Approximation of Quasi-Monte Carlo worst case error in weighted spaces of infinitely times smooth functions

Journal of Computational and Applied Mathematics 330 (2018) 155–164

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Approximation of Quasi-Monte Carlo worst case error in weighted spaces of infinitely times smooth functions Makoto Matsumoto a , Ryuichi Ohori b , Takehito Yoshiki c, * a b c

Graduate School of Sciences, Hiroshima University, Hiroshima 739-8526, Japan Fujitsu Laboratories Ltd., Kanagawa 211-8588, Japan Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto 606-8561, Japan

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info

Article history: Received 30 October 2016 Received in revised form 10 August 2017 Keywords: Quasi-Monte Carlo integration Digital net Worst case error Walsh coefficients Infinitely differentiable functions

a b s t r a c t In this paper, we consider Quasi-Monte Carlo (QMC) worst case error of weighted smooth function classes in C ∞ [0, 1]s by a digital net over F2 . We show that the ratio of the worst case error to the QMC integration error of an exponential function is bounded above and below by constants. This result provides us with a simple interpretation that a digital net with small QMC integration error for an exponential function also gives the small integration error for any function in this function space. © 2017 Elsevier B.V. All rights reserved.

1. Introduction Quasi-Monte Carlo (QMC) integration is one of methods for numerical integration over the s-dimensional unit cube [0, 1)s (see [1–3] for details). We approximate the integral of a function f : [0, 1)s → R

∫ I(f ) :=

f (x) dx [0,1)s

by a quadrature rule of the form IP (f ) :=

1 ∑ N

f (x),

x∈P

where P is a point set in [0, 1)s with finite cardinality N. We define the (signed) integration error by Err (f ; P ) := IP (f ) − I(f ). In order to make the absolute integration error |Err (f ; P )| small for a class of functions, we often measure the quality of point sets P ⊂ [0, 1)s using the so-called worst case error. For a function space F with norm ∥ · ∥F , the worst case error for F by P is defined as the supremum of the absolute value of the integration error in the unit ball of F : ewor (F ; P ) := sup |Err (f ; P )| .

(1)

f ∈F

∥f ∥F ≤1

Apparently, this quantity performs as an upper bound on |Err (f ; P )| for every f ∈ F :

|Err (f ; P )| ≤ ewor (F ; P ) · ∥f ∥F .

*

(2)

Corresponding author. E-mail addresses: [email protected] (M. Matsumoto), [email protected] (R. Ohori), [email protected] (T. Yoshiki). http://dx.doi.org/10.1016/j.cam.2017.08.010 0377-0427/© 2017 Elsevier B.V. All rights reserved.

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M. Matsumoto et al. / Journal of Computational and Applied Mathematics 330 (2018) 155–164

Thus, our goal is to find a point set P with small value of ewor (F ; P ) for a given function class F . In what follows, we focus on QMC rules using digital nets over F2 as point sets (see Definition 2.1). This is a special type of construction scheme we often use for practical application of QMC. In the previous works on QMC, the function class consisting of functions f whose derivatives up to order 1 are continuous has been extensively considered. For this function class, the star-discrepancy plays an alternative role to the worst case error in (2). Various types of low discrepancy digital nets P have been developed, whose discrepancy is of order N −1+ϵ for arbitrary small ϵ > 0 (see [1,4] for the details). Recently, smoother function spaces have also become targets of research in studies of QMC rules, e.g., α -smooth Sobolev space, consisting of functions whose mixed derivatives up to order α in each coordinate are continuous for α ≥ 2. For this function space, efficient digital nets P have also been established (see e.g., [5,6]). They satisfy the higher order convergence of the worst case error N −α+ϵ . More recently, for some family of functions in C ∞ [0, 1]s , the existence or theoretical construction algorithm of digital nets P have been developed with the convergence rate of the integration error N −Cs log N for a constant Cs depending on s [7,8]. On the other hand, heuristic algorithm is also used for constructing digital nets giving efficient QMC rules in [9], in which Walsh Figure of Merit (WAFOM) is of great importance. WAFOM performs as an upper bound on the worst case error of a digital net P. Since WAFOM is computable quickly on computers, we can obtain low-WAFOM point sets by computer search (see e.g., [9–11]). As a continuous work, Suzuki [12] considered more general function spaces

⏐ }  (α ,...,α )  ⏐ f 1 s  1 ⏐ L sup := f ∈ C [0, 1] ⏐ ∥f ∥Fs,u := ∏ αj < ∞ , ⏐ (α1 ,...,αs )∈(N∪{0})s 1≤j≤s uj {

Fs,u

s

∞

with a sequence of positive weights u = (uj )j≥1 . (Actually he considered another function space including this smooth function space.) Here f (α1 ,...,αs ) is the (α1 , . . . , αs )th mixed partial derivative of f . When we set uj = 2 for all j, this space corresponds to the original space considered in the above works. Suzuki [12] gave the existence of digital nets which achieve the convergence rate N −Cs,u log N of the worst case error for a constant Cs,u depending on s and u. Furthermore, under certain ′

E′

conditions on the weights u, this upper bound on the worst case error becomes C ′ ·N −D (log N) for absolute constants C ′ , D′ , E ′ , which is a dimension-independent convergence rate. Computer search algorithm is also effective for finding good point sets in this space as in the above case. For this function space, we can introduce a generalization of WAFOM as an upper bound on the worst case error (see [12, Remark 6.4]). ( ) In this paper, we give feasible upper and lower bounds on the worst case error ewor Fs,u ; P for the function space Fs,u by using well-known exponential functions:

(

Ls,u ≤

Err exp(−

∑ (

1≤j≤s

ewor Fs,u ; P

uj xj ); P

)

) ≤ Us,u ,

for any digital net P and some constants Ls,u and Us,u depending on s and u but not on P (see Theorem 3.1 for the detailed description of the main theorem). Although this is not an equality for the worst case error and we restrict the range of P to the class of digital nets, this gives us a simple interpretation for the worst case error. In the proof of the above inequalities, we use a figure of merit Wu (P) of P, which( is a generalization of WAFOM (see ) Definition 3.3 for Wu (P)) as mentioned above. It is proved that Wu (P) approximates ewor Fs,u ; P (see Lemma 3.6) and bounds the worst case error: L′s,u ≤

Wu (P)

(

ewor Fs,u ; P

) ≤ Us′,u ,

where L′s,u and Us′,u depends not on P but on s and u (see Corollary 3.11 for the explicit statement). This ( result ∑ verifies that) these two criteria have essentially the same role in the QMC error analysis. Since both Wu (P) and Err exp(− 1≤j≤s uj xj ); P are computable criteria for the quality of digital nets, we can consider computer search for finding efficient QMC rules (see first three items of Remark 3.13). The remainder of this article is organized as follows. In Section 2 we recall some definitions for QMC integration by digital nets, and a relation Walsh coefficients. In Section 3 we prove the main result. In Section 4, we show the numerical ( with∑ ) properties on Err exp(− 1≤j≤s uj xj ); P and Wu (P) in terms of the ratio of these two quantities. 2. Preliminaries We denote N0 = N ∪ {0} in this paper. In this section we briefly recall the notion of digital nets and Walsh coefficients. 2.1. Digital nets We first introduce digital nets over the two-element field F2 = {0, 1}, which are defined as follows.

M. Matsumoto et al. / Journal of Computational and Applied Mathematics 330 (2018) 155–164

157

Definition 2.1 (Digital Net Over F2 [2]). Let n, m ≥ 1 be integers with n ≥ m. Let 0 ≤ h < 2m be an integer and C1 , . . . , Cs be n × m matrices∑ over the finite field F2 . We define a map π : F2 → {0, 1} ⊂ Z by π (0) = 0, π (1) = 1. We write the dyadic m i−1 . For 1 ≤ j ≤ s, we define a vector expansion h = i=1 hi 2 (yh,1,j , . . . , yh,n,j )⊤ = Cj · (π −1 (h1 ), . . . , π −1 (hm ))⊤ and a real number xj (h) =

∑

π (yh,i,j ) · 2−i ∈ [0, 1) .

1≤i≤n

Then we define a digital net P by {x0 , . . . , x2m −1 } where xh = (xj (h))1≤j≤s . 2.2. Error analysis via Walsh coefficients We express QMC integration error for functions in the normed function space Fs,u through Walsh coefficients, which are introduced using Walsh functions as follows. Definition 2.2 (Walsh Functions and Walsh Coefficients Over F2 ). Let f : [0, 1)s → R and k = (k1 , . . . , ks ) ∈ Ns0 . We denote x = (x1 , . . . , xs ) ∈ [0, 1)s . We define the kth dyadic Walsh function walk : [0, 1)s → {±1} by

∏

walk (x) :=

(−1)

i≥1 κi,j xi,j

∑

.

1≤j≤s i−1 Here we write the dyadic expansion of kj by kj = and xj by xj = i≥1 κi,j 2 are 0. Using Walsh functions, we define the kth dyadic Walsh coefficient ˆ f (k) as

∑

∫ ˆ f (k) :=

[0,1)s

∑

−i

i≥1 xi,j 2

, where infinitely many digits xi,j

f (x)walk (x) dx.

For general information on Walsh analysis, see [13,14] for example. To exploit the linear structure of digital nets, we need the notion of dual nets [15,16]. Definition 2.3 (Dual Net Over F2 [16]). We define the dual net of a digital net P by

⃗1 C1 + · · · + k⃗s Cs = 0 ∈ Fm P ⊥ := {k = (k1 , . . . , ks ) ∈ Ns0 | k 2 }, ⃗j = (π −1 (κ1,j ), . . . , π −1 (κn,j )) for kj (1 ≤ j ≤ s) with dyadic expansion kj = where k C1 , . . . , Cs used in Definition 2.1.

∑

κ

i−1 , i≥1 i,j 2

the map π and the matrices

The following is the key lemma in analyzing QMC integration error by digital nets (see [1, Chapter 15] or [17, Lemma 18]): Proposition 2.4. Let f ∈ C 0 [0, 1]s with by P is given by Err (f ; P ) =

∑

∑

k ∈Ns0

⏐ ⏐ ⏐ˆ f (k)⏐ < +∞ and P ⊂ [0, 1)s be a digital net. The QMC integration error of f

ˆ f (k).

k ∈P ⊥ \{0}

Remark 2.5. If f has continuous derivative f (α1 ,...,αs ) for any nonnegative integers αj ≤ 2, it holds that (see [18, Section 2]). Note that any function in the function class Fs,u satisfies this assumption.

∑

k ∈Ns0

⏐ ⏐ ⏐ˆ f (k)⏐ < +∞

3. Worst case error for the weighted smooth function space Fs,u

(

)

In this section, we give the main theorem, which shows a simple approximation of the worst case error ewor Fs,u ; P . 3.1. Main result In this subsection, we prove our main result Theorem 3.1. Theorem 3.1. Let u = (uj )1≤j≤s ∈ Rs be a vector with all positive components and gs,u (x) := exp(− net P, we have 1 wor 1 A− Fs,u ; P ≤ 2s B− s,u · Err gs,u ; P ≤ e s,u · Err gs,u ; P ,

(

)

(

)

(

)

∑

1≤j≤s uj xj ).

For any digital

(3)

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M. Matsumoto et al. / Journal of Computational and Applied Mathematics 330 (2018) 155–164

where As,u =

∏ 1 − exp(−uj ) uj

1≤j≤s

( Bs,u =

∏ 1≤j≤s

,

1 − exp(−2−w uj ) ∏ 1 − exp(−2−i uj )

inf

2−w uj

1≤w

1≤i≤w

)

2−i uj

̸= 0.

In particular if we choose the normalized function g˜s,u (x) := gs,u (x)/∥gs,u ∥Fs,u , we have 2s As,u

Err g˜s,u ; P ≤ ewor Fs,u ; P ≤

(

(

)

)

Bs,u

( ) · Err g˜s,u ; P .

(4)

Remark 3.2. This theorem implies that the integration error of the exponential function gs,u by a digital net is always positive. This fact is shown by (11), which is the combination of Lemma 3.10 and Proposition 2.4. Before starting the proof, we introduce the following criteria on a digital net P. Definition 3.3. Let P ⊂ [0, 1)s be a digital net. For u ∈ Rs with all positive components, we define the following function of P Wu (P) :=

∑

2−µu (k) ,

k ∈P ⊥ \{0}

where for k = (k1 , . . . , ks ) with kj =

µu (k) =

κ

i−1 i≥1 i,j 2

∑

∈ N0 , we define

∑∑

(i + 1 − log2 uj )κi,j .

1≤j≤s 1≤i

When kj = 0, we set κi,j = 0 for all i. Remark 3.4. This quantity is introduced in [12] (see [12, Definition 2.5 and Remark 6.4] for the details). If we set uj = 2 for 1 ≤ j ≤ s, Wu (P) corresponds to the original WAFOM treated in [9]. We first give the upper bound on the worst case error by this function Wu (P). Lemma 3.5. For any f ∈ Fs,u and a digital net P, we have

|Err (f ; P )| ≤ 2s · ∥f ∥Fs,u · Wu (P).

(5)

In particular, this inequality implies that ewor Fs,u ; P ≤ 2s · Wu (P).

(

)

(6)

Proof. In Proposition 2.4, the QMC integration error by a digital net P is rewritten as a sum of Walsh coefficients. Using the upper bounds on Walsh coefficients in [18, Theorem 1.3] or [19, Theorem 3.9], it holds that for a function f ∈ C ∞ [0, 1]s ,

⏐ ⏐ ⏐ˆ f (k)⏐ ≤ 2s · 2−µ(1,...,1) (k) · ∥f (α1 ,...,αs ) ∥L1 ∑ i−1 where k = (k1 , . . . , ks ) with kj = and we denote the cardinality #{i ≥ 1 | κi,j = 1} by αj . In this condition, we i≥1 κi,j 2 have for u = (u1 , . . . , us ) with uj > 0, ∑∑ µu (k) = (i + 1 − log2 uj )κi,j 1≤j≤s 1≤i

=

∑∑

(i + 1)κi,j −

1≤j≤s 1≤i

= µ(1,...,1) (k) −

∑∑

(log2 uj )κi,j

1≤j≤s 1≤i

∑

αj

log2 uj .

1≤j≤s

Thus, for a function f ∈ Fs,u , we have

⏐ ⏐ ⏐ˆ f (k)⏐ ≤ ∥f ∥Fs,u · 2s · 2−µu (k) .

M. Matsumoto et al. / Journal of Computational and Applied Mathematics 330 (2018) 155–164

159

Combining this upper bound and Proposition 2.4, we obtain the following evaluation on the QMC error by a digital net P.

∑

|Err (f ; P )| ≤ 2s · ∥f ∥Fs,u ·

2−µu (k) = 2s · ∥f ∥Fs,u · Wu (P).

k∈P ⊥ \{0}

The second inequality follows from this inequality as ewor Fs,u ; P =

(

)

|Err (f ; P )| ≤ 2s · 1 · Wu (P).

sup f ∈Fs,u

□

∥f ∥Fs,u ≤1

(

)

We next obtain the upper bound on Wu (P) by Err gs,u ; P . In fact, we can obtain both of lower and upper bounds as follows. Lemma 3.6. For gs,u (x) := exp(−

∑

1≤j≤s uj xj )

and any digital net P, it holds that

−1 1 A− s,u · Err gs,u ; P ≤ Wu (P) ≤ Bs,u · Err gs,u ; P ,

(

(

)

)

where As,u and Bs,u are the same as in Theorem 3.1. A proof of Lemma 3.6 is given in the next subsection. Proof of Theorem 3.1. We have∏only to prove the first statement. In fact, the second statement (4) follows from the first statement (3) since ∥gs,u ∥Fs,u = 1≤j≤s (1 − exp(−uj ))/uj = As,u and thus 1 ˜s,u ; P . A− s,u · Err gs,u ; P = Err g

(

)

(

)

(7)

In what follows, we provide the proof of (3). ( ) The left inequality in (3) is followed by (7) and the definition of ewor Fs,u ; P . We move on to the proof on the right inequality in (3). Combining Lemmas 3.5 and 3.6, we obtain the right inequality in (3) as follows: 1 ewor Fs,u ; P ≤ 2s · Wu (P) ≤ 2s · B− s,u · Err gs,u ; P .

(

)

(

(

3.2. Approximation on Wu (P) by Err gs,u ; P

)

□

)

Lemma 3.6 is proven by calculating the Walsh coefficients of exponential functions. We first consider the case s = 1, in which we explicitly calculate the Walsh coefficients of exponential functions as follows. Lemma 3.7. Let a be a nonzero real number and∑g : [0, 1) → R; x ↦ → exp(ax). Let k be a nonnegative integer and (κj )1≤j≤w be i−1 the binary representation with κw = 1, i.e., k = κi . 1≤i≤w 2 Then, the kth Walsh coefficient ˆ g(k) of g is calculated as

ˆ g(k) =

g(2−w ) − 1 ∏ a

(1 + (−1)κi g(2−i ))

1≤i≤w

for k > 0 and

ˆ g(0) =

g(1) − 1 a

.

Proof. Since the case of k = 0 is easily proven by direct calculation, we assume [ k > 0 hereafter.) In the following calculation, we divide the interval [0, 1) into 2w intervals 2−w i, 2−w (i + 1) for 0 ≤ i < 2w , in each of which the kth Walsh function remains constant. We expansion) (ζj )1≤j≤w of l = 2w−1 ζ1 + · · · + 20 ζw , ∑use the−ibinary [inverse −w which corresponds to the binary expansion of x = if x ∈ 2 l, 2−w (l + 1) . The following calculation proves the i≥1 ζi 2 lemma: 1

∫ ˆ g(k) = =

walk (x) exp(ax) dx 0

∑

( walk

0≤l<2w

=

∑

∏

0≤l<2w 1≤i≤w

l 2w

)

( exp

(−1)κi ζi

l·a

∏ 1≤i≤w

2−w

)∫

exp(ax) dx

2w

0

( exp

2w−i ζi · a 2w

)

1

· (exp(2−w a) − 1) a

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M. Matsumoto et al. / Journal of Computational and Applied Mathematics 330 (2018) 155–164

∑

=

∏ (

(−1)κi exp 2−i a

(

))ζi 1 · (exp(2−w a) − 1) a

0≤l<2w 1≤i≤w

exp(2−w a) − 1

=

∑

a exp(2

a) − 1 ∏

a

(1 + (−1)κi exp(2−i a))

1≤i≤w

g(2−w ) − 1 ∏

=

)ζi

(−1)κi exp(2−i a)

(ζi )1≤i≤w ∈{0,1}w 1≤i≤w

−w

=

∏ (

a

(1 + (−1)κi g(2−i )).

□

1≤i≤w

Using this lemma, we give the upper and lower bounds on the Walsh coefficients of exp(−ux) with u > 0. Lemma 3.8. Let u > 0 and gu : [0, 1) → R; x ↦ → exp(−ux). It holds that Bu 2−µu (k) ≤ ˆ gu (k) ≤ Au 2−µu (k) for all nonnegative k, where Au , Bu equal A1,u , B1,u appearing in Theorem 3.1, respectively. If k = 0, we have Au 2−µu (0) = ˆ gu (0). i−1 Proof. We write the dyadic expansion k = κi for a positive integer k as we mentioned at the beginning of the 1≤i≤w 2 proof of Lemma 3.7. In the case k = 0, it follows from the above theorem directly. We assume that k ̸ = 0. Let I := {i ≤ w | κi = 1} and J := {i < w | κi = 0}. From Lemma 3.7 we have

∑

) 1 − gu (2−w ) ∏ ( 1 + (−1)κi gu (2−i ) u

ˆ gu (k) =

1≤i≤w

1 − gu (2−w ) ∏ (

=

·

u

= 2−w ·

) ∏(

1 − gu (2−i ) ·

i∈I

·

2−w u

i∈I

1 − gu (2−i ) ∏ 1 + gu (2−i ) 2−i u · 2 2−i u 2 i∈J

⎞

= ⎝2−w ·

∏

2−i u ·

∏

2⎠ ·

i∈J

i∈I

⎝

)

i∈J

1 − gu (2−w ) ∏

⎛

⎛

1 + gu (2−i )

1 − gu (2−w ) ∏ 1 − gu (2−i ) ∏ 1 + gu (2−i )

·

2−w u

i∈I

·

2−i u

2

i∈J

⎞ ⎠.

The former term can be calculated as follows. 2−w ·

∏

2−i u ·

i∈I

∏

2=

i∈J

∏

2−i−1 u

i∈I

=

w ∏

∏

1

j∈ J

2−(i+1−log2 u)κi = 2−µu (k) .

i=1

Thus, it suffices to give bounds of the form Bu ≤

1 − gu (2−w ) ∏ 1 − gu (2−i ) ∏ 1 + gu (2−i )

·

2−w u

i∈I

2−i u

·

i∈J

2

≤ Au .

(8)

To obtain the bounds (8), we first provide the following inequality: q(z) :=

z(1 + exp(−z)) 2

− (1 − exp(−z)) ≥ 0 for z ≥ 0.

(9)

By definition, it holds that q(0) = 0, and we can easily check that its derivatives satisfy q(1) (0) = 0 and q(2) (z) ≥ 0. Combining these, we obtain (9). Applying (9) to the middle term in (8), we have

∏ 1 − gu (2−i ) 1≤i≤w

2−i u

≤

∏ 1 − gu (2−i ) ∏ 1 + gu (2−i ) i∈I

2−i u

j∈J

2

≤

∏ 1 + gu (2−i ) 1≤i≤w

2

.

(10)

M. Matsumoto et al. / Journal of Computational and Applied Mathematics 330 (2018) 155–164

161

The right hand side in (10) is calculated as

∏ 1 + gu (2−i ) 2

1≤i≤w

= 2−w

∏ 1≤i≤w

=2

∏

(1 + exp(−2−i u)) = 2−w

(1 + exp(−2−w u · 2i−1 ))

1≤i≤w

−w u · 2w ) −w 1 − exp(−2

1 − exp(−2−w u)

=

2−w 1 − gu (2−w )

(1 − exp(−u)).

By inserting this in (10) and multiplying by (1 − gu (2−w ))/2−w u we obtain 1 − gu (2−w ) ∏ 1 − gu (2−i ) ∏ 1 + gu (2−i )

·

2−w u

≤

1 − gu (2−w ) 2−w u

·

2−i u

i∈I

2−w 1 − gu

2

i∈J

(2−w )

1 − exp(−u)

(1 − exp(−u)) =

u

.

Thus, Au := (1 − exp(−u))/u satisfies the upper bound in (8). Next we move on to the lower bound of (8). Since (1 − exp(−x))/x < 1 holds for all x > 0 and (1 − gu (2−w ))/(2−w u) goes to 1 as w goes to infinity, it is enough to show that the infinite product

∏ 1 − gu (2−i ) 2−i u

1≤i

converges to a nonzero value. Since exp(−x) < (1 − exp(−x))/x < 1 for all x > 0,

∏

exp(−2−i u) ≤

1≤i

∏ 1 − gu (2−i ) 2−i u

1≤i

≤1

holds, and the left hand side converges since

( 0 < Bu := inf

∑

1≤i

− 2−i u = −u. Thus there exists )

1 − gu (2−w ) ∏ 1 − gu (2−i ) 2−w u

1≤w

1≤i≤w

,

2−i u

which satisfies the claim. □ Remark 3.9. For u = 2, which corresponds to the case of the one-dimensional original WAFOM, the values of B2 and A2 are approximately 0.388 and 0.432. Next we give bounds on the Walsh coefficients of exp(− dimensional result.

∑

1≤j≤s uj xj ),

which is easily calculated using the above one-

Lemma 3.10. Let (uj )1≤j≤s ∈ Rs be a vector with all positive components and gs,u (x) = exp(− Bs,u 2 for all k ∈

−µu (k)

Ns0 ,

∑

1≤j≤s uj xj ).

It holds that

−µu (k)

≤ˆ gs,u (k) ≤ As,u 2

where As,u and Bs,u are the same as in Theorem 3.1.

Proof. Since gs,u (x) and walk (x) are written by the products of univariate functions exp(−uj xj ) and walkj (xj ), respectively, we have

∫ ˆ gs,u (k) = =

∏ [0,1)s 1≤j≤s

∏ ∫ 1≤j≤s

[0,1)

exp(−uj xj )walkj (xj ) dx1 . . . dxs

exp(−uj xj )walkj (xj ) dxj =

∏ 1≤j≤s

Combining this with Lemma 3.8, we obtain Bs,u · 2−µu (k) =

∏

−µuj (kj )

Buj 2

1≤j≤s

≤ˆ gs,u (k) ≤

∏ 1≤j≤s

−µuj (kj )

Auj 2

= As,u · 2−µu (k) .

□

ˆ guj (kj ).

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M. Matsumoto et al. / Journal of Computational and Applied Mathematics 330 (2018) 155–164

This inequality and Proposition 2.4 lead to Lemma 3.6 as

(

)

Bs,u · Wu (P) ≤ Err gs,u ; P =

∑

ˆ gs,u (k) ≤ As,u · Wu (P).

(11)

k ∈P ⊥ \{0}

3.3. Approximation for the worst case error by the quantity Wu (P)

(

)

As we mentioned in Section 1, ewor Fs,u ; P is also bounded below and above by Wu (P) as follows. Corollary 3.11. Let u = (uj )1≤j≤s be a vector with all positive components. For any digital net P, we have Bs,u As,u

( ) · Wu (P) ≤ ewor Fs,u ; P ≤ 2s · Wu (P),

(12)

where As,u and Bs,u are the same as in Theorem 3.1. Proof. The left inequality is deduced as Bs,u As,u

( ) ( ) 1 wor · Wu (P) ≤ A− Fs,u ; P , s,u Err gs,u ; P ≤ e

(13)

where we used Lemma 3.6 in the first inequality and Theorem 3.1 in the second one. The right inequality directly follows from Lemma 3.5. □ Moreover, we have the following formula as in the case of the original WAFOM. Corollary 3.12. For any digital net P with N = |P |, we have Wu (P) =

1 ∑ ∏ ∏( N

1 + (−1)ζi,j 2−(i+1−log2 uj ) − 1,

)

x∈P 1≤j≤s 1≤i

where we write x = (

∑

−i

i≥1 2

ζi,j )1≤j≤s .

We omit the proof, which is almost the same as in the case of the original WAFOM [9, Corollary 4.2]. Remark 3.13.

• To calculate the value Wu (P) on computers, we consider the following discretized approximation: ) 1 ∑ ∏ ∏ ( Wun (P) = 1 + (−1)ζi,j 2−(i+1−log2 uj ) − 1, N

x∈P 1≤j≤s 1≤i≤n

where we write x = ( i≥1 2−i ζi,j )1≤j≤s . In fact, this discretized version in the case uj = 2 is originally introduced in [9, Definition 3.5] as a definition of WAFOM. They introduced this value as a computable quantity on digital nets for QMC integration error. • This computable formula appears in [20, Proposition 3] for so-called ‘polynomial lattice rules’, which is a special type of digital nets. ( ) • In finding efficient QMC-rules, costs for computing figure of merits such as Wu (P) and Err gs,u ; P are important. n Harase [11] proposed a method to accelerate the computation of Wu (P) using lookup tables. Though the timings will depend on many factors including code quality, compiler, optimization and CPU, his method is faster than a direct ( ) computation of Err gs,u ; P in [11, Table 1]. See [11] for detailed information. • There are several WAFOM-type figure of merits for digital nets, for example the base b > 2 case [21] and root-meansquare type [22]. We could not give approximation of such generalizations in terms of exponential functions.

∑

(

4. Numerical comparison between Wun (P) and Err gs,u ; P

)

In previous sections we have shown that the following are approximately equivalent up to constant multiplication (see Theorem 3.1, Lemma 3.6 and Corollary 3.11): 1. the worst-case integration error of the function space Fs,u by P, 2. the quantity Wu of P, and ( ) 3. the integration error of the exponential function Err gs,u ; P by P.

(

)

As shown in Lemma 3.6, the ratio of Err gs,u ; P to Wu (P) lies between As,u and Bs,u . While this is a theoretical bound, this section is to observe the distribution of the ratio through the experiment.

M. Matsumoto et al. / Journal of Computational and Applied Mathematics 330 (2018) 155–164

(

Fig. 1. The values W232 (P) and Err exp(−2

∑

1≤j≤s xj )

163

) ; P /W232 (P) for randomly generated digital nets P.

The method of the experiment is as follows. 1. Fix the dimensionality s, the weights u, the size of digital nets m, the number of precision digits n and the number of trials q. Here we use the same n, m as in Definition 2.1. 2. Generate q digital nets P by uniform random choice of their matrices C1 , . . . , Cs (see Definition 2.1 for how to generate P by C1 , . . . , Cs ). ( ) 3. Compute the integration error Err gs,u ; P , WAFOM value Wu (P) and their ratio. We fix uj = 2 (which corresponds to the original WAFOM), n = 32 and q = 1024. Fig. 1 shows the approximation accuracy for randomly generated digital nets with (s, m) = (2, 8), (4, 8), (4, 16) and (8, 16). Here we use the notation u = 2 meaning that uj = 2 for any j. ( ) In each figure, the horizontal axis corresponds to W232 (P) and the vertical axis to the ratio Err gs,u ; P /W232 (P), on both of which we use base-2 log-scale. The horizontal lines represent the lower and upper bounds Bs,u and As,u . The value for As,u is computed by straightforward numerical approximation. The value for Bs,u is computed by taking infimum for w ≤ n.

164

M. Matsumoto et al. / Journal of Computational and Applied Mathematics 330 (2018) 155–164

(

)

The figures show that the ratio is between the upper and lower bounds, and that Err gs,u ; P is a moderate approximation of W232 (P), and thus a moderate approximation of W2 (P) in each case. 5. Conclusion

(

)

We show that the QMC integration error of exp(− 1≤j≤s uj xj ) by a digital net P bounds the worst case error ewor Fs,u ; P . Here Fs,u is some smooth function class in C ∞ [0, 1]s . This fact implies that the exponential function is most difficult to approximate the integration ) space. Also, a WAFOM-like figure of merit Wu (P) is shown to approximate ( value ∑ ∑in this function the integration error Err exp(− 1≤j≤s uj xj ); P . Thus, a digital net with small QMC integration error for exp(− 1≤j≤s uj xj ) must have small Wu (P).

∑

Acknowledgments The first author was supported by JSPS/MEXT Grant in Aid 23244002, 26310211, 15K13460. The second and the third author were supported by the Program for Leading Graduate Schools, MEXT, Japan. The first and third authors were supported by JST CREST. The third author was supported under the Australian Research Councils Discovery Projects funding scheme (project number DP150101770) and Grant-in- Aid for JSPS Fellows Grant number 17J02651. References [1] J. Dick, F. Pillichshammer, Digital Nets and Sequences, Discrepancy Theory and Quasi-Monte Carlo Integration, Cambridge University Press, Cambridge, 2010. [2] H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods, CBMS-NSF, Philadelphia, Pennsylvania, 1992. [3] I. Sloan, S. Joe, Lattice Methods for Multiple Integration, Clarendon Press, Oxford, 1994. [4] L. Kuipers, H. Niederreiter, Uniform Distribution of Sequences, John Wiley, New York, 2006 1974, Reprint, Dover Publications, Mineola, NY. [5] J. Dick, Explicit constructions of quasi-Monte Carlo rules for the numerical integration of high-dimensional periodic functions, SIAM J. Numer. Anal. 45 (2007) 2141–2176. [6] J. Dick, Walsh spaces containing smooth functions and quasi-Monte Carlo rules of arbitrary high order, SIAM J. Numer. Anal. 46 (2008) 1519–1553. [7] M. Matsumoto, T. Yoshiki, Existence of Higher Order Convergent Quasi-Monte Carlo Rules via Walsh Figure of Merit, Springer Proceedings in Mathematics & Statistics 65 (2013) 569–579. [8] K. Suzuki, An explicit construction of point sets with large minimum Dick weight, J. Complexity 30 (2014) 347–354. [9] M. Matsumoto, M. Saito, K. Matoba, A computable figure of merit for Quasi-Monte Carlo point sets, Math. Comp. 83 (2014) 1233–1250. [10] S. Harase, Quasi-Monte Carlo point sets with small t-values and WAFOM, Appl. Math. Comput. 254 (2015) 318–326. [11] S. Harase, A search for extensible low-WAFOM point sets, Monte Carlo Methods and Applications 22 (2016) 349–357. [12] K. Suzuki, Super-polynomial convergence and tractability of multivariate integration for infinitely times differentiable functions, J. Complexity 39 (2017) 51–68. [13] N. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949) 372–414. [14] F. Schipp, W. Wade, P. Simon, Walsh Series. An Introduction to Dyadic Harmonic Analysis, With the collaboration of J. Pál, Adam Hilger Ltd., Bristol, 1990. [15] J. Dick, F. Pillichshammer, Multivariate integration in weighted Hilbert spaces based on Walsh functions and weighted Sobolev spaces, J. Complexity 21 (2005) 149–195. [16] H. Niederreiter, G. Pirsic, Duality for digital nets and its applications, Acta Arith. 97 (2001) 173–182. [17] T. Goda, K. Suzuki, T. Yoshiki, The b-adic tent transformation for quasi-Monte Carlo integration using digital nets, J. Approx. Theory 194 (2015) 62–86. [18] T. Yoshiki, Bounds on Walsh coefficients by dyadic difference and a new Koksma-Hlawka type inequality for Quasi-Monte Carlo integration, Hiroshima Math. J. 47 (2017) 155–179. [19] K. Suzuki, T. Yoshiki, Formulas for the Walsh coefficients of smooth functions and their application to bounds on the Walsh coefficients, J. Approx. Theory 205 (2016) 1–24. [20] J. Dick, T. Goda, K. Suzuki, T. Yoshiki, Construction of interlaced polynomial lattice rules for infinitely differentiable functions. arXiv:160200793. [21] K. Suzuki, WAFOM over abelian groups for quasi-Monte Carlo point sets, Hiroshima Math. J. 45 (2015) 341–364. [22] T. Goda, R. Ohori, K. Suzuki, T. Yoshiki, The mean square quasi-monte carlo error for digitally shifted digital nets, Springer Proceedings in Mathematics & Statistics 163 (2016) 331–350.