Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square

We prove lower bounds for the error of optimal cubature formulae for dvariate functions from Besov spaces of mixed smoothness Bα p,θ(Gd ) in the case 1 ≤ p ≤ ∞, 0 < θ ≤ ∞ and α > 1p, where Gd is either the ddimensional torus T d or the ddimensional unit cube I d . We prove upper bounds for QMC methods of integration on the Fibonacci lattice for bivariate periodic functions from Bα p,θ(T 2 ) in the case 1 ≤ p ≤ ∞, 0 < θ ≤ ∞ and α > 1p. A nonperiodic modification of this classical formula yields upper bounds for Bα p,θ(I 2 ) if 1p < α < 1 + 1p. In combination these results yield the correct asymptotic error of optimal cubature formulae for functions from Bα p,θ(G2 ) and indicate that a corresponding result is most likely also true in case d > 2. This is compared to the correct asymptotic of optimal cubature formulae on Smolyak grids which results in the observation that any cubature formula on Smolyak grids can never achieve the optimal worstcase error.

Lower bounds for the integration error for multivariate functions with mixed smoothness and

optimal Fibonacci cubature for functions on the square

Dinh D˜unga∗, Tino Ullrichb

a Vietnam National University, Hanoi, Information Technology Institute

144, Xuan Thuy, Hanoi, Vietnam

bHausdorff-Center for Mathematics and Institute for Numerical Simulation

53115 Bonn, GermanyMay 5, 2014 Revised version R4

is either the d-dimensional torus T d

or the d-dimensional unit cube I d We prove upper bounds for QMC methods of integration on the Fibonacci lattice for bivariate periodic functions from B α

p,θ (T 2 ) in the case 1 ≤ p ≤ ∞, 0 < θ ≤ ∞ and α > 1/p A non-periodic modification of this classical formula yields upper bounds for B α

p,θ (I 2 ) if 1/p < α < 1 + 1/p In combination these results yield the correct asymptotic error of optimal cubature formulae for functions from B α

p,θ (G 2 ) and indicate that a corresponding result is most likely also true in case

d > 2 This is compared to the correct asymptotic of optimal cubature formulae on Smolyak grids which results in the observation that any cubature formula on Smolyak grids can never achieve the optimal worst-case error.

Keywords Quasi-Monte-Carlo integration; Besov spaces of mixed smoothness; Fibonacci tice; B-spline representations; Smolyak grids.

lat-Mathematics Subject Classifications (2000) 41A15 · 41A05 · 41A25 · 41A58 · 41A63.

1 Introduction

This paper deals with optimal cubature formulae of functions with mixed smoothness defined either

on the d-cube Id = [0, 1]d or the d-torus Td = [0, 1]d, where in each component interval [0, 1] the

∗

Corresponding author Email: dinhzung@gmail.com

points 0 and 1 are identified Functions defined on Td can be also considered as functions on Rd

which are 1-periodic in each variable A general cubature formula is given by

In(Xn, f ) := Λn(Xn, f ) The worst-case error of an optimal cubature formula with respect to the class Fd is given by

Our main focus lies on integration in Besov-Nikol’skij spaces Bp,θα (Gd) of mixed smoothness

α, where 1 ≤ p ≤ ∞, 0 < θ ≤ ∞ and α > 1/p Let Up,θα (Gd) denote the unit ball in Bp,θα (Gd).The present paper is a continuation of the second author’s work [27] where optimal cubature

of bivariate functions from Uα

p,θ(T2) on Hammersley type point sets has been studied Indeed,here we investigate the asymptotic of the quantity Intn(Uα

p,θ(Gd)) where, in contrast to [27], thesmoothness α can now be larger or equal to 2 This by now classical research topic goes back to thework of Korobov [12], Hlawka [11], and Bakhvalov [2] in the 1960s In contrast to the quadrature

of univariate functions, where equidistant point grids lead to optimal formulas, the multivariateproblem is much more involved In fact, the choice of proper sets Xn⊂ Td of integration knots isconnected with deep problems in number theory, already for d = 2

Spaces of mixed smoothness have a long history in the former Soviet Union, see [1, 7, 16, 23]and the references therein, and continued attracting significant interest also in the last 5 years[28, 26, 8] Cubature formulae in Sobolev spaces Wpα(Td) and their optimality were studied in[10, 20, 22, 23, 24] We refer the reader to [23, 24] for details and references there on the relatedresults Temlyakov [22] studied optimal cubature in the related Sobolev spaces Wpα(T2) of mixedsmoothness as well as in Nikol’skij spaces Bαp,∞(T2) by using formulae based on Fibonacci numbers(see also [23, Thm IV.2.6]) This highly nontrivial idea goes back to Bakhvalov [2] and indicatesonce more the deep connection to number theoretical issues In the present paper, we extend thoseresults to values θ < ∞ In fact, for 1 ≤ p ≤ ∞, 0 < θ ≤ ∞ and α > 1/p we prove the relation

Intn(Up,θα (T2))  n−α(log n)(1−1/θ)+, 2 ≤ n ∈ N (1.4)

As one would expect, also Fibonacci quasi-Monte-Carlo methods are optimal and yield the correctasymptotic of Intn(Up,θα (T2)) in (1.4) Note, that the case 0 < θ ≤ 1 is not excluded and the log-term disappears Thus, the optimal integration error decays as quickly as in the univariate case In

fact, this represents one of the motivations to consider the third index θ Unfortunately, Fibonaccicubature formulae so far do not have a proper extension to d dimensions Hence, the method inCorollary 3.2 below does not help for general d > 2 For a partial result in case 1/p < α ≤ 1 andarbitrary d let us refer to [13, 14, 15].

Not long ago, Triebel [25, Thm 5.15] proved that if 1 ≤ p, θ ≤ ∞ and 1/p < α < 1 + 1/p, then

n−α(log n)(d−1)(1−1/θ) Intn(Up,θα (Id)) n−α(log n)(d−1)(α+1−1/θ), 2 ≤ n ∈ N , (1.5)

by using integration knots from Smolyak grids [19] The gap between upper and lower bound in(1.5) has been recently closed by the second named author [27] in case d = 2 by proving that thelower bound is sharp if 1/p < α < 2 Let us point out that, although we have established herethe correct asymptotic (1.4) for Intn(Up,θα (T2)) in the periodic setting for all α > 1/p, it is still notknown for Intn(Uα

p,θ(I2)) and large α ≥ 2

Another main contribution of this paper is the lower bound

Intn(Up,θα (Gd)) & n−α(log n)(d−1)(1−1/θ)+, 2 ≤ n ∈ N , (1.6)for general d and all α > 1/p with 1 ≤ p ≤ ∞, 0 < θ ≤ ∞ As the main tool we use the B-splinerepresentations of functions from Besov spaces with mixed smoothness based on the first author’swork [8] To establish (1.4) we exclusively used the Fourier analytical characterization of bivariateBesov spaces of mixed smoothness in terms of a decomposition of the frequency domain

The results in the present paper (1.4) and (1.6) as well as other particular results in [23],[13, 14, 15] lead to the strong conjecture that

Intn(Up,θα (Gd))  n−α(log n)(d−1)(1−1/θ)+, 2 ≤ n ∈ N , (1.7)for all α > 1/p, 1 ≤ p ≤ ∞, 0 < θ ≤ ∞ and all d > 1 In fact, the main open problem is the upperbound in (1.7) for d > 2 and α > 1/p In some special cases, namely the conjecture (1.7) has beenalready proved by Frolov [10] for p = θ = ∞, 0 < α < 1 and Gd= Td, and by Bakhvalov [3] (thelower bound) and Dubinin [6] (the upper bound) for 1 < p ≤ ∞, θ = ∞, α > 1 and Gd= Td (seealso Temlyakov [23, Thms IV.1.1, IV.3.3 and IV.4.6] for details) Recently, Markhasin [13, 14, 15]has proven (1.7) in case 1/p < α ≤ 1 for the slightly smaller classes Up,θα (Id)q with vanishingboundary values on the “upper” and “right” boundary faces of Id= [0, 1]d

Moreover, in the present paper we are also concerned with the problem of optimal cubature onso-called Smolyak grids [19], given by

Gd(m) := [

k 1 + +kd≤m

Ik1 × × Ikd (1.8)

where Ik := {2−k` : ` = 0, , 2k − 1} If Λm = (λξ)ξ∈Gd (m), we consider the cubature formula

Λsm(f ) := Λm(Gd(m), f ) on Smolyak grids Gd(m) given by

For 1 ≤ p ≤ ∞, 0 < θ ≤ ∞ and α > 1/p, we obtain the correct asymptotic behavior

Intsn(Up,θα (Gd))  n−α(log n)(d−1)(α+(1−1/θ)+ ), 2 ≤ n ∈ N , (1.10)which, in combination with (1.4), shows that cubature formulae Λsm(f ) on Smolyak grids Gd(m) cannever be optimal for Intn(Up,θα (T2)) The upper bound of (1.10) follows from results on samplingrecovery in the L1-norm proved in [8] For surveys and recent results on sampling recovery onSmolyak grids see, for example, [5], [8], [17], and [18] To obtain the lower bound we constructtest functions based on B-spline representations of functions from Bp,θα (Td) In fact, it turns outthat the errors of sampling recovery and numerical integration on Smolyak grids asymptoticallycoincide

The paper is organized as follows In Section 2 we introduce the relevant Besov spaces Bp,θα (Gd)and our main tools, their B-spline representation as well as a Fourier analytical characterization

of bivariate Besov spaces Bαp,θ(T2) in terms of a dyadic decomposition of the frequency domain.Section 3 deals with the cubature of bivariate periodic and non-periodic functions from Up,θα (G2)

on the Fibonacci lattice In particular, we prove the upper bound of (1.4), whereas in Section 4

we establish the lower bound (1.6) for general d and all α > 1/p Section 5 is concerned with therelation (1.10) as well the asymptotic behavior of the quantity of optimal sampling recovery onSmolyak grids

Notation Let us introduce some common notations which are used in the present paper Asusual, N denotes the natural numbers, Z the integers and R the real numbers The set Z+collectsthe nonnegative integers, sometimes we also use N0 We denote by T the torus represented as theinterval [0, 1] with identification of the end points For a real number a we put a+ := max{a, 0}.The symbol d is always reserved for the dimension in Zd, Rd, Nd, and Td For 0 < p ≤ ∞and x ∈ Rd we denote |x|p = (Pd

i=1|xi|p)1/p with the usual modification in case p = ∞ Theinner product between two vectors x, y ∈ Rd is denoted by x · y or hx, yi In particular, we have

|x|2

2 = x · x = hx, xi For a number n ∈ N we set [n] = {1, , n} If X is a Banach space, the norm

of an element f in X will be denoted by kf kX For real numbers a, b > 0 we use the notation a b

if it exists a constant c > 0 (independent of the relevant parameters) such that a ≤ cb Finally,

a  b means a b and b a

2 Besov spaces of mixed smoothness

Let us define Besov spaces of mixed smoothness Bp,θα (Gd), where Gd denotes either Td or Id

In order to treat both situations, periodic and non-periodic spaces, simultaneously, we use theclassical definition via mixed moduli of smoothness Later we will add the Fourier analyticalcharacterization for spaces on T2 in terms of a decomposition in frequency domain Let us firstrecall the basic concepts For univariate functions f : [0, 1] → C the `th difference operator ∆`h isdefined by

∆`h(f, x) :=

( P` j=0(−1)`−j `j
f (x + jh) : x + `h ∈ [0, 1],

0 : otherwise

Let e be any subset of [d] For multivariate functions f : Id → C and h ∈ Rd the mixed (`, e)thdifference operator ∆`,eh is defined by

∆`,eh := Y

i∈e

∆`hi and ∆`,∅h = Id,

where Id f = f and the univariate operator ∆`h

i is applied to the univariate function f by ering f as a function of variable xi with the other variables kept fixed In case d = 2 we slightlysimplify the notation and use ∆`

consid-(h 1 ,h 2 ):= ∆`,{1,2}h , ∆`

h 1 ,1:= ∆`,{1}h , and ∆`

h 2 ,2:= ∆`,{2}h For 1 ≤ p ≤ ∞, denote by Lp(Gd) the Banach space of functions on Gdwith finite pth integralnorm k · kp:= k · kLp(Gd ) if 1 ≤ p < ∞, and sup-norm k · k∞:= k · kL∞(Gd ) if p = ∞ Let

ωe`(f, t)p:= sup

|hi|<ti,i∈e

k∆`,eh (f )kp , t ∈ Id,

be the mixed (`, e)th modulus of smoothness of f ∈ Lp(Gd) (in particular, ω`∅(f, t)p = kf kp) Let

us turn to the definition of the Besov spaces Bp,θα (Gd) For 1 ≤ p ≤ ∞, 0 < θ ≤ ∞, α > 0 and

` > α we introduce the semi-quasi-norm |f |Bα,e

Id

hQ

i∈et−αi ω`e(f, t)piθQ

i∈et−1i dt1/θ : θ < ∞ ,supt∈Id Qi∈et−αi ω`e(f, t)p : θ = ∞(in particular, |f |Bα,∅

e⊂[d]

|f |Bα,e p,θ (G d ).The space of periodic functions Bp,θα (Td) can be considered as a subspace of Bp,θα (Id)

and the d-variate B-spline Nk,s(x), k ∈ Zd

with the corresponding change when p = ∞

We extend the notation x+:= max{0, x} to vectors x ∈ Rdby putting x+:= ((x1)+, , (xd)+) Furthermore, for a subset e ⊂ {1, , d} we define the subset Zd+(e) ⊂ Zd by Zd+(e) := {s ∈ Zd+ :

si = 0, i /∈ e} For a proof of the following lemma we refer to [8, Lemma 2.3]

Lemma 2.2 Let 1 ≤ p ≤ ∞ and δ = r − 1 + 1/p If the continuous function g on Id is represented

by the series g =P

k∈Z d + gk with convergence in C(Id), where gk ∈ Σd

r(k), then we have for any

whenever the sum on the right-hand side is finite The constant C is independent of g and `

As a next step, we obtain as a consequence of Lemma 2.2 the following result Its proof issimilar to the one in [8, Theorem 2.1(ii)] (see also [9, Lemma 2.5]) The main tool is an application

of the discrete Hardy inequality, see [8, (2.28)–(2.29)]

Lemma 2.3 Let 1 ≤ p ≤ ∞, 0 < θ ≤ ∞ and 0 < α < r − 1 + 1/p Let further g be a continuousfunction on Id which is represented by a series

g = X

k∈Z d +

2.2 The tensor Faber basis in two dimensions

Let us collect some facts about the important special case r = 2 of the cardinal B-spline system.The resulting system is called “tensor Faber basis” In this subsection we will mainly focus on aconverse statement to Lemma 2.3 in two dimensions

To simplify notations let us introduce the set N−1 = N0∪ {−1} Let further D−1 := {0, 1} and

Dj := {0, , 2j− 1} if j ≥ 0 Now we define for j ∈ N−1 and m ∈ Dj

Lemma 2.4 Let 1 ≤ p ≤ ∞, 0 < θ ≤ ∞ and 1/p < α < 2 Then we have for any f ∈ Bp,θα (I2),

p,θ (I 2 ) (2.7)

The following lemma is a periodic version of Lemma 2.3 for the tensor Faber basis For a proof

we refer to [27, Prop 3.6]

Lemma 2.5 Let 1 ≤ p ≤ ∞, 0 < θ ≤ ∞ and 1/p < α < 1 + 1/p Then we have for all f ∈ C(T2),

kf kBα p,θ (T 2 ).h X

2.3 Decomposition of the frequency domain

We consider the Fourier analytical characterization of bivariate Besov spaces of mixed smoothness.The characterization comes from a partition of the frequency domain The following assertionshave counterparts also for d > 2, see [26] Here, we will need it just for d = 2

Definition 2.6 Let Φ(R) be defined as the collection of all systems ϕ = {ϕj(x)}∞j=0 ⊂ C0∞(R)satisfying

Remark 2.7 The class Φ(R) is not empty Consider the following example Let ϕ0(x) ∈ C0∞(R)

be smooth function with ϕ0(x) = 1 on [−1, 1] and ϕ0(x) = 0 if |x| > 2 For j > 0 we define

ϕj(x) = ϕ0(2−jx) − ϕ0(2−j+1x)

Now it is easy to verify that the system ϕ = {ϕj(x)}∞j=0 satisfies (i) - (iv)

Now we fix a system {ϕj}∞

j=0∈ Φ(R) For j = (j1, j2) ∈ Z2 let the building blocks fj be given by

Lemma 2.8 Let 1 ≤ p ≤ ∞, 0 < θ ≤ ∞ and α > 0 Then Bα

p,θ(T2) is the collection of all

f ∈ Lp(T2) such that

kf |Bα p,θ(T2)k := X

j∈N 2

2|j|1 αθkδj(f )kθp1/θ (2.9)

is finite (usual modification in case q = ∞) Moreover, the quasi-norms k·kBα

p,θ (T 2 )and k·|Bp,θα (T2)kare equivalent

Proof For the bivariate case we refer to [16, 2.3.4] See [26] for the corresponding characterizations

of Besov-Lizorkin-Triebel spaces with dominating mixed smoothness on Rd and Td

3 Integration on the Fibonacci lattice

In this section we will prove upper bounds for Intn(Up,θα (G2)) which are realized by Fibonaccicubature formulas If G = T we obtain sharp results for all α > 1/p whereas we need theadditional condition 1/p < r < 1 + 1/p if G = I The restriction to d = 2 is due the concept ofthe Fibonacci lattice rule which so far does not have a proper extension to d > 2 The Fibonaccinumbers given by

b0 = b1 = 1 , bn= bn−1+ bn−2 , n ≥ 2 , (3.1)play the central role in the definition of the associated integration lattice In the sequel, the symbol

bn is always reserved for (3.1) For n ∈ N we are going to study the Fibonacci cubature formula

3.1 Integration of periodic functions

We are going to prove the theorem below which extends Temlyakov’s results [23, Thm IV.2.6]

on the spaces Bp,∞α (T2), to the spaces Bp,θα (T2) with 0 < θ ≤ ∞ By using simple embedding

properties, our results below directly imply Temlyakov’s earlier results [23, Thm IV.2.1], [4, Thm.1.1] on Sobolev spaces Wpr(T2) Let us denote by

Rn(f ) := Φn(f ) − I(f )the Fibonacci integration error

Theorem 3.1 Let 1 ≤ p ≤ ∞, 0 < θ ≤ ∞ and α > 1/p Then there exists a constant c > 0depending only on α, p and θ such that

sup

f ∈U α p,θ (T 2 )

|Rn(f )| ≤ c b−αn (log bn)(1−1/θ)+ , 2 ≤ n ∈ N

We postpone the proof of this theorem to Subsection 3.2

Corollary 3.2 Let 1 ≤ p ≤ ∞, 0 < θ ≤ ∞ and α > 1/p Then there exists a constant c > 0depending only on α, p and θ such that

Note that the case 0 < θ ≤ 1 is not excluded here In this case we obtain the upper bound

n−α without the log term Consequently, optimal cubature for this model of functions behaveslike optimal quadrature for Bp,θα (T) We conjecture the same phenomenon for d-variate functions.This gives one reason to vary the third index θ in (0, ∞]

3.2 Proof of Theorem 3.1

Let us divide the proof of Theorem 3.1 into several steps The first part of the proof followsTemlyakov [23, pages 220,221] To begin with we will consider the integration error Rn(f ) for atrigonometric polynomial f on T2 Let f (x) =P

k∈Z 2f (k)eˆ 2πik·x be the Fourier series of f Thenclearly, Φn(f ) =P

k∈Z 2f (k)Φˆ n(e2πik·) and I(f ) = ˆf (0) Therefore, we obtain

Rn(f ) = X

k∈Z 2 k6=0

ˆ

where Φn(k) := Φn(e2πik·) , k ∈ Z2 By definition, we have that

Γ(η) = {(k1, k2) ∈ Z2: max{1, |k1|} · max{1, |k2|} ≤ η} , η > 0

The following two Lemmas are essentially Lemma IV.2.1 and Lemma IV.2.2, respectively, in [23].They represent useful number theoretic properties of the set L(n) For the sake of completeness

we provide a detailed proof of Lemma 3.4 below

Lemma 3.3 There exists a universal constant γ > 0 such that for every n ∈ N,

Γ(γbn) ∩ L(n) \ {0}
= ∅ (3.8)

Proof See Lemma IV.2.1 in [23]

Lemma 3.4 For every n ∈ N the set L(n) can be represented in the form

L(n) =n ubn−2− vbn−3, u + 2v) : u, v ∈ Zo (3.9)

Proof Let ˜L(n) =(ubn−2− vbn−3, u + 2v) : u, v ∈ Z

Step 1 We prove ˜L(n) ⊂ L(n) For k ∈ ˜L(n) we have to show that k1 + bn−1k2 = `bn forsome ` ∈ Z Indeed, ubn−2− vbn−3+ bn−1(u + 2v) = ubn+ vbn−2+ vbn−1= bn(u + v)

Step 2 We prove L(n) ⊂ ˜L(n) For k = (k1, k2) ∈ L(n) we have to find u, v ∈ Z such thatthe representation k1 = ubn−2− vbn−3 and k2 = u + 2v holds true Indeed, since k ∈ L(n), wehave that k1+ bn−1k2 = k1+ (bn−3+ bn−2)k2 = `bn = `(bn−3+ 2bn−2) for some ` ∈ Z The last

identity implies k1= (` − k2)bn−3+ (2` − k2)bn−2 Putting v = k2− ` and u = 2` − k2 yields thedesired representation.

In the following, we will use a different argument than the one used by Temlyakov to deal withthe case θ = ∞ We will modify the definition of the functions χs introduced in [23] before (2.37)

on page 229 This allows for the an alternative argument in order to incorporate the case p = 1 inthe proof of Lemma 3.5 below Let us also mention, that the argument to establish the relationbetween (2.25) and (2.26) in [23] on page 226 requires some additional work, see Step 3 of theproof of Lemma 3.5 below

For s ∈ N0 we define the discrete set ρ(s) = {k ∈ Z : 2s−2 ≤ |k| < 2s+2} if s ∈ N andρ(s) = [−4, 4] if s = 0 Accordingly, let v0(·), v(·), vs(·), s ∈ N, be the piecewise linear functionsgiven by

Our next goal is to estimate kχskp for 1 ≤ p ≤ ∞

Lemma 3.5 Let 1 ≤ p ≤ ∞, s ∈ N20, and n ∈ N Then there is a constant c > 0 depending only

It is obvious that det Bn = bn, which will be important in the sequel Clearly, if ε > 0 is smallenough we obtain

Step 2 Let us deal with the case p = 1 By (3.11) we have that χs(·) = ηs(Bn∗·), where ηs

is the trigonometric polynomial given by

and v depending on s Since v0 and v are continuous, piecewise linear and compactly supportedunivariate functions we obtain from (3.13) the relation kηsk1 1

Step 3 It remains to show kηs(Bn∗·)k1 kηsk1 which implies (3.10) in case p = 1 In fact,

1

bnZ

B ∗

n (0,1) 2

|ηs(x)| dx ≤ m

bnZ

T2

|ηs(x)| dx = m

bnkηsk1. (3.15)