Stability of lattice rules and polynomial lattice rules constructed by the component-by-component algorithm

We study quasi-Monte Carlo (QMC) methods for numerical integration of multivariate functions defined over the high-dimensional unit cube. Lattice rules and polynomial lattice rules, which are special classes of QMC methods, have been intensively studied and the so-called component-by-component (CBC) algorithm has been well-established to construct rules which achieve the almost optimal rate of convergence with good tractability properties for given smoothness and set of weights. Since the CBC algorithm constructs rules for given smoothness and weights, not much is known when such rules are used for function classes with different smoothness and/or weights. In this paper we prove that a lattice rule constructed by the CBC algorithm for the weighted Korobov space with given smoothness and weights achieves the almost optimal rate of convergence with good tractability properties for general classes of smoothness and weights. Such a stability result also can be shown for polynomial lattice rules in weighted Walsh spaces. We further give bounds on the weighted star discrepancy and discuss the tractability properties for these QMC rules. The results are comparable to those obtained for Halton, Sobol and Niederreiter sequences.

Authors

• 10 publications
• 12 publications
01/09/2020

Digit-by-digit and component-by-component constructions of lattice rules for periodic functions with unknown smoothness

Lattice rules are among the most prominently studied quasi-Monte Carlo m...
09/15/2020

Weighted integration over a cube based on digital nets and sequences

Quasi-Monte Carlo (QMC) methods are equal weight quadrature rules to app...
05/24/2016

Convergence guarantees for kernel-based quadrature rules in misspecified settings

Kernel-based quadrature rules are becoming important in machine learning...
09/23/2021

Component-by-component construction of randomized rank-1 lattice rules achieving almost the optimal randomized error rate

We study a randomized quadrature algorithm to approximate the integral o...
04/17/2018

Numerical Integration in Multiple Dimensions with Designed Quadrature

We present a systematic computational framework for generating positive ...
09/01/2017

Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings

This paper presents convergence analysis of kernel-based quadrature rule...
03/30/2020

Numerical integration without smoothness assumption

We consider numerical integration in classes, for which we do not impose...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.

1 Introduction

We study numerical integration of multivariate functions defined over the -dimensional unit cube . For an integrable function we denote the integral of by

 I(f)=∫[0,1)sf(x)dx.

For an -element set , we consider approximating by

 QP(f)=1N∑x∈Pf(x).

Such an equal-weight quadrature rule is called a quasi-Monte Carlo (QMC) rule. We refer to [1, 2, 3, 4] for comprehensive information on QMC rules.

For a Banach space with norm , the worst-case error by the algorithm is defined by

 ewor(P;W):=supf∈W∥f∥W≤1|I(f)−QP(f)|.

Our interest is then to construct a good point set such that the algorithm makes small for a given . However, it is often difficult to know whether one point set constructed for a certain function space works well for different function spaces as well. In this paper we show some positive results on this question for lattice rules in weighted Korobov spaces and also for polynomial lattice rules in weighted Walsh spaces.

Lattice rules and polynomial lattice rules are defined respectively as follows (here and in what follows, we denote the set of positive integers by ):

Definition 1 (lattice rules).

For , let . An -element lattice point set is given by

 P(z)={({nz1N},…,{nzsN})∣n=0,1,…,N−1},

where denotes the fractional part of a non-negative real numbers . The resulting QMC algorithm

is called a lattice rule with generating vector

.

Definition 2 (polynomial lattice rules).

For a prime and , let be a polynomial of degree over the finite field of order and let where we write . A -element polynomial lattice point set is given by

 P(p,q)={(νm(n(x)q1(x)p(x)),…,νm(n(x)qs(x)p(x)))∣n∈Gm},

where we write

 νm(∞∑i=wtix−i)=m∑i=max(w,1)tib−i∈[0,1).

The resulting QMC algorithm is called a polynomial lattice rule with modulus and generating vector .

For , there is no known explicit construction of generating vectors for lattice rules or for polynomial lattice rules. Instead, the so-called component-by-component (CBC) algorithm, a greedy algorithm which iteratively searches for one component (or ) with earlier ones (or, , respectively) kept unchanged, has been well-established, see for instance [5, 6, 7, 8, 9, 10, 11] among many others.

After the seminal work of Sloan and Woźniakowski [12], it has been standard to consider weighted function spaces when constructing point sets, where a set of weight parameters is introduced in the definition of function spaces to play a role in moderating the relative importance of different variables or groups of variables. It can be shown under some conditions on the weights that the worst-case error bound for good (polynomial) lattice rules depends only polynomially on the dimension , or even, that the error bound is dimension-independent, see for instance [13, 8, 5, 6]. To prove such tractability results for lattice rules or polynomial lattice rules, not only the smoothness parameter of the function space but also a set of weight parameters are required as inputs in the CBC algorithm. In general, it is unknown whether one QMC rule constructed by the CBC algorithm for given smoothness and weights does also work well for different smoothness and weights.

In this paper we prove that a lattice rule constructed by the CBC algorithm for the weighted Korobov space with certain smoothness and weights achieves the almost optimal rate of convergence with good tractability properties for any smoothness and general weights. The result with respect to the smoothness parameter is well understood and can readily be derived from [1, Chapter 5] and Jensen’s inequality, however the stability result with respect to the weights is new. Moreover, we show a similar result for polynomial lattice rules in weighted Walsh spaces. We also give bounds on the weighted star discrepancy and discuss the tractability properties of lattice rules and polynomial lattice rules.

2 Stability of lattice rules with respect to changes in smoothness and weights

In this section, we study stability of lattice rules in weighed Korobov spaces.

2.1 Weighted Korobov space

Let be periodic and given by its Fourier series

 f(x)=∑k∈Zs^f(k)exp(2πik⋅x),

where the dot denotes the usual inner product of two vectors and denotes the -th Fourier coefficient defined by

 ^f(k)=∫[0,1)sf(x)exp(−2πik⋅x)dx.

We measure the smoothness of periodic functions by a parameter . A set of weight parameters with is considered to moderate the relative importance of different variables or groups of variables. For a non-empty subset and , we define

 rα(γ,ku):=γu∏j∈u1|kj|2α.

Then the weighted Korobov space with smoothness , denoted by , is a reproducing kernel Hilbert space with the reproducing kernel [14]

 KKorα,γ(x,y)=1+∑∅≠u⊆{1,…,s}∑ku∈(Z∖{0})|u|rα(γ,ku)exp(2πiku⋅(xu−yu)),

where we write and . The inner product of the space is given by

 ⟨f,g⟩Korα,γ=^f(0)^g(0)+∑∅≠u⊆{1,…,s}∑ku∈(Z∖{0})|u|^f(ku,0)^g(ku,0)rα(γ,ku).

Here, for a non-empty subset such that , we assume that the corresponding inner sum equals 0 and we set . The induced norm is then given by

 ∥f∥Korα,γ=  ⎷|^f(0)|2+∑∅≠u⊆{1,…,s}∑ku∈(Z∖{0})|u||^f(ku,0)|2rα(γ,ku).

2.2 CBC algorithm for lattice rules

In order to construct a good lattice rule which works for the weighted Korobov space with certain and , we consider the worst-case error as a quality criterion. Since the worst-case error depends only on the generating vector for fixed , we simply write . It follows from the reproducing property of that we have

 (ewor(z;HKorα,γ))2 =∫[0,1)s∫[0,1)sKKorα,γ(x,y)dzdy −2N∑x∈P(z)∫[0,1)sKKorα,γ(x,y)dy+1N2∑x,y∈P(z)KKorα,γ(x,y).

Define the dual lattice for by

 P⊥(z)={k∈Zs∣k⋅z≡0(modN)}.

Then the following property is well known.

Lemma 1.

For and , we have

 1N∑x∈P(z)exp(2πik⋅z)={1if k∈P⊥(z),0otherwise.

For a non-empty subset , define

 P⊥u(z):={ku∈(Z∖{0})|u|∣(ku,0)∈P⊥(z)}.

Then the squared worst-case error is given by

 (ewor(z;HKorα,γ))2=∑∅≠u⊆{1,…,s}∑ku∈P⊥u(z)rα(γ,ku)=:Pα,γ(z).

There is a concise computable form of the criterion when is a natural number

 Pα,γ(z)=1N∑x∈P(z)∑∅≠u⊆{1,…,s}γu∏j∈u((2π)2α(−1)α+1(2α)!B2α(xj)),

where is the Bernoulli polynomial of degree .

Now the CBC algorithm for lattice rules proceeds as follows:

Algorithm 1 (CBC for lattice rules).

Let , and be given.

1. Let and .

2. Compute for all and let

 z∗ℓ+1=argminzℓ+1Pα,γ(z∗1,…,z∗ℓ,zℓ+1).
3. If , let and go to Step 2.

Remark 1.

For special types of weights

, the necessary computational cost for the CBC algorithm can be made small by using the fast Fourier transform

[11]. In the case of product weights, i.e.,

 γu=∏j∈uγj

for , the set of non-negative real numbers, we only need arithmetic operations with memory. In the case of POD weights, i.e.,

 γu=Γ|u|∏j∈uγj

for and , we need arithmetic operations with memory.

As shown, for instance, in [4, Theorem 5.12], the worst-case error for lattice rules constructed by the CBC algorithm can be bounded as follows.

Proposition 1.

Let , and be given. The generating vector constructed by Algorithm 1 satisfies

 Pα,γ(z)≤⎛⎝1φ(N)∑∅≠u⊆{1,…,s}γλu(2ζ(2αλ))|u|⎞⎠1/λ,

for any , where denotes the Riemann zeta function and denotes the Euler totient function defined by

 φ(N)=∑1≤n≤Ngcd(n,N)=11.
Remark 2.

For any , let us write . Then it follows from Jensen’s inequality that

 (Pα/δ,γ1/δ(z))δ =⎛⎜⎝∑∅≠u⊆{1,…,s}∑ku∈P⊥u(z)rα/δ(γ1/δ,ku)⎞⎟⎠δ ≤∑∅≠u⊆{1,…,s}∑ku∈P⊥u(z)(rα/δ(γ1/δ,ku))δ =∑∅≠u⊆{1,…,s}∑ku∈P⊥u(z)rα(γ,ku)=Pα,γ(z).

Thus, the generating vector constructed by Algorithm 1 based on the criterion also works for weighted Korobov spaces with special types of smoothness and weights, i.e.,

 α′=αδ,γ′=γ1/δ,

for any . In the next subsection, we prove a more general stability result than what we obtain simply from Jensen’s inequality.

2.3 Stability result

First we note that is bounded below by another quality criterion, the Zaremba index or also called figure of merit ([1, Chapter 5], [2, Chapter 4])

 ρα,γ(z):=max∅≠u⊆{1,…,s}maxku∈P⊥u(z)rα(γ,ku).

This criterion turns out to be very useful in our context (it has for instance recently also been used in [15]).

As one of the main results of this paper, we prove the following upper bound on the squared worst-case error (cf. [1, Theorem 5.34], [17, 16]).

Theorem 1.

Let and . For any and such that whenever , we have

 Pα′,γ′(z)≤cα′(ρα,γ(z))α′/α∑∅≠u⊆{1,…,s}γ′uγα′/αu(22α′+122α′−1−1)|u|(log2N)|u|−1,

with

 cα′=(1+ζ(2α′))+(22α′+ζ(2α′))22α′−1−124α′.

We defer the proof of this theorem to the end of this section.

Theorem 1 implies that, if we can construct a lattice rule with small value for given and , the same lattice rule also does work for weighted Korobov spaces with different smoothness and weights. As mentioned in Remark 2, applying Jensen’s inequality leads to a kind of stability result, but it works only for higher smoothness and restrictive form of the weights.

Let be the generating vector constructed by the CBC algorithm based on the criterion for given and . Applying the result from Proposition 1, we have

 Pα′,γ′(z) ≤cα′⎛⎝1φ(N)∑∅≠u⊆{1,…,s}γλu(2ζ(2αλ))|u|⎞⎠α′/(αλ) ×∑∅≠u⊆{1,…,s}γ′uγα′/αu(22α′+122α′−1−1)|u|(log2N)|u|−1, (1)

for any . We can show under some conditions on the weights and that the worst-case error depends only polynomially on the dimension , or even, that the bound is independent of the dimension.

Corollary 1.

Let , and such that whenever . Let be constructed by Algorithm 1 based on the criterion . Then the following holds true:

1. For general weights and , assume that there exist such that , ,

 sups∈N1sq∑∅≠u⊆{1,…,s}γλu(2ζ(2αλ))|u|<∞,

and

 sups,N∈N1sq′Nδ∑∅≠u⊆{1,…,s}γ′uγα′/αu(22α′+122α′−1−1)|u|(log2N)|u|−1<∞.

Then the worst-case error depends only polynomially on and is bounded by

 Pα′,γ′(z)≤Cδsqα′/(αλ)+q′N−α′/(αλ)+δ,

for some constant which is independent of and . If the above conditions hold for , the worst-case error is bounded independently of .

2. In particular, in the case of product weights and , assume that there exists such that

 ∞∑j=1γλj<∞and∞∑j=1γ′jγα′/αj<∞.

Then the worst-case error is independent of and bounded by

 Pα′,γ′(z)≤CδN−α′/(αλ)+δ,

for arbitrarily small .

Proof.

The result for the first item immediately follows from the bound (2.3). The second item can be proven by combining arguments used in [8, Theorem 4] and [18, Lemma 3]. ∎

One of the most important indications from the first item of Corollary 1 is that by choosing of product form such that the conditions given in item of Corollary 1 are satisfied, then the CBC algorithm can construct a lattice rule in arithmetic operations with memory, which achieves the almost optimal rate of convergence in with good tractability properties even for general weights. As far as the authors know, such a constructive result for general weights has been not known in the literature.

In the proof of Theorem 1, we shall use the following elementary inequality. We refer to [3, Lemma 13.24] for the proof.

Lemma 2.

For any real and any , we have

 ∞∑t=t0b−t(t+k−1k−1)≤b−t0(t0+k−1k−1)(1−1b)−k.
Proof of Theorem 1.

Recalling the definition of , we have

 Pα′,γ′(z)=∑∅≠u⊆{1,…,s}γ′u∑ku∈P⊥u(z)∏j∈u1|kj|2α′.

Let us define , for which we have . Moreover, let

 ϕu,0(z):=minku∈P⊥u,0(z)∏j∈umax(1,|kj|)

and

 ϕu(z):=minku∈P⊥u(z)∏j∈u|kj|.

Then it is straightforward to see that

 ρα,γ(z)=max∅≠u⊆{1,…,s}γumaxku∈P⊥u(z)∏j∈u1|kj|2α=max∅≠u⊆{1,…,s}γu(ϕu(z))2α.

Hence it holds for any non-empty subset that

 ϕu(z)≥(γuρα,γ(z))1/(2α).

Moreover, by assuming whenever , we obtain a lower bound

 ϕu,0(z)=min∅≠v⊆uϕv(z)≥min∅≠v⊆u(γvρα,γ(z))1/(2α)=(γuρα,γ(z))1/(2α). (2)

For a non-empty subset , we denote by the largest integer such that holds. It follows from the proof of [1, Theorem 5.34] that the inner sum on the expression of for a given with is bounded above by

 ∑ku∈P⊥u(z)∏j∈u1|kj|2α′ ≤∑ku∈P⊥u,0(z)∏j∈u1max(1,|kj|)2α′ ≤2|u|(ϕu,0(z))2α′[(1+ζ(2α′))(μu+|u|−1|u|−1) +(22α′+ζ(2α′))∞∑k=12(1−2α′)k(k+μu+|u|−2|u|−2)].

For the first term in the parenthesis, we have

 (μu+|u|−1|u|−1)=|u|−1∏i=1μu+ii≤(μu+1)|u|−1.

For the second term in the parenthesis, applying Lemma 2 with and gives

 ∞∑k=12(1−2α′)k(k+μu+|u|−2|u|−2) =2(2α′−1)μu∞∑k=μu+12−(2α′−1)k(k+|u|−2|u|−2) ≤2−(2α′−1)(μu+|u|−1|u|−2)(22α′−122α′−1−1)|u|−1 ≤22α′−1−124α′−2(22α′−122α′−1−1)|u||u|−2∏i=1μu+i+1i ≤22α′−1−124α′−2(22α′−122α′−1−1)|u|(μu+2)|u|−2 ≤22α′−1−124α′(22α′22α′−1−1)|u|(μu+1)|u|−2.

Using these bounds, we obtain

 ∑ku∈P⊥u(z)∏j∈u1|kj|2α′ ≤2|u|(ϕu,0(z))2α′[(1+ζ(2α′))(μu+1)|u|−1 +(22α′+ζ(2α′))22α′−1−124α′(22α′22α′−1−1)|u|(μu+1)|u|−2] ≤cα′(22α′+122α′−1−1)|u|(μu+1)|u|−1(ϕu,0(z))2α′.

Note that this bound on the inner sum on the expression of also applies to the case .

As shown in (2), has a lower bound. On the other hand, as proven in [1, Lemma 5.8], also has a trivial upper bound, which is . This bound directly means that . Therefore we can bound as

 Pα′,γ′(z) ≤cα′∑∅≠u⊆{1,…,s}γ′u(22α′+122α′−1−1)|u|(μu+1)|u|−1(ϕu,0(z))2α′ ≤cα′(ρα,γ(z))α′/α∑∅≠u⊆{1,…,s}γ′uγα′/αu(22α′+122α′−1−1)|u|(log2N)|u|−1.

This completes the proof. ∎

2.4 Bound on the weighted star discrepancy

Here we study tractability properties of the weighted star discrepancy for lattice rules constructed by the CBC algorithm based on the criterion .

For an -element point set , the local discrepancy function is defined by

 ΔP(y):=1N∑x∈P1[0,y)(x)−λ([0,y)),

for , where and

denotes the characteristic function of the interval

. For a non-empty subset , let us write . Then the weighted star discrepancy is defined by

 D∗γ(P)=max∅≠u⊆{1,…,s}γusupyu∈[0,1)|u||ΔPu(yu)|.

In what follows, we focus on lattice point sets and simply write instead of .

As shown in [19, Section 2], the weighted star discrepancy for a lattice point set with generating vector is bounded above by

 D∗γ(z)≤∑∅≠u⊆{1,…,s}γu[1−(1−1N)|u|+Ru(z)2],

where

 Ru(z)=∑k∈P⊥u,0(z)∩C∗N,|u|∏j∈u1max(1,|kj|),

with

 C∗N,|u|={ku∈Z|u|∖{0}∣−N2

Moreover, as proven in [1, Theorem 5.35], we have

 Ru(z)≤1ϕu,0(z)[log2(log2N)|u|+3(2log2N)|u|−1],

for any non-empty . By using the lower bound (2) on , the following result holds true.

Theorem 2.

Let and . For any and such that whenever , we have

 D∗γ′(z)≤ ∑∅≠u⊆{1,…,s}γ′u[1−(1−1N)|u| +(ρα,γ(z))1/(2α)2γ1/(2α)u[log2(log2N)|u|+3(2log2N)|u|−1]].

Applying the result from Proposition 1, we can prove the following tractability properties. For general weights we use

 1−(1−1N)|u|≤|u|N,

for any non-empty . We assume that the sum in the case of product weights to ensure a dimension independent bound on the sum

 ∑∅≠u⊆{1,…,s}γ