2 N .
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 7 / 36
QMC point sets, which achieve the optimal order worst-case error in this RKHS, also achieve the optimal order L2-discrepancy: L2(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
In this work we consider functions of higher smoothness: V = Hα,s =
s
⊗
j=1
Hα, where α ≥ 2 and Hα = { f : [0, 1] → R | f (r) : abs. conti. for 0 ≤ r ≤ α − 1, f (α) ∈ L2 } .
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 8 / 36
In this work we consider functions of higher smoothness: V = Hα,s =
s
⊗
j=1
Hα, where α ≥ 2 and Hα = { f : [0, 1] → R | f (r) : abs. conti. for 0 ≤ r ≤ α − 1, f (α) ∈ L2 } . Hα,s coincides with an RKHS with kernel Kα,s(x, y) =
s
∏
j=1
[ α ∑
r=0
Br(xj)Br(yj) (r!)2 + (−1)α+1 B2α(|xj − yj|) (2α)! ] , where Br denotes the Bernoulli poly. of degree r (Wahba, 1990).
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 8 / 36
Known results (Dick, 2008; Baldeaux and Dick, 2009)
Higher order digital nets/sequences achieve ewor(Hα,s, P) ≪ (log N)c(α,s) Nα . c(α, s) = αs 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
Known results (Dick, 2008; Baldeaux and Dick, 2009)
Higher order digital nets/sequences achieve ewor(Hα,s, P) ≪ (log N)c(α,s) Nα . c(α, s) = αs 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 ewor(Hα,s, P) ≪ (log N)(s−1)/2 Nα .
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 9 / 36
Theorem (G., Suzuki and Yoshiki, 2018)
For α ≥ 2, order 2α + 1 digital nets/sequences in prime base b achieve ewor(Hα,s, P) ≍ (log N)(s−1)/2 Nα for N = bm with m ∈ N.
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 10 / 36
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
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 L2-discrepancy bound for order 3 digital nets.
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 11 / 36
Definition (Niederreiter, 1992)
For prime b, Fb denotes the b-element field.
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 12 / 36
Definition (Niederreiter, 1992)
For prime b, Fb denotes the b-element field. For s, m, n ∈ N, let C1, . . . , Cs ∈ Fn×m
b
.
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 12 / 36
Definition (Niederreiter, 1992)
For prime b, Fb denotes the b-element field. For s, m, n ∈ N, let C1, . . . , Cs ∈ Fn×m
b
. For 0 ≤ h < bm, write h = η0 + η1b + · · · + ηm−1bm−1.
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 12 / 36
Definition (Niederreiter, 1992)
For prime b, Fb denotes the b-element field. For s, m, n ∈ N, let C1, . . . , Cs ∈ Fn×m
b
. For 0 ≤ h < bm, write h = η0 + η1b + · · · + ηm−1bm−1. For 1 ≤ j ≤ s, let xh,j = ξ1,h,j b + · · · + ξn,h,j bn ∈ [0, 1], where (ξ1,h,j, . . . , ξn,h,j)⊤ = Cj · (η0, . . . , ηm−1)⊤ ∈ Fn
b.
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 12 / 36
Definition (Niederreiter, 1992)
For prime b, Fb denotes the b-element field. For s, m, n ∈ N, let C1, . . . , Cs ∈ Fn×m
b
. For 0 ≤ h < bm, write h = η0 + η1b + · · · + ηm−1bm−1. For 1 ≤ j ≤ s, let xh,j = ξ1,h,j b + · · · + ξn,h,j bn ∈ [0, 1], where (ξ1,h,j, . . . , ξn,h,j)⊤ = Cj · (η0, . . . , ηm−1)⊤ ∈ Fn
b.
Set xh = (xh,1, . . . , xh,s) ∈ [0, 1]s.
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 12 / 36
Definition (Niederreiter, 1992)
For prime b, Fb denotes the b-element field. For s, m, n ∈ N, let C1, . . . , Cs ∈ Fn×m
b
. For 0 ≤ h < bm, write h = η0 + η1b + · · · + ηm−1bm−1. For 1 ≤ j ≤ s, let xh,j = ξ1,h,j b + · · · + ξn,h,j bn ∈ [0, 1], where (ξ1,h,j, . . . , ξn,h,j)⊤ = Cj · (η0, . . . , ηm−1)⊤ ∈ Fn
b.
Set xh = (xh,1, . . . , xh,s) ∈ [0, 1]s. We call P = {xh : 0 ≤ h < bm} a digital net over Fb with generating matrices C1, . . . , Cs.
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 12 / 36
Definition
The dual net of P with C1, . . . , Cs is defined by P⊥ = { (k1, . . . , ks) ∈ Ns
0 : C ⊤ 1 ⃗
k1 ⊕ · · · ⊕ C ⊤
s ⃗
ks = 0 ∈ Fm
b
} , where we write ⃗ k = (κ0, . . . , κn−1) ∈ Fn
b
for k = κ0 + κ1b + · · · .
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 13 / 36
For k = κ1ba1−1 + · · · + κvbav−1 with κi ∈ {1, . . . , b − 1} and a1 > a2 > · · · , the NRT metric is µ1(k) = a1 and µ1(k1, . . . , ks) =
s
∑
j=1
µ1(kj).
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 14 / 36
For k = κ1ba1−1 + · · · + κvbav−1 with κi ∈ {1, . . . , b − 1} and a1 > a2 > · · · , the NRT metric is µ1(k) = a1 and µ1(k1, . . . , ks) =
s
∑
j=1
µ1(kj).
Definition
We call P a digital (t, m, s)-net if µ1(P⊥) := min
k∈P⊥\{0} µ1(k) ≥ m − t + 1.
(Constructions: Sobol’, Faure, Niederreiter, Tezuka, Niederreiter-Xing, ...)
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 14 / 36
Let α ∈ N. For k = κ1ba1−1 + · · · + κvbav−1 with κi ∈ {1, . . . , b − 1} and a1 > a2 > · · · , the Dick metric is µα(k) = a1 + · · · + amin(v,α) and µα(k1, . . . , ks) =
s
∑
j=1
µα(kj). α = 1: the NRT metric.
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 15 / 36
Let α ∈ N. For k = κ1ba1−1 + · · · + κvbav−1 with κi ∈ {1, . . . , b − 1} and a1 > a2 > · · · , the Dick metric is µα(k) = a1 + · · · + amin(v,α) and µα(k1, . . . , ks) =
s
∑
j=1
µα(kj). α = 1: the NRT metric.
Definition
We call P an order α digital (t, m, s)-net if µα(P⊥) := min
k∈P⊥\{0} µα(k) ≥ αm − t + 1.
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 15 / 36
Define Dα : [0, 1]α → [0, 1] by x1 = (0.ξ1,1ξ2,1ξ3,1 . . .)b x2 = (0.ξ1,2ξ2,2ξ3,2 . . .)b . . . xα = (0.ξ1,αξ2,αξ3,α . . .)b. → (0. ξ1,1ξ1,2 . . . ξ1,α
ξ2,1ξ2,2 . . . ξ2,α
. . .)b. For x ∈ [0, 1]αs, we write Dα(x) = (Dα(x1, . . . , xα), . . . , Dα(xα(s−1)+1, . . . , xαs)) ∈ [0, 1]s.
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 16 / 36
Define Dα : [0, 1]α → [0, 1] by x1 = (0.ξ1,1ξ2,1ξ3,1 . . .)b x2 = (0.ξ1,2ξ2,2ξ3,2 . . .)b . . . xα = (0.ξ1,αξ2,αξ3,α . . .)b. → (0. ξ1,1ξ1,2 . . . ξ1,α
ξ2,1ξ2,2 . . . ξ2,α
. . .)b. For x ∈ [0, 1]αs, we write Dα(x) = (Dα(x1, . . . , xα), . . . , Dα(xα(s−1)+1, . . . , xαs)) ∈ [0, 1]s.
Theorem (Dick, 2007)
Let P be a digital (t, m, αs)-net. Then Dα(P) is an order α digital (tα, m, s)-net with tα ≤ α min {m, t + ⌊s(α − 1)/2⌋} .
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 16 / 36
Let P be a digital (t, m, αs)-net with C1, . . . , Cαs ∈ Fm×m
b
. We write C1 = c1,1 . . . cm,1 , . . . , Cα = c1,α . . . cm,α , Cα+1 = c1,α+1 . . . cm,α+1 . . . .
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 17 / 36
Let P be a digital (t, m, αs)-net with C1, . . . , Cαs ∈ Fm×m
b
. We write C1 = c1,1 . . . cm,1 , . . . , Cα = c1,α . . . cm,α , Cα+1 = c1,α+1 . . . cm,α+1 . . . . Dα(P) is a digital net with D1, . . . , Ds ∈ Fαm×m
b
where D1 = c1,1 . . . c1,α . . . cm,1 . . . cm,α , D2 = c1,α+1 . . . c1,2α . . . cm,α+1 . . . cm,2α , . . . .
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 17 / 36
Definition
For k = κ0 + κ1b + · · · , the k-th Walsh function walk is defined by walk(x) := exp [2πi b (κ0ξ1 + κ1ξ2 + · · · ) ] , where x = ξ1/b + ξ2/b2 + · · · . For vectors k ∈ Ns
0, we define
walk(x) :=
s
∏
j=1
walkj(xj).
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 18 / 36
The set {walk : k ∈ Ns
0} forms an ONB in L2([0, 1]s).
Therefore, in general, we have K(x, y) = ∑
k,l∈Ns
ˆ K(k, l)walk(x)wall(y), where ˆ K(k, l) is the (k, l)-th Walsh coefficient ˆ K(k, l) = ∫
[0,1]s
∫
[0,1]s K(x, y)walk(x)wall(y) dx dy.
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 19 / 36
Lemma
Let P be a digital net with C1, . . . , Cs ∈ Fn×m
b
. We have ∑
x∈P
walk(x) = { bm = N if k ∈ P⊥,
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 20 / 36
Lemma
For an RKHS V with kernel K and a digital net P, (ewor(V , P))2 = ∑
k,l∈P⊥\{0}
ˆ K(k, l).
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 21 / 36
Lemma
For an RKHS V with kernel K and a digital net P, (ewor(V , P))2 = ∑
k,l∈P⊥\{0}
ˆ K(k, l). Proof: Recall the following explicit formula (ewor(V , P))2 = ∫
[0,1]s
∫
[0,1]s K(x, y) dx dy
− 2 N ∑
x∈P
∫
[0,1]s K(x, y) dy + 1
N2 ∑
x,y∈P
K(x, y).
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 21 / 36
The first term is ∫
[0,1]s
∫
[0,1]s K(x, y) dx dy = ˆ
K(0, 0).
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 22 / 36
The first term is ∫
[0,1]s
∫
[0,1]s K(x, y) dx dy = ˆ
K(0, 0). The second term is 2 N ∑
x∈P
∫
[0,1]s K(x, y) dy
= 1 N ∑
x∈P
∫
[0,1]s K(x, y) dy + 1
N ∑
x∈P
∫
[0,1]s K(y, x) dy
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 22 / 36
The first term is ∫
[0,1]s
∫
[0,1]s K(x, y) dx dy = ˆ
K(0, 0). The second term is 2 N ∑
x∈P
∫
[0,1]s K(x, y) dy
= 1 N ∑
x∈P
∫
[0,1]s K(x, y) dy + 1
N ∑
x∈P
∫
[0,1]s K(y, x) dy
= 1 N ∑
x∈P
∑
k∈Ns
ˆ K(k, 0)walk(x) + 1 N ∑
x∈P
∑
l∈Ns
ˆ K(0, l)wall(x)
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 22 / 36
The first term is ∫
[0,1]s
∫
[0,1]s K(x, y) dx dy = ˆ
K(0, 0). The second term is 2 N ∑
x∈P
∫
[0,1]s K(x, y) dy
= 1 N ∑
x∈P
∫
[0,1]s K(x, y) dy + 1
N ∑
x∈P
∫
[0,1]s K(y, x) dy
= 1 N ∑
x∈P
∑
k∈Ns
ˆ K(k, 0)walk(x) + 1 N ∑
x∈P
∑
l∈Ns
ˆ K(0, l)wall(x) = ∑
k∈P⊥
ˆ K(k, 0) + ∑
l∈P⊥
ˆ K(0, l).
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 22 / 36
Finally, the third term is 1 N2 ∑
x,y∈P
K(x, y) = 1 N2 ∑
x,y∈P
∑
k,l∈Ns
ˆ K(k, l)walk(x)wall(y)
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 23 / 36
Finally, the third term is 1 N2 ∑
x,y∈P
K(x, y) = 1 N2 ∑
x,y∈P
∑
k,l∈Ns
ˆ K(k, l)walk(x)wall(y) = ∑
k,l∈Ns
ˆ K(k, l) 1 N ∑
x∈P
walk(x) 1 N ∑
y∈P
wall(y)
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 23 / 36
Finally, the third term is 1 N2 ∑
x,y∈P
K(x, y) = 1 N2 ∑
x,y∈P
∑
k,l∈Ns
ˆ K(k, l)walk(x)wall(y) = ∑
k,l∈Ns
ˆ K(k, l) 1 N ∑
x∈P
walk(x) 1 N ∑
y∈P
wall(y) = ∑
k,l∈P⊥
ˆ K(k, l).
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 23 / 36
Finally, the third term is 1 N2 ∑
x,y∈P
K(x, y) = 1 N2 ∑
x,y∈P
∑
k,l∈Ns
ˆ K(k, l)walk(x)wall(y) = ∑
k,l∈Ns
ˆ K(k, l) 1 N ∑
x∈P
walk(x) 1 N ∑
y∈P
wall(y) = ∑
k,l∈P⊥
ˆ K(k, l). Altogether we have (ewor(V , P))2 = ˆ K(0, 0) − ∑
k∈P⊥
ˆ K(k, 0) − ∑
l∈P⊥
ˆ K(0, l) + ∑
k,l∈P⊥
ˆ K(k, l) = ∑
k,l∈P⊥\{0}
ˆ K(k, l).
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 23 / 36
For a digital net P, (L2(P))2 = ∑
k,l∈P⊥\{0}
ˆ K ∗
1,s(k, l),
where K ∗
1,s(x, y) = s
∏
j=1
min(1 − xj, 1 − yj). Constructions of digital nets with optimal order L2-discrepancy ✄ Chen and Skriganov (2002) ✄ Dick and Pillichshammer (2014): order 3 digital nets
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 24 / 36
For a digital net P, (L2(P))2 = ∑
k,l∈P⊥\{0}
ˆ K ∗
1,s(k, l),
where K ∗
1,s(x, y) = s
∏
j=1
min(1 − xj, 1 − yj). Constructions of digital nets with optimal order L2-discrepancy ✄ Chen and Skriganov (2002) ✄ Dick and Pillichshammer (2014): order 3 digital nets
Common idea
is to exploit the decay and the sparsity of ˆ K ∗
1,s.
Decay: ˆ K ∗
1,s(k, l) ≪ b−µ1(k)−µ1(l)
Sparsity: ˆ K ∗
1,s(k, l) = 0
for many (k, l)
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 24 / 36
P: an order 3 digital (t, m, s)-net (L2(P))2 = ∑
k,l∈P⊥\{0}
ˆ K ∗
1,s(k, l)
(Dick and Pillichshammer, 2014)
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 25 / 36
P: an order 3 digital (t, m, s)-net (L2(P))2 = ∑
k,l∈P⊥\{0}
ˆ K ∗
1,s(k, l) ≪
∑
k,l∈P⊥\{0} ˆ K ∗
1,s(k,l)̸=0
b−µ1(k)−µ1(l) (Dick and Pillichshammer, 2014)
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 25 / 36
P: an order 3 digital (t, m, s)-net (L2(P))2 = ∑
k,l∈P⊥\{0}
ˆ K ∗
1,s(k, l) ≪
∑
k,l∈P⊥\{0} ˆ K ∗
1,s(k,l)̸=0
b−µ1(k)−µ1(l) =
∞
∑
z=2µ1(P⊥)
b−z
K ∗
1,s(k, l) ̸= 0}
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 25 / 36
P: an order 3 digital (t, m, s)-net (L2(P))2 = ∑
k,l∈P⊥\{0}
ˆ K ∗
1,s(k, l) ≪
∑
k,l∈P⊥\{0} ˆ K ∗
1,s(k,l)̸=0
b−µ1(k)−µ1(l) =
∞
∑
z=2µ1(P⊥)
b−z
K ∗
1,s(k, l) ̸= 0}
ms−1 b2µ1(P⊥) ≪ ms−1 b2m = (logb N)s−1 N2 . (Dick and Pillichshammer, 2014)
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 25 / 36
(ewor(Hα,s, P))2 = ∑
k,l∈P⊥\{0}
ˆ Kα,s(k, l), where Kα,s(x, y) =
s
∏
j=1
[ α ∑
r=0
Br(xj)Br(yj) (r!)2 + (−1)α+1 B2α(|xj − yj|) (2α)! ] .
Lemma (Baldeaux and Dick, 2009)
For k, l ∈ Ns
0,
ˆ Kα,s(k, l) ≪ b−µα(k)−µα(l).
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 26 / 36
P: an order α digital (t, m, s)-net (ewor(Hα,s, P))2 = ∑
k,l∈P⊥\{0}
ˆ Kα,s(k, l)
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 27 / 36
P: an order α digital (t, m, s)-net (ewor(Hα,s, P))2 = ∑
k,l∈P⊥\{0}
ˆ Kα,s(k, l) ≪ ∑
k,l∈P⊥\{0}
b−µα(k)−µα(l)
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 27 / 36
P: an order α digital (t, m, s)-net (ewor(Hα,s, P))2 = ∑
k,l∈P⊥\{0}
ˆ Kα,s(k, l) ≪ ∑
k,l∈P⊥\{0}
b−µα(k)−µα(l) =
∞
∑
z=2µα(P⊥)
b−z
Optimal order digital nets and sequences RICAM Workshop 27 / 36
P: an order α digital (t, m, s)-net (ewor(Hα,s, P))2 = ∑
k,l∈P⊥\{0}
ˆ Kα,s(k, l) ≪ ∑
k,l∈P⊥\{0}
b−µα(k)−µα(l) =
∞
∑
z=2µα(P⊥)
b−z
m2αs b2µα(P⊥) ≪ m2αs b2αm = (logb N)2αs N2α . The exponent of log N terms comes from the counting step with respect to µα-metric.
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 27 / 36
Counting along µα-metric gives
for z ≥ µα(P⊥),
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 28 / 36
Counting along µα-metric gives
for z ≥ µα(P⊥), while counting along µ1-metric gives
for z ≥ µ1(P⊥).
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 28 / 36
Counting along µα-metric gives
for z ≥ µα(P⊥), while counting along µ1-metric gives
for z ≥ µ1(P⊥). We want to
1 switch the metric from µα to µ1, and 2 exploit the sparsity of ˆ
Kα,s to deal with the double sum ∑
k,l∈P⊥\{0}.
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 28 / 36
(ewor(Hα,s, P))2 = ∑
k,l∈P⊥\{0}
ˆ Kα,s(k, l).
Exploit the decay and the sparsity of ˆ Kα,s
Decay: ˆ Kα,s(k, l) ≪ b−µα(k)−µα(l) (Baldeaux and Dick, 2009) Sparsity: ˆ Kα,s(k, l) = 0 for many (k, l) (G., Suzuki and Yoshiki, 2018)
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 29 / 36
In order to switch the metric in the counting step, we need:
Additional tricks
Propagation: P: order 2α + 1 net ⇒ P: order 1 net (Dick, 2008) Interpolation: µα(k) ≥ Aµ2α+1(k) + Bµ1(k) where A = α − 1 2α and B = α + 1 2α (G., Suzuki and Yoshiki, 2017). The interpolation inequality on µα is nothing but Jensen’s inequality with respect to α.
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 30 / 36
Let k =
v
∑
i=1
κibci−1 and l =
w
∑
i=1
λibdi−1, such that c1 > c2 > · · · , d1 > d2 > · · · and κi, λi ∈ {1, . . . , b − 1}.
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 31 / 36
Let k =
v
∑
i=1
κibci−1 and l =
w
∑
i=1
λibdi−1, such that c1 > c2 > · · · , d1 > d2 > · · · and κi, λi ∈ {1, . . . , b − 1}. For p, q ∈ N0, we write k(p) =
v
∑
i=p+1
κibci−1 and l(q) =
w
∑
i=q+1
λibdi−1 where the empty sum is set to 0.
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 31 / 36
Let k =
v
∑
i=1
κibci−1 and l =
w
∑
i=1
λibdi−1, such that c1 > c2 > · · · , d1 > d2 > · · · and κi, λi ∈ {1, . . . , b − 1}. For p, q ∈ N0, we write k(p) =
v
∑
i=p+1
κibci−1 and l(q) =
w
∑
i=q+1
λibdi−1 where the empty sum is set to 0. We say “(k, l) is of type (p, q)” if k(p) = l(q) and κpbcp−1 ̸= λqbdq−1.
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 31 / 36
Let k =
v
∑
i=1
κibci−1 and l =
w
∑
i=1
λibdi−1, such that c1 > c2 > · · · , d1 > d2 > · · · and κi, λi ∈ {1, . . . , b − 1}. For p, q ∈ N0, we write k(p) =
v
∑
i=p+1
κibci−1 and l(q) =
w
∑
i=q+1
λibdi−1 where the empty sum is set to 0. We say “(k, l) is of type (p, q)” if k(p) = l(q) and κpbcp−1 ̸= λqbdq−1.
Lemma (G., Suzuki and Yoshiki, 2018)
If there exists one index j ∈ {1, . . . , s} such that (kj, lj) is of type (pj, qj) with pj + qj > 2α, then ˆ Kα,s(k, l) = 0.
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 31 / 36
P: an order 2α + 1 digital (t, m, s)-net (ewor(Hα,s, P))2 = ∑
k,l∈P⊥\{0}
ˆ Kα,s(k, l)
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 32 / 36
P: an order 2α + 1 digital (t, m, s)-net (ewor(Hα,s, P))2 = ∑
k,l∈P⊥\{0}
ˆ Kα,s(k, l) ≪ ∑
k,l∈P⊥\{0} ˆ Kα,s(k,l)̸=0
b−µα(k)−µα(l)
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 32 / 36
P: an order 2α + 1 digital (t, m, s)-net (ewor(Hα,s, P))2 = ∑
k,l∈P⊥\{0}
ˆ Kα,s(k, l) ≪ ∑
k,l∈P⊥\{0} ˆ Kα,s(k,l)̸=0
b−µα(k)−µα(l) ≤ b−2Aµ2α+1(P⊥) ∑
k,l∈P⊥\{0} ˆ Kα,s(k,l)̸=0
b−B(µ1(k)+µ1(l))
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 32 / 36
P: an order 2α + 1 digital (t, m, s)-net (ewor(Hα,s, P))2 = ∑
k,l∈P⊥\{0}
ˆ Kα,s(k, l) ≪ ∑
k,l∈P⊥\{0} ˆ Kα,s(k,l)̸=0
b−µα(k)−µα(l) ≤ b−2Aµ2α+1(P⊥) ∑
k,l∈P⊥\{0} ˆ Kα,s(k,l)̸=0
b−B(µ1(k)+µ1(l)) = b−2Aµ2α+1(P⊥)
∞
∑
z=2µ1(P⊥)
b−Bz ×
Kα,s(k, l) ̸= 0}
Optimal order digital nets and sequences RICAM Workshop 32 / 36
P: an order 2α + 1 digital (t, m, s)-net (ewor(Hα,s, P))2 = ∑
k,l∈P⊥\{0}
ˆ Kα,s(k, l) ≪ ∑
k,l∈P⊥\{0} ˆ Kα,s(k,l)̸=0
b−µα(k)−µα(l) ≤ b−2Aµ2α+1(P⊥) ∑
k,l∈P⊥\{0} ˆ Kα,s(k,l)̸=0
b−B(µ1(k)+µ1(l)) = b−2Aµ2α+1(P⊥)
∞
∑
z=2µ1(P⊥)
b−Bz ×
Kα,s(k, l) ̸= 0}
ms−1 b2Aµ2α+1(P⊥)+2Bµ1(P⊥) ≪ ms−1 b2αm = (logb N)s−1 N2α .
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 32 / 36
Again I want to stress similarity of the proofs. In the proof of Dick and Pillichshammer (2014), we see
K ∗
1,s(k, l) ̸= 0}
In our proof, we see
Kα,s(k, l) ̸= 0}
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 33 / 36
Again I want to stress similarity of the proofs. In the proof of Dick and Pillichshammer (2014), we see
K ∗
1,s(k, l) ̸= 0}
In our proof, we see
Kα,s(k, l) ̸= 0}
On order 2α + 1 digital sequences: The optimal order error bound holds only for a geometric sequence of # of points N = b1, b2, . . .. This cannot be improved for any extensible QMC rule (Owen, 2016).
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 33 / 36
Order 2α + 1 digital nets/sequences achieve the optimal order of convergence for numerical integration in Hα,s = ⊗s
j=1 Hα.
Explicit construction of higher order digital nets and sequences is
extensible both in the number of points and the dimension. The main idea of the proof comes from the L2-discrepancy problem together with some additional tricks (interpolation and propagation).
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 34 / 36
After the work of Dick and Pillichshammer (2014), it has been proven that order 2 (instead of 3) digital sequences achieve the best possible
▶ Dick, Hinrichs, Markhasin and Pillichshammer (2017) Israel J. Math.
The proof is based on Haar functions instead of Walsh functions.
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 35 / 36
After the work of Dick and Pillichshammer (2014), it has been proven that order 2 (instead of 3) digital sequences achieve the best possible
▶ Dick, Hinrichs, Markhasin and Pillichshammer (2017) Israel J. Math.
The proof is based on Haar functions instead of Walsh functions. Can we learn further from their result? Is order 2α + 1 best possible?? Does Haar analysis help us??? How about other function spaces (e.g., Besov and Triebel-Lizorkin)? For the case of smoothness less than 2, please refer to
▶ Hinrichs, Markhasin, Oettershagen and Ullrich (2016) Numer. Math.
We do not have any progress on these questions so far...
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 35 / 36
Takashi Goda (U. Tokyo) Optimal order digital nets and sequences RICAM Workshop 36 / 36