Polynomial Lattice Point Setsjosefdick/MCQMC... · 2013. 5. 24. · lattice point sets. In the following two sections the basic de nitions of (polynomial) lattice point sets and their

Polynomial Lattice Point Sets

Friedrich Pillichshammer

Abstract Polynomial lattice point sets are special types of (t,m, s)-nets asintroduced by H. Niederreiter in the 1980s. Quasi-Monte Carlo rules usingthem as underlying nodes are called polynomial lattice rules. In their overallstructure polynomial lattice rules are very similar to usual lattice rules dueto E. Hlawka and N. M. Korobov. The main difference is that here one usespolynomial arithmetic over a finite field instead of the usual integer arith-metic. In this overview paper we give a comprehensive review of the researchon polynomial lattice rules during the last decade. We touch on topics likeextensible polynomial lattice rules, higher order polynomial lattice rules andthe weighted discrepancy of polynomial lattice point sets. Furthermore wecompare polynomial lattice rules with lattice rules and show what results forpolynomial lattice rules also have an analog for usual lattice rules and viceversa.

1 Introduction

Assume we are interested in the approximation of multivariate integrals ofthe form Is(f) =

∫[0,1]s

f(x) dx using a quasi-Monte Carlo (QMC) rule of the

form QN,s(f) = (1/N)∑N−1n=0 f(xn) where x0, . . . ,xN−1 are fixed sample

nodes from the unit cube [0, 1)s. On first sight this approach looks quitesimple but the crux of this method is the choice of underlying nodes toobtain good approximations for large classes of functions.

Generally speaking, point sets with good uniform distribution propertiesyield a small absolute integration error. This is, for example, reflected in the

Friedrich PillichshammerInstitut fur Finanzmathematik, Universitat Linz, Altenbergerstraße 69, A-4040 Linz,Austria. e-mail: friedrich.pillichshammer(AT)jku.at

1

2 Friedrich Pillichshammer

Koksma-Hlawka inequality which states that

|Is(f)−QN,s(f)| ≤ V (f)D∗N (P)

where V (f) is the variation of f in the sense of Hardy and Krause andwhere D∗N denotes the star discrepancy of the point set P = {x0, . . . ,xN−1}.The star discrepancy can be defined as follows: given a point set P ={x0, . . . ,xN−1} of N elements in [0, 1)s the discrepancy function of P isdefined by

∆P(z) :=#{0 ≤ n < N : xn ∈ [0, z)}

N− λs([0, z)) for z ∈ (0, 1]s,

where λs is the s-dimensional Lebesgue measure. The star discrepancy of Pis then the L∞-norm of ∆P , i.e.,

D∗N (P) = supz∈(0,1]s

|∆P(z)|.

This is a quantitative measure for the deviation of P from uniform distri-bution modulo one. For more information on the Koksma-Hlawka inequalityand the star discrepancy we refer to the books [22, 26, 44, 58].

For any point set P consisting of N points in [0, 1)s it is known that

D∗N (P) ≥ cs(logN)κs/N

with a positive cs independent of P and where κ2 = 1 (see [5, 71]) andκs ≥ (s− 1)/2 for s ≥ 3 which follows from a result of Roth [68]. (For s ≥ 3the lower bound on κs has recently been improved to κs ≥ (s− 1)/2 + δs forsome unknown 0 < δs < 1/2; see [6].)

On the other hand, a point set P whose star discrepancy satisfies an upperbound of the form D∗N (P) ≤ Cs(logN)αs/N with a positive Cs independentof P and where αs ≥ 0, is informally called a low discrepancy point set. Thereare several methods to construct low discrepancy point sets:

• Hammersley point sets which are based on the infinite van der Corputsequence (see, e.g., [22, 58]);

• lattice point sets (or, more general, integration lattices) which were intro-duced independently by Korobov [38] and Hlawka [36] and which are wellexplained in the books of Niederreiter [58] and of Sloan and Joe [72];

• (t,m, s)-nets in base b which were introduced by Niederreiter [56, 58] andwhich are the main topic of the recent book [22]. Very special examples ofsuch nets go back to constructions of Sobol’ [77] and Faure [27].

In this article we are concerned with a sub-class of (t,m, s)-nets which hasa close relation to lattice point sets. Before we give its definition we recallthe definition of (t,m, s)-nets in base b according to Niederreiter [56].

Polynomial Lattice Point Sets 3

Definition 1. Let b, s,m, t be integers such that s ≥ 1, b ≥ 2 and 0 ≤ t ≤ m.A point set P consisting of bm points in [0, 1)s is called (t,m, s)-net in baseb if every so-called b-adic elementary interval of the form

s∏i=1

[aibdi

,ai + 1

bdi

)⊆ [0, 1)s, where ai, di ∈ N0 for 1 ≤ i ≤ s,

of volume bt−m contains exactly bt points of P.

Some remarks on the definition of (t,m, s)-nets in base b are in order (formore information see [22, 58]).

Remark 1. 1. Definition 1 states that for every b-adic elementary interval Jvolume bt−m we have #{x ∈ P : x ∈ J} − bmλs(J) = 0.

2. The uniform distribution quality depends on the so-called quality parame-ter t ∈ {0, . . . ,m}. A small t implies good uniform distribution. This is alsoreflected in Niederreiter’s bound on the star discrepancy of a (t,m, s)-netP in base b which states that

D∗N (P) = Os,b(bt(logN)s−1/N) (1)

where N = bm; see [22, 56, 58], and where Os,b indicates that the impliedconstant depends on s and b.

3. The optimal value t = 0 is only possible if the parameters b and s satisfys ≤ b + 1. On the other hand, any point set consisting of bm elements in[0, 1)s is an (m,m, s)-net in base b since this choice of parameters makesDefinition 1 trivial (and also the discrepancy bound (1)).

As already mentioned we are concerned with a sub-class of (t,m, s)-nets.Introduced by Niederreiter [57, 58], today this sub-class is known as polyno-mial lattice point sets. This name has its origin in a close relation to ordinarylattice point sets. In fact, the research on polynomial lattice point sets andon ordinary lattice point sets often follows two parallel tracks and bears alot of similarities. It is the aim of this overview to review the, in the author’sopinion, most important results on polynomial lattice point sets during thelast decade and to point out which of these results have counterparts forlattice point sets.

In the following two sections the basic definitions of (polynomial) latticepoint sets and their duals are provided. In Sections 4–9 we present the re-sults on polynomial lattice point sets and point out their analogs for latticepoint sets. The paper closes with a short summary and further remarks inSection 10.

Notation: Throughout the paper we assume that b is a prime number. By Zbwe denote the finite field with b elements and by Zb[x] the set of polynomialsover Zb. Define Gb,m := {h ∈ Zb[x] : deg(h) < m} and G∗b,m = Gb,m \ {0}.We have |Gb,m| = bm.


The field of formal Laurent series over Zb is denoted by Zb((x−1)). Ele-ments of Zb((x−1)) are of the form

L =

∞∑`=w

t`x−`, where w ∈ Z and all t` ∈ Zb.

For n ∈ N let νn : Zb((x−1))→ [0, 1) be defined by νn(L) =∑n`=max(1,w) t`b

−`.

For x ∈ R let {x} denote the fractional part of x, and by {x} for x ∈ Rswe mean that the fractional part is applied component-wise.

In many results which we are going to present in the following sectionsthere appear constants c which are assumed to be different from case tocase. Optionally these constants may depend on the dimension s, on b or onother quantities which are then indicated as sub-scripts. In most cases theseconstants could be given explicitly.

2 Polynomial lattice point sets

On account of their close relation to polynomial lattice point sets we firstrecall the possibly more familiar concept of lattice point sets:

Definition 2. For an integer N ≥ 2 and for g ∈ Zs the point set P(g, N)consisting of the N elements

xn ={ nNg}

for all 0 ≤ n < N

is called a lattice point set (LPS). A QMC rule using P(g, N) as underlyingnode set is called a lattice rule.

Polynomial lattice point sets are in their overall structure very similarto LPSs. The main difference is that LPSs are based on number theoreticconcepts whereas polynomial lattice point sets are based on algebraic meth-ods (polynomial arithmetic over a finite field). For simplicity we only discusspolynomial lattice point sets in prime base b. For the more general case ofprime-power bases we refer to [22, 58].

Definition 3. For s,m ∈ N, p ∈ Zb[x], with deg(p) = m, and q ∈ Zb[x]s thepoint set P(q, p) consisting of the bm elements

xh = νm

(h(x)

p(x)q(x)

)for all h ∈ Gb,m

is called a polynomial lattice point set (PLPS). A QMC rule using P(q, p) asunderlying node set is called a polynomial lattice rule.


Note that we obtain an LPS when we choose b = N , m = 1 and p(x) = x.The structural similarity between Definition 2 and Definition 3 is evident.Hence let us compare the two concepts by means of some pictures.

Fig. 1 left: P(g, N) with N = 13 and g = (1, 8); right: P(q, p) with p(x) = x4+x2+1and q = (1, x3) over Z2.

The LPS P(g, N) shown in the left part of Fig. 1 shows a very regularlattice structure. Such a geometric structure cannot be observed for the PLPSP(q, p) shown in the right part of Fig. 1. However, also this point set has someinherent structure, namely the (t,m, s)-net structure. In fact, for this exampleevery 2-adic elementary interval of area 2−4 contains exactly one element ofthe point set P(q, p) and hence we have a (0, 4, 2)-net in base 2; cf. Fig. 2.

A further example of an LPS and a PLPS is shown in Fig. 3.

3 The dual net

For LPSs one has the notion of a dual lattice which plays a crucial role inthe quality analysis of such point sets.

Definition 4. The dual lattice of the LPS P(g, N) from Definition 2 is de-fined as

Lg,N = {h ∈ Zs : h · g ≡ 0 (mod N)}.

An important property of LPSs is that∑x∈P(g,N)

ek(x) =

{N if k ∈ Lg,N ,0 otherwise,


Fig. 2 P(q, p) from Fig. 1 as (0, 4, 2)-net in base 2; every 2-adic elementary intervalof area 2−4 contains exactly one point.

Fig. 3 left: P(g, N) with N = 987 and g = (1, 610); right: P(q, p) with p(x) =x10 + x8 + x4 + x2 + 1 and q = (1, x9 + x5 + x) over Z2.

where ek(x) = exp(2πik ·x). This relation is the reason why for the analysisof the integration error of lattice rules it is most convenient to consider one-periodic functions; see [58, 72].

The corresponding definition for PLPSs leads to the notion of a dual net.

Definition 5. The dual net of the PLPS P(q, p) from Definition 3 is definedas

Dq,p = {k ∈ Gsb,m : k · q ≡ 0 (mod p)}.

An important property of PLPSs is that (see [22, Lemmas 4.75 and 10.6])

Polynomial Lattice Point Sets 7∑x∈P(q,p)

bwalk(x) =

{bm if k ∈ Dq,p,0 otherwise,

where bwalk(x) is the kth b-adic Walsh function defined by bwalk(x) :=∏si=1 bwalki(xi) for k = (k1, . . . , ks) ∈ Ns0 and x = (x1, . . . , xs) ∈ [0, 1)s.

The one-dimensional kth b-adic Walsh function is defined by bwalk(x) :=exp(2πi(ξ1κ0 + · · · + ξa+1κa)/b) for k = κ0 + κ1b + · · · + κab

a with κi ∈{0, . . . , b−1} and x = ξ1b

−1+ξ2b−2+· · · with infinitely many digits ξi 6= b−1.

Many properties of Walsh functions are summarized in [22, Appendix A].The above relation is the reason why for the analysis of the integration

error of polynomial lattice rules it is most convenient to consider Walsh series.We will come back to this issue in Section 6.

4 Quality measures and existence results

Based on the dual net one can introduce two quality measures for PLPSs (see[58, Chapter 4] or [22, Chapter 10]): for p ∈ Zb[x] and q ∈ Zb[x]s define

ρ(q, p) = s− 1 + minh∈Dq,p\{0}

s∑i=1

deg(hi)

and

Rb(q, p) =∑

h∈Dq,p\{0}

s∏i=1

rb(hi),

where rb(0) = 1 and rb(h) = b−r−1 sin−2(πκr/b) for h ∈ Gb,m of the formh = κ0 + κ1b+ · · ·+ κrx

r, κr 6= 0.We remark here that analogous quality measures also exist for LPSs; see

[58, Chapter 5]. Based on these quality measures Niederreiter [58] proved thefollowing results:

Theorem 1. The PLPS P(q, p) is a (t,m, s)-net in base b with m = deg(p),t = m− ρ(q, p) and

D∗bm(P(q, p)) ≤ s

bm+Rb(q, p).

For example for p = x4 + x2 + 1 and q = (1, x3) over Z2 the “minimal”element of Dq,p is (h1, h2) = (x2 + 1, x) and hence ρ(q, p) = 4 in this case.Theorem 1 then shows that P(q, p) is a (0, 4, 2)-net in base 2; cf. Fig. 2.Theorem 1 also gives a bound on the star discrepancy of PLPSs which iseasier to handle than D∗bm itself (note that the exact computation of the stardiscrepancy of a given point set is an NP-hard problem, see [28]). For ananalogous discrepancy bound for LPSs we refer to [58, Chapter 5] or [22,


Proposition 3.49]. Based on Theorem 1 one can use averaging arguments toobtain the following existence results:

Theorem 2. Let p ∈ Zb[x] with deg(p) = m.

1. If p is irreducible, then there exists q ∈ Gsb,m such that

t ≤ (s− 1) logbm− (s− 2)− logb(s− 1)!

(b− 1)s−1.

Hence D∗bm(P(q, p)) = Os,b(m2s−2b−m

).

2. For 0 ≤ ε < 1 there are more than ε|Gsb,m| vectors q ∈ Gsb,m with

D∗bm(P(q, p)) ≤ s

bm+Rb(q, p) = Os,b,ε

(ms

bm

).

Part 1 of Theorem 2 for b = 2 has been shown by Larcher et al. [51];see also [70] or [22, Chapter 10] for general b. Part 2 has been shown byNiederreiter [58, Chapter 4] and also by Dick et al. [14] and [18]. For ananalogous discrepancy bound for LPSs we refer to [58, Chapter 5] or [22,Theorem 3.51].

The bound on Rb in Theorem 2 is best possible in the order of magni-tude in m. This was shown recently by Kritzer and the author in [42]. Acorresponding result for LPSs has been shown by Larcher [49].

Theorem 3. There exists cs,b > 0 such that for any p ∈ Zb[x] with deg(p) =m and any q ∈ Gsb,m, qi 6= 0, 1 ≤ i ≤ s, we have

Rb(q, p) ≥ cs,bbdeg(δs)(m− deg(δs))

s

bmwhere δs := gcd(q1, . . . , qs, p).

On the other hand, the bound on D∗bm in Theorem 2 is not best possiblein the order of magnitude in m. For example, in dimension s = 2 the so-called Fibonacci PLPS has a star discrepancy of order Ob(mb

−m); see [58,Chapter 4] or [22, Chapter 10]. For arbitrary dimension s it was shown byLarcher [50] that for any m ≥ 2 there exists q ∈ Gsb,m with

D∗bm(P(q, xm)) = Os,b(ms−1(logm)b−m

);

see also [43] for an extension of this result to more general polynomials q. Acounterpart of Larcher’s result for LPSs is known for dimension s = 2 only;see [48, Corollary 3].


5 CBC construction of polynomial lattice point sets

According to Theorem 2, for any given irreducible polynomial p ∈ Zb[x] thereexist a sufficiently large number of “good” vectors q of polynomials whichyield PLPSs with reasonably low star discrepancy. Now one aims at findingsuch vectors by computer search. Unfortunately a full search is not possible(except maybe for small values of m, s) since one has to check bms vectors ofpolynomials.

At this point one gets a cue from the analogy between PLPSs and LPSswhere the component-by-component (CBC) construction approach worksvery well. This approach was introduced by Korobov [39] for LPSs andlater it was re-invented by Sloan and Reztsov [73]. The same idea appliesto PLPSs. Here we use the more general weighted star discrepancy as intro-duced by Sloan and Wozniakowski [74] as underlying quality criterion: letγ = (γ1, γ2, . . .) be a sequence of weights in R+. Let Is = {1, . . . , s} and foru ⊆ Is let γu =

∏i∈u γi. The weighted star discrepancy of an N -element point

set P in [0, 1)s is given by

D∗N,γ(P) = supz∈(0,1]s

max∅6=u⊆Is

γu|∆P((zu, 1))|.

The weights γ are additional parameters which model the importance ofthe different coordinate projections. For the weights γ = 1 := (1, 1, . . .) onehas D∗N,γ(P) = D∗N (P) for any point set P. In the weighted setting theCBC construction has the advantage that the quadrature points P can beoptimized with respect to γ.

The weighted Koksma-Hlawka inequality then states that

|Is(f)−QN,s(f)| ≤ D∗N,γ(P)‖f‖s,γ

with a certain norm ‖ · ‖s,γ ; see [37, 74] or [22, Chapter 2] for details.Let p ∈ Zb[x] with deg(p) = m and let q ∈ Gsb,m. Then it can be shown

(see [22, Corollary 10.16]) that

D∗bm,γ(P(q, p)) ≤∑∅6=u⊆Is

γu

(1−

(1− 1

bm

)|u|)+Rb,γ(q, p),

where

Rb,γ(q, p) =∑

h∈Dq,p\{0}

s∏i=1

rb(hi, γi)

and where for h ∈ Gb,m we put rb(0, γ) = 1 + γ and rb(h, γ) = γrb(h) ifh 6= 0, where rb(h) is as in Section 4. An analogous bound for the weightedstar discrepancy of LPSs can be found in [37].


Now we deal with the quantity Rb,γ(q, p) which can be computed inO(bms) operations (see [22, Proposition 10.20]).

Algorithm 1 CBC-algorithm for PLPSs

Require: b a prime, s,m ∈ N, p ∈ Zb[x], with deg(p) = m, and weights γ = (γi)i≥1.1: Choose q1 = 1.2: for d = 2 to s do3: find qd ∈ G∗b,m which minimises the quantity Rb,γ((q1, . . . , qd−1, z), p) as a

function of z.4: end for5: return q = (q1, . . . , qs).

Theorem 4. Let p be irreducible. If q ∈ Gsb,m is constructed with Algo-rithm 1, then

Rb,γ(q, p) ≤ 1

bm − 1

s∏i=1

(1 + γi

(1 +m

b2 − 1

3b

)),

A proof can be found in [18]. A similar result for not necessarily irre-ducible p has been shown in [14] and a corresponding result for LPSs is [37,Theorem 3].

Using an argument from [19, Section 7] one can deduce the following resultfrom Theorem 4; see also [22, Corollary 10.30].

Corollary 1. Let p be irreducible. If∑∞i=0 γi <∞, then for any δ > 0 there

exists cγ,δ > 0, such that for q ∈ Gsb,m constructed with Algorithm 1 we have

D∗bm,γ(P(q, p)) ≤ cγ,δb−m(1−δ).

Let N ∈ N with 2-adic expansion N = 2m1 + · · · + 2mk , where 0 ≤m1 < m2 < . . . < mk. For 1 ≤ j ≤ k choose p(j) ∈ Z2[x] irreduciblewith deg(p(j)) = mj and construct P(q(j), p(j)) with Algorithm 1. Then setPN = P(q(1), p(1)) ∪ . . . ∪ P(q(k), p(k)). In [35] the following is shown:

Corollary 2. If∑∞i=0 γi <∞, then for any δ > 0 there exists Cγ,δ > 0, such

that

D∗N,γ(PN ) ≤ Cγ,δN−1+δ for any N ∈ N.

The weighted star discrepancy is strongly polynomial tractable with ε-exponentequal to one.

The cost of the CBC-algorithm is of O(b2ms2) operations. This is compa-rable with the CBC construction cost of LPSs; cf. [37, Section 3]. However,in this form the CBC-algorithm can only be used for not too large cardinal-ity bm. A breakthrough for this problem was obtained by Nuyens and Cools


[64, 65] when they introduced — first for LPSs and then for PLPSs — the fastCBC construction with a significant reduction of cost to O(smbm) operationsusing O(bm) memory space. Only through this reduction of the constructioncost does the CBC-algorithm become applicable for the generation of PLPSs(and of LPSs) with reasonably large cardinality. See also [22, Section 10.3].

6 Integration of Walsh series

As already mentioned in Section 3 it is most convenient for the error anal-ysis of polynomial lattice rules to consider Walsh series. Let α > 1 and letHwal,s,α,γ be the weighted Hilbert function space with reproducing kernelgiven by

Kwal,s,α,γ(x,y) =∑k∈Ns0

ρα(k,γ) bwalk(x) bwalk(y),

where for k = (k1, . . . , ks) ∈ Ns0 we put ρα(k,γ) =∏sj=1 ρα(kj , γj) with

ρα(0, γ) = 1 and ρα(k, γ) = γb−αv if bv ≤ k < bv+1 for v ∈ N0. The norm inthis function space is given by

‖f‖Hwal,s,α,γ=∑k∈Ns0

ρα(k,γ)−1|fwal(k)|2

where fwal(k) =∫[0,1]s

f(x) bwalk(x) dx. For more information on Hwal,s,α,γ

we refer to [20]. The counterpart to the function space Hwal,s,α,γ for the anal-ysis of LPSs is the so-called Korobov space ([25, 75, 63] or [62, Appendix A.1])whose reproducing kernel looks similar to Kwal,s,α,γ but with the main differ-ence that the Walsh function system is replaced by the trigonometric functionsystem and Walsh coefficients are replaced by Fourier coefficients.

The worst-case integration error of a QMC rule is defined as the worst per-formance of the QMC algorithm over the unit ball of the function space underconsideration, i.e., in our case e(Hwal,s,α,γ ,P) := sup‖f‖Hwal,s,α,γ

≤1 |Is(f) −Qbm,s(f)|. For PLPSs it can be shown that

e2(q, p) := e2(Hwal,s,α,γ ,P(q, p)) =∑

k∈Ns0\{0}trum(k)(x)∈Dq,p

ρα(k,γ)

where trum(k) :≡ k (mod bm) (component-wise) and where

k = κ0 + κ1b+ · · ·+ κm−1bm−1 ∈ N0

is identified with


k(x) = κ0 + κ1x+ · · ·+ κm−1xm−1 ∈ Zb[x].

For the worst-case integration error of a polynomial lattice rule for integra-tion in Hwal,s,α,γ we have the following result which was first proved in [16]for irreducible p and later generalized in [41] to not necessarily irreducible p.The corresponding result for LPSs was shown by Korobov [39] for γ = 1 andby Kuo [45] for general weights (see also [10]).

Theorem 5. For any p ∈ Zb[x] with deg(p) = m one can construct q ∈ Gsb,musing a CBC algorithm such that (with N = bm)

e(q, p) ≤ cs,α,γ,δN−α/2+δ for all 0 < δ ≤ α−12 .

If∑∞i=1 γ

1/(α−2δ)i < ∞, then cs,α,γ,δ ≤ c∞,α,γ,δ < ∞, i.e., the above bound

can be made independent of the dimension s.

7 Extensible polynomial lattice point sets

A disadvantage of the CBC algorithm as used so far is that the generatedvectors q depend on p and hence onN = bdeg(p). If one changes p, then one hasto construct a new vector q ∈ Zb[x]s. The same problem appears for the CBCconstruction of LPSs. For this reason several authors have independently fromeach other introduced the concept of extensible LPSs, see [32, 33, 34, 40, 55].Niederreiter [59] was the first who considered extensible PLPSs. A specialcase will be explained below.

For p ∈ Zb[x] with m = deg(p) ≥ 1, let Yp be the set of all p-adic polyno-mials

∑∞k=0 akp

k with deg(ak) < m. Any Q ∈ Yp reduced modulo pn givesa polynomial in Zb[x] of degree less than nm, i.e., Yp/(p

n) = Gb,nm. LetQ ∈ Y sp and for n ∈ N let qn ≡ Q (mod pn). Then

P(q1, p) ⊆ P(q2, p2) ⊆ P(q3, p

3) ⊆ . . . .

Definition 6. An extensible PLPS is defined as the formal union P(Q, p) :=⋃k≥1 P(qk, p

k).

For P(qn, pn) only the first n “digits” in the p-adic expansion of each

component of Q are important. This observation is used in the followingconstruction algorithm which uses ideas from Korobov [40] for LPSs.

Theorem 6. If qn ∈ Gsb,m is constructed according to Algorithm 2, then

e2(qn, pn) ≤ cs,b,γ,αb−nm.

If∑∞i=1 γi < ∞, then cs,α,γ,δ ≤ c∞,α,γ,δ < ∞, i.e., the above bound can be

made independent of the dimension s.


Algorithm 2 Construction of extensible PLPSs

Require: b a prime, s,m ∈ N, p ∈ Zb[x] monic and irreducible with deg(p) = m,and weights γ = (γi)i≥1.

1: Find q1 := q by minimizing e2(q, p) over all q ∈ Gsb,m.2: for n = 2, 3, . . . do3: find qn := qn−1 + pn−1q by minimizing e2(qn−1 + pn−1q, pn) over all q ∈

Gsb,m.4: return qn.5: end for

A proof of this result and also a corresponding result for LPSs can befound in [61]; see also [22]. A disadvantage of the above error bound isthat the worst-case error converges only with order O(N−1/2) compared toO(N−α/2+δ) from Theorem 5 for not necessarily extensible PLPSs.

There exists another algorithm — first introduced for LPSs in [23] andthen for PLPSs in [11] — which is called CBC sieve algorithm (see [22, Sec-tion 10.4]) and which yields better error bounds, but with the disadvantagethat the generated PLPSs (and LPSs respectively) are only finitely extensi-ble. In this context one also speaks about embedded PLPSs (and embeddedLPSs respectively). For embedded LPSs we also refer to [7]. A pure existenceresult for extensible PLPSs with small star discrepancy is due to Niederre-iter [59]. For existence results for extensible LPSs we refer to Hickernell andNiederreiter [34].

8 Integration in Sobolev spaces

For x = x1b−1 + x2b

−2 + · · · and σ = σ1b−1 + σ2b

−2 + · · · with xi, σi ∈{0, . . . , b − 1} the digitally shifted point y = x ⊕ σ is given by y = y1b

−1 +y2b−2 + · · · , where yi = xi + σi (mod b). For vectors x and σ we define the

digitally shifted point y = x ⊕ σ component-wise. This digital shift can beused to randomize a PLPS.

Definition 7. For σ ∈ [0, 1)s the point set Pσ(q, p) := P(q, p)⊕ σ is calleda digitally shifted PLPS.

In the context of LPSs one often uses a “geometric” shift instead of thedigital shift to randomize the point set and speaks then about shifted LPSs.

Similar results to those from Section 6 hold for the mean square worst-case error of digitally shifted polynomial lattices for integration in the Sobolev

space H(1)sob,s,γ with reproducing kernel

K(1)sob,s,γ(x,y) =

s∏i=1

(1 + γiB1(xi)B1(yi) +

γi2B2(|xi − yi|)

),


where Bi is the ith Bernoulli polynomial. The function space H(1)sob,s,γ con-

tains all functions f : [0, 1]s → R whose mixed partial derivatives up toorder one in each variable are square integrable. See [24, 76] and [62, Ap-

pendix A.2.3.] for more information on H(1)sob,s,γ .

The mean square worst-case error of digitally shifted PLPSs for integration

in H(1)sob,s,γ is defined by

e2(q, p) =

∫[0,1]s

e2(H(1)sob,s,γ ,Pσ(q, p)) dσ.

We have the following result the proof of which can be found in [22, Theo-rem 12.14]; see also [16]. The corresponding result for shifted LPSs was shownby Kuo [45] (and follows in the case γ = 1 also from [39]).

Theorem 7. For any p ∈ Zb[x] with deg(p) = m we can construct q ∈ Gsb,musing a CBC algorithm such that (with N = bm)

e(q, p) ≤ cs,b,γ,εN−1+ε for all 0 < ε ≤ 1/2.

If∑si=1 γ

1/(2(1−ε))i < ∞, then cs,b,γ,ε ≤ c∞,b,γ,ε < ∞, i.e., the above bound

can be made independent of the dimension s.

Remark 2. Baldeaux and Dick [1] showed that in the randomized setting onecan obtain an improved error bound by using Owen’s scrambling (see [66] or[22, Chapter 13]). For scrambled PLPSs one has

E[|Is(f)−QN,s(f)|2

]≤ cs,b,γ,εN−3+ε for ε > 0

where N = bm and where the expectation is with respect to all randomscramblings of a PLPS. Such a result is not known for LPSs.

Now we assume more smoothness for integrands. Consider the Sobolev

space H(2)sob,s,γ with reproducing kernel

K(2)sob,s,γ(x,y) =s∏i=1

(1 + γiB1(xi)B1(yi) +

γ2i4B2(xi)B2(yi)−

γ2i24B4(|xi − yi|)

),

where Bi is the ith Bernoulli polynomial. The function space H(2)sob,s,γ con-

tains all functions f : [0, 1]s → R whose mixed partial derivatives up to ordertwo in each variable are square integrable. See [22, Section 14.6] for moreinformation.

Using an idea from Hickernell [31] we use the tent transformation φ(x) =1− |2x− 1|. For vectors x we apply φ component-wise and for a point set P,φ(P) means that the tent transformation is applied to every element of P.


We call φ(P) the folded point set P. Define the mean square worst-case errorof folded digitally shifted PLPSs by

e2φ(q, p) =

∫[0,1]s

e2(H(2)sob,s,γ , φ(Pσ(q, p))) dσ.

The following result, proved in [9], shows that one can obtain an improvedconvergence rate for the mean square worst-case error of folded digitally

shifted PLPSs for functions f ∈ H(2)sob,s,γ as integrands. A corresponding

result for LPSs has been shown by Hickernell [31].

Theorem 8. For any p ∈ Z2[x] with deg(p) = m we can construct q ∈ Gs2,musing a CBC algorithm such that (with N = 2m)

eφ(q, p) ≤ cs,γ,εN−2+ε for all 0 < ε ≤ 3/2.

If∑si=1 γ

1/(2(2−ε))i <∞, then cs,γ,ε ≤ c∞,γ,ε <∞, i.e., the above bound can

be made independent of the dimension s.

9 Higher order polynomial lattice rules

Now we go a step further and consider functions with arbitrary smoothnessas integrands. For a more detailed definition of the function spaces underconsideration we need some notation:

For k = κ1ba1−1 +κ2b

a2−1 + · · ·+κvbav−1, where 1 ≤ av < · · · < a1, v ∈ N

and κ1, . . . , κv ∈ {1, . . . , b− 1}, and for α ≥ 1 define

µα(k) := a1 + · · ·+ amin(v,α).

Furthermore, for γ > 0 put rα(0, γ) = 1 and rα(k, γ) = γb−µα(k) fork ∈ N. For k = (k1, . . . , ks) ∈ Ns0 and γ = (γ1, γ2, . . .), set rα(k,γ) :=∏si=1 rα(ki, γi).Let Wα,s,γ ⊆ L2([0, 1]s) be the space consisting of all Walsh series f =∑k∈Ns0

fwal(k) bwalk for which

‖f‖Wα,s,γ := supk∈Ns0

|fwal(k)|rα(k,γ)

<∞.

For α ≥ 2 the function space Wα,s,γ contains all functions f : [0, 1]s → Rwhose mixed partial derivatives up to order α in each variable are squareintegrable; see [12]. We call α the smoothness parameter of the function space.

Of course one would expect that the higher smoothness of integrands isreflected in the convergence rate of the integration error. Higher smoothnessshould lead to improved convergence rates. However, it turns out that this is


not the case when the concept of (digitally shifted) PLPSs, as introduced inDefinition 3, is used as underlying nodes. For this reason the following suitablegeneralization has been introduced in [21]; see also [22, Section 15.7].

Definition 8. For s,m, n ∈ N, m ≤ n, p ∈ Zb[x], with deg(p) = n, andq ∈ Zb[x]s the point set Pm,n(q, p) consisting of the bm points

xh = νn

(h(x)

p(x)q(x)

)for all h ∈ Gb,m

is called a higher order polynomial lattice point set (HOPLPS). A QMC ruleusing Pm,n(q, p) is called a higher order polynomial lattice rule.

Remark 3. For m = n we have Pm,m(q, p) = P(q, p).

Definition 9. The dual net of the HOPLPS Pm,n(q, p) from Definition 8 isdefined as

Dq,p = {k ∈ Gsb,n : k · q ≡ u (mod p) with deg(u) < n−m}.

Similar as in Section 4 one can introduce a generalization of the qualitymeasure ρ for HOPLPSs which can then be related to the worst-case integra-tion error of HOPLPSs. This was done in [15] (see also [22, Definition 15.27]).Instead of following this track here we study the worst-case error of HOPLPSsin Wα,s,γ more directly.

For α ≥ 2 the worst-case error for integration in Wα,s,γ using Pm,n(q, p)is given by (see [2, Proposition 2.1])

e2α(q, p) := e2α(Wα,s,γ ,Pm,n(q, p)) =∑

k∈Ns0\{0}trun(k)(x)∈Dq,p

rα(k,γ).

The following result has been shown in [2].

Theorem 9. For any irreducible p ∈ Zb[x] with deg(p) = n we can constructq ∈ Gsb,n using a CBC algorithm such that

eα(q, p) ≤ cs,α,γ,τ b−min(τm,n) for all 1 ≤ τ < α.

If∑∞i=1 γ

1/τi <∞ then cs,α,γ,τ ≤ c∞,α,γ,τ <∞, i.e., the above bound can be

made independent of the dimension s.

Remark 4. Choosing n large we obtain a convergence order of N−α+ε forε > 0 where N = bm. By a result of Sarygin [69] this convergence rate isessentially best possible. For a fast version of the CBC algorithm mentionedin Theorem 9 we refer to [4].

The result from Theorem 9 holds for a fixed smoothness parameter α ≥ 2.However, in practical applications the smoothness parameter is in general not


known a priori. Hence it is reasonable to ask for constructions of HOPLPSswhich achieve almost optimal convergence rates for a range of smoothnessparameters and which adjust themselves to the smoothness of a given inte-grand.

The basic idea in [2] can be roughly explained as follows. Assume thatp ∈ Zb[x] is given. If there exists a large enough amount of HOPLPSs P(q, p)which perform well for the smoothness parameter α and if there exists a largeenough amount of HOPLPSs P(q, p) which perform well for the smoothnessparameter α′, then there must be a HOPLPS P(q, p) which performs wellfor both smoothness parameters α and α′. The underlying mathematicalargument is the following “sieve principle”: let X be some finite set andA,B ⊆ X. If |A|, |B| > |X|/2, then |A ∩B| > 0.

Algorithm 3 Sieve Algorithm for HOPLPSs

Require: b a prime, s,m, β ∈ N, β ≥ 2, p ∈ Zb[x] irreducible with deg(p) = m,weights γ = (γi)i≥1,

1: Set n = βm.2: Find b(1− β−1)bβmsc+ 1 vectors q in Gsb,βm which satisfy

e2(q, p) ≤ cs,b,γ,m,β,2,τ2b−τ2m for all 1 ≤ τ2 < 2,

and label this set T2.3: for α = 3, . . . , β do4: find b(1− (α− 1)β−1)bβmsc+ 1 vectors q in Tα−1 which satisfy

eα(q, p) ≤ cs,b,γ,m,β,α,ταb−ταm for all 1 ≤ τα < α

and label this set Tα.5: end for6: return Select q∗ to be any vector from Tβ .

Algorithm 3 only presents the basic idea of a construction for HOPLPSwhich perform well for a range of smoothness parameters. In practice thisalgorithm would not be applicable since it is much too time consuming. How-ever, in [2, Algorithm 2] it has been show how Algorithm 3 can be combinedwith the CBC approach. This leads then to the following result which is [2,Theorem 4.2]:

Theorem 10. Let s,m, β ∈ N, β ≥ 2, then one can construct a vector q ∈Gsb,βm such that

eα(q, p) ≤ cs,b,α,β,γ,ταb−ταm for all 1 ≤ τα < α

and for all 2 ≤ α ≤ β.If∑∞i=1 γ

1/ταi < ∞, then cs,b,α,β,γ,τα ≤ c∞,b,α,β,γ,τα < ∞, i.e., the above

bound can be made independent of the dimension s.

There exists no counterpart of the results from this section for LPSs.


10 Summary and further comments

In this paper we have reviewed the main progress in the analysis of PLPSsover the last decade and we pointed out several connections to the theory ofLPSs.

For both concepts we have comparable discrepancy bounds and tractabil-ity properties, and the worst-case error analysis in several reproducing kernelHilbert spaces follows parallel tracks. PLPSs and LPSs can both be con-structed with the (fast) CBC approach and both can be made extensible inthe number of elements. The tent transformation together with a suitablerandomization leads in both cases to improved error bounds for smootherintegrands.

However, there are also some differences. For example, with a slight gen-eralization of the concept of PLPSs one can achieve almost optimal conver-gence rates for smooth integrands (even with varying smoothness from a finiterange) together with strong tractability, which means that the error bound isindependent of the dimension. Such a result is not known for LPSs until now.(But it is known that with LPSs one can obtain almost optimal convergencerates together with strong tractability for smooth periodic functions from aKorobov space.)

A further difference is that for PLPSs it makes sense to apply Owen’sscrambling scheme since this preserves the (t,m, s)-net structure of a pointset but not the geometric lattice structure. This leads to an improved errorbound in the randomized setting, a result which is not known for LPSs.

Also the consideration of the quality parameter t of LPSs makes in generallittle sense since these point sets are not constructed to have a good (t,m, s)-net structure. Nevertheless, the analog of the quality measure ρ(q, p) = m− tfrom Section 4 has some interpretation, namely it is the enhanced trigonomet-ric degree of a lattice rule [8, 54]. A cubature rule of enhanced trigonometricdegree δ is one that integrates all trigonometric polynomials of degree lessthen δ exactly. However, in this vein ρ(q, p) = m− t from Section 4 can alsobe interpreted as the enhanced Walsh degree of a polynomial lattice rule sinceany (t,m, s)-net in base b integrates all Walsh polynomials of degree ≤ m− texactly (this follows from [30, Lemma 1]).

A further point which was not discussed so far but which is worth to bementioned is that with LPSs one can even obtain exponential convergencefor the worst-case error of infinitely times differentiable periodic functions;see [17]. This should also be possible with PLPSs.

LPSs and PLPSs can also be applied for the problem of function approxi-mation. More information in this direction can be found in [46, 47] for LPSsand in [3, 13] for PLPSs.

We close this paper with an outlook to more general constructions: a moregeneral form of LPSs as given in Definition 2 is the concept of integrationlattices which are presented in [58, Section 5.3] and in [72]. An integrationlattice is a discrete subset of Rs which is closed under addition and sub-


traction, and which contains Zs as a subset. In the same vein Lemieux andL’Ecuyer [52, 53] introduced so-called polynomial integration lattices whichgeneralize the concept of PLPSs from Definition 3. Results on the star dis-crepancy and the t-parameter of such point sets can be found in [29].

A very general construction of point sets in [0, 1)s for which PLPSs serveas special cases is the concept of cyclic nets due to Niederreiter [60] and,even more general, of hyperplane nets due to Pirsic et al. [67]. Cyclic andhyperplane nets are constructions of digital (t,m, s)-nets which are inspiredby a close connection between coding theory and the theory of digital nets. Infact, the cyclic net construction is the analog to the construction of so-calledcyclic codes which are well known in coding theory. For more information werefer to [22, Chapter 11] and the references therein.

Acknowledgements The author is partially supported by the Austrian ScienceFoundation (FWF), Project S9609, that is part of the Austrian National ResearchNetwork “Analytic Combinatorics and Probabilistic Number Theory”. The authoralso thanks Josef Dick and Peter Kritzer for many remarks and suggestions.

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