Lattice rules with random $n$ achieve nearly the optimal $\mathcal{O}(n^{-\alpha-1/2})$ error independently of the dimension

TL;DR

This paper introduces a random lattice rule algorithm for high-dimensional integration that achieves near-optimal error rates independent of dimension, using a simple and implementable approach based on Bakhvalov's randomization technique.

Contribution

It demonstrates that random rank-1 lattice rules can attain nearly optimal convergence rates for weighted Sobolev spaces, with error bounds independent of dimension, and simplifies implementation compared to existing methods.

Findings

Achieves error rate of (n^{-\u03b1-1/2}) for <

Error bounds are independent of the problem dimension under certain conditions

Method is easier to implement than Frolov's method in high dimensions

Abstract

We analyze a new random algorithm for numerical integration of $d$-variate functions over $[0,1]^d$ from a weighted Sobolev space with dominating mixed smoothness $\alpha\ge 0$ and product weights $1\ge\gamma_1\ge\gamma_2\ge\cdots>0$, where the functions are continuous and periodic when $\alpha>1/2$. The algorithm is based on rank-$1$ lattice rules with a random number of points~$n$. For the case $\alpha>1/2$, we prove that the algorithm achieves almost the optimal order of convergence of $\mathcal{O}(n^{-\alpha-1/2})$, where the implied constant is independent of the dimension~$d$ if the weights satisfy $\sum_{j=1}^\infty \gamma_j^{1/\alpha}<\infty$. The same rate of convergence holds for the more general case $\alpha>0$ by adding a random shift to the lattice rule with random $n$. This shows, in particular, that the exponent of strong tractability in the randomized setting equals $1/(\alpha+1/2)$, if the weights decay fast enough. We obtain a lower bound to indicate that our results are essentially optimal. This paper is a significant advancement over previous related works with respect to the potential for implementation and the independence of error bounds on the problem dimension. Other known algorithms which achieve the optimal error bounds, such as those based on Frolov's method, are very difficult to implement especially in high dimensions. Here we adapt a lesser-known randomization technique introduced by Bakhvalov in 1961. This algorithm is based on rank-$1$ lattice rules which are very easy to implement given the integer generating vectors. A simple probabilistic approach can be used to obtain suitable generating vectors.

Equations

$$I_d(f):={\int }_{[0,1{]}^d}f(\mathbf{x})\mathrm{d}\mathbf{x}$$

$$Q_{d,p,\mathbf{z}}(f):=\frac{1}{p}\sum _{k=0}^{p-1}f\left(\left\{\frac{k\mathbf{z}}{p}\right\}\right),$$

$$e^{\rm ran}_{d,\alpha,{\boldsymbol{\gamma}}}(M_{n})\,:=\,\sup_{\begin{subarray}{c}f\in\mathcal{H}_{d,\alpha,{\boldsymbol{\gamma}}}\\ \|f\|_{d,\alpha,{\boldsymbol{\gamma}}}\leq 1\end{subarray}}\mathbb{E}\Big{[}\big{|}M_{n}(f)-I_{d}(f)\big{|}\Big{]}.$$

$$e^{\rm ran}_{d,\alpha,{\boldsymbol{\gamma}}}(M_{n})\,=\,{\mathcal{O}}(n^{-a-1/2})\qquad\mbox{for $a<\alpha$ arbitrarily close to $\alpha$},$$

$$\sum _{j=1}^{\infty }{\gamma }_j^{1/\alpha }<\infty .$$

$$\widetilde{M}_{n}(f)\,:=\,M_{n}\bigl{(}f(\{\cdot+\boldsymbol{U}\})\bigr{)},$$

$$e^{\rm ran}_{d,\alpha,{\boldsymbol{\gamma}}}(\widetilde{M}_{n})\,=\,{\mathcal{O}}(n^{-a-1/2})\qquad\mbox{for $a<\alpha$ arbitrarily close to $\alpha$},$$

$$\frac{{\gamma }_1}{2}\frac{\sqrt{\log n}}{n^{\alpha +1/2}}$$

$$\|f\|_{d,\alpha,{\boldsymbol{\gamma}}}\,:=\,\left(\sum_{\boldsymbol{h}\in{\mathbb{Z}}^{d}}\big{|}r_{\alpha,{\boldsymbol{\gamma}}}(\boldsymbol{h})\,\hat{f}(\boldsymbol{h})\big{|}^{2}\right)^{1/2}\,,$$

$$r_{\alpha ,\mathbf{\gamma }}(\mathbf{h})=\prod _{j=1}^d{r}_{\alpha ,{\gamma }_j}(h_j),$$

$$r_{\alpha ,{\gamma }_j}(h_j):=\max \left\{1,\frac{{\left|h_j\right|}^{\alpha }}{{\gamma }_j}\right\}.$$

$$f(\mathbf{x})=\sum _{\mathbf{h}\in {\mathbb{Z}}^d}\hat{f}(\mathbf{h})e^{2\pi \mathtt{i}\mathbf{h}\cdot \mathbf{x}}\text{mbox}forf\in {\mathcal{H}}_{d,\alpha ,\mathbf{\gamma }}.$$

$$\|f\|_{d,\alpha,{\boldsymbol{\gamma}}}^{2}\,=\,\sum_{{\mathfrak{u}}\subseteq\{1:d\}}(2\pi)^{-2\alpha|{\mathfrak{u}}|}\Bigg{(}\prod_{j\in{\mathfrak{u}}}\,\gamma_{j}^{-2}\Bigg{)}\Bigg{(}\int_{[0,1]^{|{\mathfrak{u}}|}}\Bigg{|}\int_{[0,1]^{d-|{\mathfrak{u}}|}}\Bigg{(}\prod_{j\in{\mathfrak{u}}}\frac{\partial}{\partial x_{j}}\Bigg{)}^{\alpha}f(\boldsymbol{x})\,\,\mathrm{d}\boldsymbol{x}_{\{1:d\}\setminus{\mathfrak{u}}}\Bigg{|}^{2}\,\mathrm{d}\boldsymbol{x}_{{\mathfrak{u}}}\Bigg{)},$$

$$r_{\alpha ,{\gamma }_j}^{\ast }(h_j):=\sqrt{1+\frac{|2\pi h_j{|}^{2\alpha }}{{\gamma }_j^2}},$$

$$\|f\|_{*}^{2}\,=\,\sum_{{\mathfrak{u}}\subseteq\{1:d\}}\Bigg{(}\prod_{j\in{\mathfrak{u}}}\gamma_{j}^{-2}\Bigg{)}\;\Bigg{\|}\Bigg{(}\prod_{j\in{\mathfrak{u}}}\frac{\partial}{\partial x_{j}}\Bigg{)}^{\alpha}f\,\Bigg{\|}_{L_{2}([0,1]^{d})}^{2}\,=\,\sum_{\boldsymbol{b}\in\{0,1\}^{d}}{\boldsymbol{\gamma}}^{-2\boldsymbol{b}}\,\|D^{\alpha\boldsymbol{b}}f\|_{L_{2}([0,1]^{d})}^{2},$$

$$Q_{d,p,\mathbf{z}}(f)-I_d(f)=\sum _{\stackrel{\mathbf{h}\in {\mathbb{Z}}^d\setminus \{0\}}{\mathbf{h}\cdot \mathbf{z}{\equiv }_{p}0}}\hat{f}(\mathbf{h}),$$

$$P_{\alpha ,\mathbf{\gamma }}(p,\mathbf{z}):=\sum _{\stackrel{\mathbf{h}\in {\mathbb{Z}}^d\setminus \{0\}}{\mathbf{h}\cdot \mathbf{z}{\equiv }_{p}0}}\frac{1}{r_{\alpha ,\mathbf{\gamma }}(\mathbf{h})},$$

$$\begin{split}\left|Q_{d,p,\boldsymbol{z}}(f)-I_{d}(f)\right|\,&\leq\,\|f\|_{d,\alpha,{\boldsymbol{\gamma}}}\,\Bigg{(}\sum_{\stackrel{{\scriptstyle\scriptstyle{\boldsymbol{h}\in{\mathbb{Z}}^{d}\setminus\{\boldsymbol{0}\}}}}{{\scriptstyle{\boldsymbol{h}\cdot\boldsymbol{z}\equiv_{p}0}}}}\frac{1}{[r_{\alpha,{\boldsymbol{\gamma}}}(\boldsymbol{h})]^{2}}\Bigg{)}^{1/2}\\ \,&=\,\|f\|_{d,\alpha,{\boldsymbol{\gamma}}}\,\sqrt{P_{2\alpha,{\boldsymbol{\gamma}}^{2}}(p,\boldsymbol{z})},\end{split}$$

$$e_{d,\alpha ,\mathbf{\gamma }}^{\mathrm{wor}}(Q_{d,p,\mathbf{z}}):=\underset{\begin{array}{c}f\in {\mathcal{H}}_{d,\alpha ,\mathbf{\gamma }}\\ \Vert f{\Vert }_{d,\alpha ,\mathbf{\gamma }}\le 1\end{array}}{\sup }\left|Q_{d,p,\mathbf{z}}(f)-I_d(f)\right|=\sqrt{P_{2\alpha ,{\mathbf{\gamma }}^2}(p,\mathbf{z})}.$$

$${\rho }_{\alpha ,\mathbf{\gamma }}(p,\mathbf{z}):=\underset{\stackrel{\mathbf{h}\in {\mathbb{Z}}^d\setminus \{0\}}{\mathbf{h}\cdot \mathbf{z}{\equiv }_{p}0}}{\min }{r}_{\alpha ,\mathbf{\gamma }}(\mathbf{h}),$$

$$\frac{1}{{\rho }_{\alpha ,\mathbf{\gamma }}(p,\mathbf{z})}<P_{\alpha ,\mathbf{\gamma }}(p,\mathbf{z}).$$

$$r_{\alpha,{\boldsymbol{\gamma}}}(\boldsymbol{h})\,=\,\prod_{j=1}^{d}\max\Bigl{\{}1,\,|h_{j}|^{\alpha}/\gamma_{j}\Bigr{\}}.$$

$$\sum _{\mathbf{h}\in {\mathbb{Z}}^d}\frac{1}{r_{\beta ,\mathbf{\gamma }}(\mathbf{h})}=\prod _{j=1}^d\left(1+2{\gamma }_j\zeta (\beta )\right),$$

$$V_d(\beta ,\mathbf{\gamma }):=3\sum _{\mathbf{h}\in {\mathbb{Z}}^d}\frac{1}{r_{\beta ,\mathbf{\gamma }}(\mathbf{h})}=3\prod _{j=1}^d\left(1+2{\gamma }_j\zeta (\beta )\right)\le 3\exp \left(2\zeta (\beta )\sum _{j=1}^d{\gamma }_j\right),$$

$${\mathcal{A}}_{\beta ,\mathbf{\gamma }}(T):=\{\mathbf{h}\in {\mathbb{Z}}^d:r_{\beta ,\mathbf{\gamma }}(\mathbf{h})\le T\}.$$

$$\left|{\mathcal{A}}_{\beta ,\mathbf{\gamma }}(T)\right|\le T{V}_d(\beta ,\mathbf{\gamma }).$$

$$V_d(\beta ,\mathbf{\gamma })\ge \sum _{\mathbf{h}\in {\mathcal{A}}_{\beta ,\mathbf{\gamma }}(T)}\frac{1}{r_{\beta ,\mathbf{\gamma }}(\mathbf{h})}\ge \frac{|{\mathcal{A}}_{\beta ,\mathbf{\gamma }}(T)|}{T}.$$

$$\#\bigl{\{}\boldsymbol{z}\in\{1:p-1\}^{d}\colon\,\boldsymbol{h}\cdot\boldsymbol{z}\equiv_{p}0\bigl{\}}\;\leq\begin{cases}(p-1)^{d}&\text{ if }\,\boldsymbol{h}\equiv_{p}\boldsymbol{0},\\ (p-1)^{d-1}&\text{ otherwise,}\end{cases}$$

$$\frac{1}{(p-1{)}^d}\sum _{\mathbf{z}\in \{1:p-1{\}}^d}{P}_{\beta ,\mathbf{\gamma }}(p,\mathbf{z})\le \frac{V_d(\beta ,\mathbf{\gamma })}{p}.$$

$$\frac{1}{(p-1{)}^d}\sum _{\mathbf{z}\in \{1:p-1{\}}^d}{P}_{\beta ,\mathbf{\gamma }}(p,\mathbf{z})$$

Full text

This paper is a significant advancement over previous related works with respect to the potential for implementation and the independence of error bounds on the problem dimension. Other known algorithms which achieve the optimal error bounds, such as those based on Frolov’s method, are very difficult to implement especially in high dimensions. Here we adapt a lesser-known randomization technique introduced by Bakhvalov in 1961. This algorithm is based on rank-$1$ lattice rules which are very easy to implement given the integer generating vectors. A simple probabilistic approach can be used to obtain suitable generating vectors.

1 Introduction

We study the problem of numerical integration of $d$-variate functions, i.e., the approximation of

[TABLE]

for $f$ in a weighted Sobolev space $\mathcal{H}_{d,\alpha,{\boldsymbol{\gamma}}}$ with smoothness parameter $\alpha\geq 0$ and product weights ${\boldsymbol{\gamma}}=(\gamma_{j})_{j\geq 1}$ (details are provided below), where the functions are continuous and periodic when $\alpha>1/2$ (also known in this case as a weighted “Korobov space” in recent literature). To this end we use a randomized algorithm $M_{n}$ that employs at most $n\in\mathbb{N}$ function evaluations. The basic building blocks of the algorithm $M_{n}$ are so-called rank-$1$ lattice rules of the form

[TABLE]

where $p$ is a prime number, $\boldsymbol{z}\in{\mathbb{Z}}^{d}$ is known as the generating vector, and $\{x\}$ denotes the fractional part of a real number $x$ and is applied componentwise to a vector. Lattice rules as in (1) are very well studied in the field of quasi-Monte Carlo methods (see [4] and [23] for overviews on lattice rules, and [5, 15] for further introductions to the field of quasi-Monte Carlo methods). Here, however, we do not study one fixed quasi-Monte Carlo rule, but an algorithm based on randomly choosing one of a certain set of lattice rules. Indeed, the random algorithm $M_{n}$, for given $n\in\mathbb{N}$, is defined by $Q_{d,p,\boldsymbol{z}}$ with randomly chosen prime $p\in\{\lceil n/2\rceil+1,\dots,n\}$ and $\boldsymbol{z}\in\{1,\dots,p-1\}^{d}$. To be precise, we choose a random prime number $p$ in the given range and then a random generating vector $\boldsymbol{z}$ from a certain set ${\mathcal{Z}}_{p}$ of “good” generating vectors (see (12) below). We call such an algorithm $M_{n}$ a randomized lattice algorithm.

An algorithm of such a form was first analyzed by Bakhvalov in [1] (see also [2] for general results on randomized algorithms) for generating vectors of a special form commonly known as the “Korobov type”, i.e., $\boldsymbol{z}=(1,z,\dots,z^{d-1})$ for some $z\in\mathbb{N}$. Bakhvalov proved that this algorithm has almost the optimal order of convergence in a Sobolev space with dominating mixed smoothness, but the error bound depends on the dimension $d$.

A significant advancement of this paper is that we prove the existence of good generating vectors, not restricted to the Korobov type, such that our randomized lattice algorithm $M_{n}$ achieves almost the optimal order of convergence in weighted Sobolev spaces, with the error bound independent of $d$ under a summability condition on the weights ${\boldsymbol{\gamma}}$ that is common in this field of research.

The advantage over previous related works lies in the potential for implementation. Other known algorithms which achieve the optimal error bounds, such as those based on Frolov’s method (see, e.g., [6, 9, 11, 28, 29, 30]), are very difficult to implement especially in high dimensions. For our algorithm, a simple probabilistic approach can be used to obtain suitable generating vectors, see Remark 10 below.

More precisely, we analyze the randomized (worst case) error for our randomized lattice algorithm $M_{n}$ in the unit ball of $\mathcal{H}_{d,\alpha,{\boldsymbol{\gamma}}}$, defined by

[TABLE]

The norm $\|\cdot\|_{d,\alpha,{\boldsymbol{\gamma}}}$ will be defined in Section 2.1. The details of the expectation will be made clear when we formally specify our randomized lattice algorithm in Section 3. We have three main theorems in this paper. In Theorem 9 below, we prove for $\alpha>1/2$ (thus all functions are continuous and periodic) and $n$ sufficiently large that

[TABLE]

where the implied constant is independent of $d$ if

[TABLE]

(In the literature, the exponent in (3) sometimes differs by a factor of two, depending on how $\alpha$ and ${\boldsymbol{\gamma}}$ enter the definition of the norm, see Section 2.1.)

Additionally we will analyze the randomized lattice algorithm with shift, which is defined by

[TABLE]

where $\boldsymbol{U}$ is a random variable that is uniformly distributed on $[0,1]^{d}$, and the braces mean to take the fractional part as in (1). Again, the algorithm uses at most $n$ function evaluations. The advantage of the algorithm $\widetilde{M}_{n}$ is that it can treat the case $\alpha\in(0,1/2]$, i.e., when the integrands are not necessarily continuous. In Theorem 11 below, we prove for $\alpha>0$ that

[TABLE]

where again the implied constant is independent of $d$ if (3) holds. Moreover, the result also holds with the randomized error replaced by the root-mean-squared error of $\widetilde{M}_{n}$, which is larger.

Then, we show that the upper bounds for the algorithms $M_{n}$ and $\widetilde{M}_{n}$ are essentially best possible, by proving in Theorem 12 the lower bound

[TABLE]

for the randomized errors of the above algorithms.

The presented upper bounds are almost optimal because the optimal order of convergence that can be achieved by any randomized algorithm which uses only function values is $\Theta(n^{-\alpha-1/2})$, even in the unweighted situation. This has been proven only recently by one of the authors [29] using the algorithm of [11]. However, the algorithm in [29] is quite impractical in high dimensions, in contrast to the algorithm that will be analyzed in the following. This mainly comes from the fact that the involved point sets, i.e., certain subsets of irrational lattices in $\mathbb{R}^{d}$, seem to be very hard to implement, see, e.g., [9, 30]. Moreover, the presented upper bound in [29] is at least exponential in $d$ and probably not improvable, see [29, Remark 3.2].

The optimal order $\Theta(n^{-\alpha-1/2})$ in the randomized setting, which holds for arbitrary $\alpha\geq 0$, should be compared to the optimal order $\Theta(n^{-\alpha}(\log n)^{(d-1)/2})$ in the deterministic setting, which can only hold if $\alpha>1/2$, see [3, 6] or [27, 30] and the references therein. For $\alpha\leq 1/2$, deterministic methods do not converge at all in the worst case setting, as the functions are possibly not continuous. A tutorial on the proof of the upper bound can be found in [28]. By now there are many known constructions for optimal algorithms if $\alpha\leq 1$, see, e.g., [5, 15, 18], while for $\alpha>1$ they are still rare, see [6, 7]. By introducing weights in these function spaces, one can achieve the rate ${\mathcal{O}}(n^{-a})$ for $a<\alpha$, with the implied constant independent of $d$, see [26, Theorem 3]. For a component-by-component construction that achieves this bound, see, e.g., [4, 12].

The rest of the paper is structured as follows. In Section 2, we define the function space, summarize some known results from the deterministic error setting and show a few auxiliary results that will be needed in this paper. In Section 3 we state and prove the main results, Theorems 9 and 11, while in Section 4 we show a lower bound for the specific algorithms considered in this paper, see Theorem 12. The independence of the error bounds on the dimension is the essence of strong tractability, and we will provide a brief discussion about this in Section 5. In particular, again under a suitable summability condition on the weights, the exponent of strong tractability in the randomized setting equals the optimal value of $1/(\alpha+1/2)$. We end the paper with a short conclusion in Section 6.

2 Previous and auxiliary results

In this section we define the function space and review some known results for the error of lattice rules in the deterministic setting (this includes the definitions necessary for the analysis), and we also prove a few helpful results.

2.1 Function spaces

Let us now define the function spaces under consideration in this paper. These spaces are so-called Sobolev spaces with dominating mixed smoothness, where we assume (product) weights as introduced by Sloan and Woźniakowski in [25]. The basic idea of weighted spaces is that the weights, occurring in the norm of the space, allow us to model different influence of different coordinates on the integration problem, where larger weights mean more influence and smaller weights mean less influence.

Let $\alpha\geq 0$, let ${\boldsymbol{\gamma}}=(\gamma_{j})_{j\geq 1}$ be a non-increasing sequence of positive weights bounded by 1, and let $d\in\mathbb{N}$. We denote by $\mathcal{H}_{d,\alpha,{\boldsymbol{\gamma}}}$ the Sobolev space of functions defined on $[0,1]^{d}$, with finite norm

[TABLE]

where $\hat{f}(\boldsymbol{h}):=\int_{[0,1]^{d}}f(\boldsymbol{x})\,e^{-2\pi i\boldsymbol{h}\cdot\boldsymbol{x}}\,\mathrm{d}\boldsymbol{x}$, with “$\cdot$” denoting the usual Euclidean inner product, and

[TABLE]

with

[TABLE]

It is known that $\mathcal{H}_{d,\alpha,{\boldsymbol{\gamma}}}\subseteq L_{2}([0,1]^{d})$ (with equality for $\alpha=0$), and that its elements can be expressed in terms of their Fourier series, i.e.,

[TABLE]

Note that the convergence of the Fourier series holds pointwise if $\alpha>1/2$, and in this case $\mathcal{H}_{d,\alpha,{\boldsymbol{\gamma}}}$ consists of continuous and periodic functions. In the recent literature on lattice rules, $\mathcal{H}_{d,\alpha,{\boldsymbol{\gamma}}}$ with $\alpha>1/2$ is often called weighted Korobov space. If $\alpha\leq 1/2$, then the Fourier series of a function in $\mathcal{H}_{d,\alpha,{\boldsymbol{\gamma}}}$ does not necessarily converge pointwise, and the above equation is to be understood almost everywhere, which is enough for our purposes.

It is straightforward to check that, for $\alpha\in\mathbb{N}$,

[TABLE]

where $\{1:d\}:=\{1,\dots,d\}$ and $\boldsymbol{x}=(x_{1},\dots,x_{d})$ with $\boldsymbol{x}_{{\mathfrak{u}}}$ and $\boldsymbol{x}_{\{1:d\}\setminus{\mathfrak{u}}}$ denoting the projection of $\boldsymbol{x}$ onto the coordinates in ${\mathfrak{u}}$ and $\{1:d\}\setminus{\mathfrak{u}}$, respectively.

Remark 1**.**

If we replace the definition of $r_{\alpha,\gamma_{j}}(h_{j})$ by

[TABLE]

then, for $\alpha\in\mathbb{N}$, we obtain the norm

[TABLE]

with ${\boldsymbol{\gamma}}^{-2\boldsymbol{b}}=\prod_{j=1}^{d}\gamma_{j}^{-2b_{j}}$, which recovers the standard norm for the Sobolev space when ${\boldsymbol{\gamma}}\equiv\boldsymbol{1}$. We could obtain the latter upper error bounds for this norm in the same way, since $r_{\alpha,\gamma_{j}}(h_{j})\leq r^{*}_{\alpha,\gamma_{j}}(h_{j})$, which implies $\|f\|_{d,\alpha,{\boldsymbol{\gamma}}}\leq\|f\|_{*}$.

2.2 Lattice rule error

For a prime $p$ and $\boldsymbol{z}=(z_{1},\ldots,z_{d})\in\{1:p-1\}^{d}$, where $\{1:p-1\}:=\{1,2,\ldots,p-1\}$, it easily follows that the error of a (rank-1) lattice rule $Q_{d,p,\boldsymbol{z}}$ is given by

[TABLE]

provided that $\sum_{\boldsymbol{h}\in\mathbb{Z}^{d}}|\hat{f}(\boldsymbol{h})|<\infty$, where $\boldsymbol{h}\cdot\boldsymbol{z}\equiv_{p}0$ means that $\boldsymbol{h}\cdot\boldsymbol{z}$ is congruent to zero modulo $p$. Hence, the performance of a lattice rule $Q_{d,p,\boldsymbol{z}}$ depends solely on the structure of the dual lattice, i.e., the set of all $\boldsymbol{h}\in\mathbb{Z}^{d}$ such that $\boldsymbol{h}\cdot\boldsymbol{z}\equiv_{p}0$. There are several measures, or figures of merit, for the quality of a lattice rule, see, e.g. [14, 15].

One figure of merit is

[TABLE]

for which the unweighted version was defined independently by Hlawka in [8] and Korobov in [10]. This quantity is the worst case error in the class of functions $E_{\alpha,{\boldsymbol{\gamma}}}$, which consists of all functions $f$ with $|\hat{f}(\boldsymbol{h})|\leq r_{\alpha,{\boldsymbol{\gamma}}}(\boldsymbol{h})^{-1}$ for all $\boldsymbol{h}\in\mathbb{Z}^{d}$. For the weighted Sobolev space that is considered in this paper, we clearly obtain by the Cauchy–Schwarz inequality that

[TABLE]

where we used that $[r_{\alpha,{\boldsymbol{\gamma}}}(\boldsymbol{h})]^{2}=r_{2\alpha,{\boldsymbol{\gamma}}^{2}}(\boldsymbol{h})$. It is easily seen that equality holds in (5) for some (worst case) function $f$, and hence we conclude that the (deterministic) worst case error of a single lattice rule $Q_{d,p,\boldsymbol{z}}$ is precisely

[TABLE]

Another relevant figure of merit is

[TABLE]

which is a weighted version of the Zaremba index for higher smoothness. Clearly,

[TABLE]

2.3 The existence of good generating vectors

Here, we show by a standard averaging argument that there exist generating vectors which make the worst case error of the corresponding lattice rule small. In addition, we show that many such vectors exist, which will be essential for the proof of our main result. Recall that, for $\boldsymbol{h}\in\mathbb{Z}^{d}$, we have

[TABLE]

We start with some auxiliary results.

Lemma 2**.**

Let $d\in\mathbb{N}$, $\beta>1$, and ${\boldsymbol{\gamma}}\in(0,1]^{\mathbb{N}}$. Then we have

[TABLE]

where $\zeta$ denotes the Riemann zeta function.

Proof.

The result follows easily by the definition of the function $r_{\beta,{\boldsymbol{\gamma}}}$. $\Box$

The quantity on the right-hand side of (8) will be crucial in the following computations. To simplify notation, we define, for $d\in\mathbb{N}$, $\beta>1$, and ${\boldsymbol{\gamma}}\in(0,1]^{\mathbb{N}}$,

[TABLE]

where we used $1+x\leq e^{x}$ for $x\in\mathbb{R}$. Note that $V_{d}(\beta,{\boldsymbol{\gamma}})$ is bounded independently of $d$ for all $\beta>1$ if $\sum_{j=1}^{\infty}\gamma_{j}<\infty$.

As a direct consequence of Lemma 2 we obtain a bound on the number of $\boldsymbol{h}\in\mathbb{Z}^{d}$ such that $r_{\beta,{\boldsymbol{\gamma}}}(\boldsymbol{h})$ is small.

Corollary 3**.**

Let $d\in\mathbb{N}$, $\beta>1$, $T>0$, and ${\boldsymbol{\gamma}}\in(0,1]^{\mathbb{N}}$, and define

[TABLE]

Then we have

[TABLE]

Proof.

From Lemma 2 we obtain

[TABLE]

This proves the result. $\Box$

The next lemma is useful in bounding the number of points in the dual lattice for a given generating vector.

Lemma 4**.**

For every prime number $p$, every $d\in\mathbb{N}$, and $\boldsymbol{h}\in\mathbb{Z}^{d}$ we have

[TABLE]

with equality in the first case, where by $\boldsymbol{h}\equiv_{p}\boldsymbol{0}$ we mean that each component of $\boldsymbol{h}$ is congruent to zero modulo $p$.

Proof.

Recall that $\boldsymbol{h}\cdot\boldsymbol{z}=\sum_{j=1}^{d}h_{j}z_{j}$. If $\boldsymbol{h}\equiv_{p}\boldsymbol{0}$, then every $\boldsymbol{z}\in\{1:p-1\}^{d}$ is a solution to $\boldsymbol{h}\cdot\boldsymbol{z}\equiv_{p}0$. Hence, there are $(p-1)^{d}$ solutions in this case. Otherwise, there is an $\ell\in\{1:d\}$ such that $h_{\ell}\not\equiv_{p}0$. Now, for any fixed $z_{j}$, $j\neq\ell$, we have $\boldsymbol{h}\cdot\boldsymbol{z}\equiv_{p}0$ if and only if $h_{\ell}z_{\ell}\equiv_{p}-\sum_{j\neq\ell}h_{j}z_{j}$. Since $h_{\ell}\not\equiv_{p}0$ and $p$ is prime there is at most one such $z_{\ell}\in\{1:p-1\}$. This shows that there are no more than $(p-1)^{d-1}$ solutions to $\boldsymbol{h}\cdot\boldsymbol{z}\equiv_{p}0$ in this case. $\Box$

A similar result to the following proposition can be found in many papers, see, e.g., [26, Lemma 2].

Proposition 5**.**

Let $p$ be prime, $d\in\mathbb{N}$, and ${\boldsymbol{\gamma}}\in(0,1]^{\mathbb{N}}$. For any $\beta>1$, we have

[TABLE]

Proof.

By definition we have

[TABLE]

We split this sum into a sum over all $\boldsymbol{h}$ with $\boldsymbol{h}\equiv_{p}\boldsymbol{0}$ and the remaining $\boldsymbol{h}$, and bound the sums separately. By Lemma 4 we have for the sum over $\boldsymbol{h}\equiv_{p}\boldsymbol{0}$,

[TABLE]

The penultimate inequality easily follows from the definition of $r_{\beta,{\boldsymbol{\gamma}}}(\boldsymbol{h})$ and $\boldsymbol{h}\neq\boldsymbol{0}$, and the last inequality follows from Lemma 2. For the sum over $\boldsymbol{h}\not\equiv_{p}\boldsymbol{0}$ we use again Lemma 2 and Lemma 4, and obtain

[TABLE]

Combining the last two estimates with $1/(p-1)\leq 2/p$ leads to the desired result. $\Box$

Clearly, there must be one choice of $\boldsymbol{z}$ such that $P_{\beta,{\boldsymbol{\gamma}}}(p,\boldsymbol{z})$ is as good as the average. By this argument, and using (5), it is typically shown that there exists a generating vector that makes the worst case error small. Although this is clearly not a constructive argument, there are methods to generate such vectors in an efficient way, in particular component-by-component algorithms. These constructions, dating back to Korobov, were re-invented in 2002 in [24], and proven to yield optimal results in [12]. Moreover, there exists a fast variant due to [21] which is heavily used nowadays.

For the upcoming analysis, it is not enough to have a single “good” generating vector. However, as the next corollary shows, there are actually many of them.

Corollary 6**.**

Let $p$ be prime, $d\in\mathbb{N}$, and ${\boldsymbol{\gamma}}\in(0,1]^{\mathbb{N}}$. For any $\beta>1$ and $\tau\in(0,1)$, there exist at least $\lceil\tau(p-1)^{d}\rceil$ generating vectors $\boldsymbol{z}\in\{1:p-1\}^{d}$ such that

[TABLE]

Proof.

Suppose to the contrary that $P_{\beta,{\boldsymbol{\gamma}}}(p,\boldsymbol{z})$ is smaller than or equal to the right-hand side above for $\chi$ choices of $\boldsymbol{z}$, where $\chi\leq\lceil\tau(p-1)^{d}\rceil-1<\tau(p-1)^{d}$. The number of $\boldsymbol{z}$ such that $P_{\beta,{\boldsymbol{\gamma}}}(p,\boldsymbol{z})$ is larger, i.e., $(p-1)^{d}-\chi$, therefore satisfies $(p-1)^{d}-\chi>(1-\tau)(p-1)^{d}$. Then the average over all $\boldsymbol{z}$ is bigger than $V_{d}(\beta,{\boldsymbol{\gamma}})/p$, which contradicts Proposition 5. $\Box$

Most of the results discussed so far in this section hold for arbitrary $\beta>1$. However, for the further analysis we need a bound on $\rho_{\alpha,{\boldsymbol{\gamma}}}(p,\boldsymbol{z})$ for many $\boldsymbol{z}$ for all $\alpha>0$. It is known that the classical Zaremba index, corresponding to $\alpha=1$ and ${\boldsymbol{\gamma}}=\boldsymbol{1}$, has a growth behavior like $p$ (with additional logarithmic terms), see, e.g., [15], so we expect a growth behavior of $\rho_{\alpha,{\boldsymbol{\gamma}}}(p,\boldsymbol{z})$ comparable to $p^{\alpha}$. This is made more precise in the following lemma.

Lemma 7**.**

Let $p$ be prime, $d\in\mathbb{N}$, $\alpha>0$, and ${\boldsymbol{\gamma}}\in(0,1]^{\mathbb{N}}$. For any $\tau\in(0,1)$, there exist at least $\lceil\tau(p-1)^{d}\rceil$ generating vectors $\boldsymbol{z}\in\{1:p-1\}^{d}$ such that

[TABLE]

Proof.

Using (7) and $r_{\alpha,{\boldsymbol{\gamma}}}(\boldsymbol{h})=(r_{\alpha/\lambda,{\boldsymbol{\gamma}}^{1/\lambda}}(\boldsymbol{h}))^{\lambda}$ for all $\lambda\in(0,\alpha)$, we obtain for any $\boldsymbol{z}\in\{1:p-1\}^{d}$ that

[TABLE]

Now for each $\lambda\in(0,\alpha)$, we know from Corollary 6 that the inequality

[TABLE]

holds for at least $\lceil\tau(p-1)^{d}\rceil$ different generating vectors $\boldsymbol{z}=\boldsymbol{z}(\lambda)$. To obtain the set of generating vectors that satisfies (11) for all $\lambda\in(0,\alpha)$, we consider the infimum of the right-hand side over $\lambda$. If this infimum is attained for some $\lambda^{*}\in(0,\alpha)$, then those generating vectors $\boldsymbol{z}(\lambda^{*})$ which correspond to $\lambda^{*}$ will satisfy the desired bound (10) for all other values of $\lambda$. If it is not attained in the interval $(0,\alpha)$, then since the limit as $\lambda\to\alpha$ is clearly infinity, the infimum is the limit as $\lambda\to 0$. However, this limit equals 1, which makes the statement of the lemma trivial. $\Box$

Remark 8**.**

For a given prime $p$, let us choose a generating vector $\boldsymbol{z}\in\{1:p-1\}^{d}$ uniformly at random. By Lemma 7 we have that the probability of obtaining, by this procedure, a “good” generating vector, i.e., a vector $\boldsymbol{z}$ that satisfies the bound (10), is at least $\tau$. Besides possible implementation issues, we did not find any advantage of adjusting $\tau$. Hence, to ease the notation, we set $\tau=1/2$ in the following.

3 New results in the randomized setting

3.1 The randomized lattice algorithm without shift

Let $n\in\mathbb{N}$, $n\geq 2$, and

[TABLE]

Let $d\in\mathbb{N}$, $\alpha>0$, and ${\boldsymbol{\gamma}}\in(0,1]^{\mathbb{N}}$. For each $p\in\mathcal{P}_{n}$, let ${\mathcal{Z}}_{p}$ denote the set of good generating vectors $\boldsymbol{z}$ in the sense of Lemma 7, with $\tau=1/2$, that is,

[TABLE]

We know from Lemma 7 that $|{\mathcal{Z}}_{p}|\,\geq\,\lceil(p-1)^{d}/2\rceil$.

Our random algorithm $M_{n}$ is defined by first randomly and uniformly selecting a prime $p\in\mathcal{P}_{n}$ and then randomly and uniformly selecting a generating vector $\boldsymbol{z}\in{\mathcal{Z}}_{p}$. The randomized error (2) for the lattice algorithm $M_{n}$ is then given precisely by

[TABLE]

We stress that the randomized error is not the same as the average of the worst case errors (6) of a set of deterministic lattice rules. (Here the averaging occurs inside the supremum rather than outside.)

Theorem 9**.**

For $\alpha>1/2$, $\lambda\in(1/2,\alpha)$, ${\boldsymbol{\gamma}}\in(0,1]^{\mathbb{N}}$, and

[TABLE]

the randomized error of the randomized lattice algorithm $M_{n}$ satisfies

[TABLE]

for arbitrary $\delta$ satisfying $0<\delta<\min(\lambda-1/2,1)$, where $V_{d}$ is defined as in (9) and $C_{\lambda,\delta}$ is a constant depending only on $\lambda$ and $\delta$. The upper bound is independent of $d$ if $\sum_{j=1}^{\infty}\gamma_{j}^{1/\lambda}<\infty$.

Remark 10**.**

A simple probabilistic approach to implement this algorithm is to first randomly select a prime $p\in\mathcal{P}_{n}$, and then randomly select a vector $\boldsymbol{z}\in\{1:p-1\}^{d}$ repeatedly until the condition in (12) is satisfied. The chance of getting a vector $\boldsymbol{z}\in{\mathcal{Z}}_{p}$ after $k$ tries is $1-\tau^{k}=1-2^{-k}$.

Note that the inequality in (12) should hold simultaneously for all values of $\lambda\in(0,\alpha)$, and so in a practical implementation it is not easy to verify whether a given vector $\boldsymbol{z}$ belongs to ${\mathcal{Z}}_{p}$. An easy way around this is to relax the definition of ${\mathcal{Z}}_{p}$ to a set of generating vectors ${\mathcal{Z}}_{p,\lambda}$ which assumes the given bound only for a fixed parameter $\lambda\in(0,\alpha)$, and consider the corresponding random algorithm $M_{n,\lambda}$ which depends on the parameter $\lambda$. In this case, Theorem 9 holds for $M_{n,\lambda}$ for this parameter $\lambda$, and the implementation of the algorithm is straightforward and simple.

We leave it for future research if sampling from the set ${\mathcal{Z}}_{p}$ itself can be done efficiently, e.g., by a component-by-component-type algorithm.

• Proof of Theorem 9.

We first define

[TABLE]

Then the condition (13) ensures that $B_{n}\geq 1$.

It follows from the definition of ${\mathcal{Z}}_{p}$ in (12) that for all $p\in\mathcal{P}_{n}$ and $\boldsymbol{z}\in{\mathcal{Z}}_{p}$ we have $p>n/2$ and $\rho_{\alpha,{\boldsymbol{\gamma}}}(p,\boldsymbol{z})>B_{n}$; and consequently for every $\boldsymbol{h}\in\mathbb{Z}^{d}\setminus\{\boldsymbol{0}\}$ with $\boldsymbol{h}\cdot\boldsymbol{z}\equiv_{p}0$ we have $r_{\alpha,{\boldsymbol{\gamma}}}(\boldsymbol{h})>B_{n}$. From (4) we obtain

[TABLE]

where

[TABLE]

with $\mathbb{I}(\cdot)$ denoting the indicator function. The last inequality was obtained by multiplying and dividing the penultimate expression by $r_{\alpha,{\boldsymbol{\gamma}}}(\boldsymbol{h})$ and then applying the Cauchy–Schwarz inequality. Hence we conclude that

[TABLE]

We now proceed to obtain a bound on $\omega_{n}(\boldsymbol{h})$. For fixed $p$, if $\boldsymbol{h}\equiv_{p}\boldsymbol{0}$ then $\boldsymbol{h}\cdot\boldsymbol{z}\equiv_{p}0$ holds for all $\boldsymbol{z}\in{\mathcal{Z}}_{p}$. On the other hand, if $\boldsymbol{h}\not\equiv_{p}\boldsymbol{0}$ then we may bound the last sum in (15) by the number of all $\boldsymbol{z}\in\{1:p-1\}^{d}$ with $\boldsymbol{h}\cdot\boldsymbol{z}\equiv_{p}0$. We already computed this number in Lemma 4. Thus we have

[TABLE]

and therefore

[TABLE]

Note that any number $h\in{\mathbb{N}}$ has at most $\log_{M}(h)$ prime divisors greater than $M\in{\mathbb{N}}$. So for $\boldsymbol{h}\neq\boldsymbol{0}$ the number of primes $p\geq\lceil n/2\rceil+1$ for which $\boldsymbol{h}\equiv_{p}\boldsymbol{0}$ holds is at most $\log_{\lceil n/2\rceil+1}(|\boldsymbol{h}|_{\infty})$, that is,

[TABLE]

for all $n\geq 2$. Combining this with the estimate $|\mathcal{P}_{n}|>c^{\prime}n/\log(n)$ for some $c^{\prime}>0$, see, e.g., [22], we conclude that

[TABLE]

for some $c<\infty$, which yields

[TABLE]

with $\beta:=\alpha/\lambda>1$ and ${\boldsymbol{\gamma}}^{\prime}={\boldsymbol{\gamma}}^{1/\lambda}$. Here we used $r_{\alpha,{\boldsymbol{\gamma}}}(\boldsymbol{h})=(r_{\alpha/\lambda,{\boldsymbol{\gamma}}^{1/\lambda}}(\boldsymbol{h}))^{\lambda}$. Now we apply $|\boldsymbol{h}|_{\infty}\leq r_{\beta,{\boldsymbol{\gamma}}^{\prime}}(\boldsymbol{h})$ since $\beta>1$ and ${\boldsymbol{\gamma}}\in(0,1]^{\mathbb{N}}$, and $\log(1+x)\leq x^{\delta}/\delta$ which holds for all $\delta\in(0,1]$ and $x>0$, and obtain

[TABLE]

where we further restrict $\delta$ to be such that $2(\lambda-\delta)>1$, which is equivalent to $\delta<\lambda-1/2$.

To complete the proof we will show a suitable upper bound on

[TABLE]

For $T>0$, we define the set

[TABLE]

Recall from Corollary 3 that $\left|\mathcal{A}(T)\right|\leq T\,V_{d}(\alpha/\lambda,{\boldsymbol{\gamma}}^{1/\lambda})$. Moreover, we have for $u>1$ and $T\geq 1$,

[TABLE]

Recall also that $B_{n}^{1/\lambda}\geq 1$. Since $\boldsymbol{h}\notin\mathcal{A}(T)$ implies $1/r_{\beta,{\boldsymbol{\gamma}}^{\prime}}(\boldsymbol{h})<1/T$, we obtain

[TABLE]

where we used $\lfloor x\rfloor\geq x/2$ for $x\geq 1$ in the second to last inequality. Together with (14) and (17), this shows

[TABLE]

Note that the coefficient of $n^{-\lambda-1/2+\delta}$ in the latter term is bounded under the assumptions we made, i.e., $\alpha>\lambda>1/2$ and $0<\delta<\lambda-1/2$, as well as $V_{d}(\alpha/\lambda,{\boldsymbol{\gamma}}^{1/\lambda})<\infty$ independently of $d$ if $\sum_{j=1}^{\infty}\gamma_{j}^{1/\lambda}<\infty$, see (9). This completes the proof. $\hfill\Box$

3.2 The randomized lattice algorithm with shift

We are now going to prove in Theorem 11 below that $\widetilde{M}_{n}$, i.e., the random lattice rule with an additional shift, satisfies a similar upper bound on its randomized error as $M_{n}$, but also for functions from $\mathcal{H}_{d,\alpha,{\boldsymbol{\gamma}}}$ with $0<\alpha\leq 1/2$. Note that, by allowing general $\alpha>0$ in Theorem 11, we are considering a larger function class than in Theorem 9.

The randomized error (2) for the algorithm $\widetilde{M}_{n}$ is given by

[TABLE]

where $\mathbb{E}_{\boldsymbol{U}}$ denotes expectation with respect to the random shift $\boldsymbol{U}\in[0,1]^{d}$. For this algorithm we can even bound the root-mean-square error (or standard deviation)

[TABLE]

Clearly, e^{\rm ran}_{d,\alpha,{\boldsymbol{\gamma}}}\bigl{(}\widetilde{M}_{n}\bigr{)}\leq e^{\rm rms}_{d,\alpha,{\boldsymbol{\gamma}}}\bigl{(}\widetilde{M}_{n}\bigr{)}.

Theorem 11**.**

For $\alpha>0$, $\lambda\in(0,\alpha)$, and ${\boldsymbol{\gamma}}\in(0,1]^{\mathbb{N}}$, the randomized worst case error of the randomized lattice algorithm with shift $\widetilde{M}_{n}$ satisfies

[TABLE]

for arbitrary $\delta$ satisfying $0<\delta<\min(1/\alpha,2)$, where again $V_{d}$ is defined as in (9) and $c$ is an absolute constant. The upper bound is independent of $d$ if $\sum_{j=1}^{\infty}\gamma_{j}^{1/\lambda}<\infty$.

Proof.

By a variation of Poisson’s summation formula, we have

[TABLE]

and therefore

[TABLE]

with $\omega_{n}(\boldsymbol{h})$ from (15). From (16) we know that

[TABLE]

We apply $(|\boldsymbol{h}|_{\infty})^{\alpha}\leq r_{\alpha,{\boldsymbol{\gamma}}}(\boldsymbol{h})$ and $\log(1+x)\leq x^{\alpha\delta}/(\alpha\delta)$ for all $\delta\in(0,1/\alpha]$ to obtain

[TABLE]

where we further restrict $\delta$ to be such that $\delta<2$. Together with (14) this yields

[TABLE]

which proves the result. $\Box$

4 Lower bound on the error of $M_{n}$ and $\widetilde{M}_{n}$

In the last section we proved that the algorithms $M_{n}$ and $\widetilde{M}_{n}$ can achieve an order of convergence that is arbitrarily close to the optimal $n^{-\alpha-1/2}$, see [29]. However, we will show in Theorem 12 that the exact optimal order cannot be achieved by these algorithms, no matter how we choose the weights ${\boldsymbol{\gamma}}$. This shows that the main order in our results in Theorems 9 and 11 is essentially best possible. Note that the lower bound also holds for the case $\alpha=0$. However, it would be interesting to find an upper bound of the form $C(\log n)^{q}/n^{\alpha+1/2}$ with $C,q$ independent of $n$ and $d$. It is not clear if such a bound exists.

Theorem 12**.**

Let $\alpha\geq 0$ and ${\boldsymbol{\gamma}}\in(0,1]^{\mathbb{N}}$. Then the randomized errors of the random algorithms $M_{n}$ and $\widetilde{M}_{n}$ are bounded from below by

[TABLE]

and

[TABLE]

Proof.

The bound $e^{\rm rms}_{d,\alpha,{\boldsymbol{\gamma}}}(M_{n})\geq e^{\rm ran}_{d,\alpha,{\boldsymbol{\gamma}}}(M_{n})$ is obvious. To prove the lower bound on $e^{\rm ran}_{d,\alpha,{\boldsymbol{\gamma}}}(M_{n})$ it is enough to construct, for each $n\in\mathbb{N}$, $n\geq 2$, a function, say $f_{n}$, that satisfies the lower bound. To this end, we define $f_{n}$ such that

[TABLE]

Clearly, $\|f_{n}\|_{d,\alpha,{\boldsymbol{\gamma}}}=1$ and $I_{d}(f_{n})=0$. Moreover, for $p\in\mathcal{P}_{n}$, we have

[TABLE]

where we used $\left|\mathcal{P}_{n}\right|\leq\frac{2n}{\log n}$ from [22]. This proves

[TABLE]

Using

[TABLE]

we can proceed exactly as above to prove the lower bound on e^{\rm rms}_{d,\alpha,{\boldsymbol{\gamma}}}\bigl{(}\widetilde{M}_{n}\bigr{)}. $\Box$

5 Results on tractability

Let us now briefly comment on tractability results. Suppose that for fixed $d\in{\mathbb{N}}$ and $\varepsilon\in(0,1)$, we would like to achieve $e_{d,\alpha,{\boldsymbol{\gamma}}}^{\rm ran}(M_{n})\leq\varepsilon$. Then from Theorem 9 we conclude that for $\mathcal{H}_{d,\alpha,{\boldsymbol{\gamma}}}$ with $\alpha>1/2$, provided $n$ satisfies (13), it is sufficient to choose $n$ such that

[TABLE]

Hence, the information complexity $n(\varepsilon,d)$, i.e., the minimal number of function evaluations that is required by any kind of random algorithm to achieve a randomized error within the threshold $\varepsilon$, satisfies

[TABLE]

More generally, for $\mathcal{H}_{d,\alpha,{\boldsymbol{\gamma}}}$ with $\alpha>0$, we obtain from Theorem 11 that

[TABLE]

In both cases, we have $n(\varepsilon,d)={\mathcal{O}}(\varepsilon^{-\beta})$, for $\beta$ arbitrarily close to $1/(\lambda+1/2)$, where the implied constant is independent of $d$ and $\varepsilon$ if $\sum_{j=1}^{\infty}\gamma_{j}^{1/\lambda}<\infty$. The integration problem is then said to be strongly tractable in the randomized setting, with the exponent of strong tractability being the infimum of those $\beta$.

We summarize these results in the following theorem.

Theorem 13**.**

Let $\alpha\geq 0$, let ${\boldsymbol{\gamma}}=(\gamma_{j})_{j\geq 1}$ be a non-increasing sequence of positive weights bounded by 1, and let $d\in\mathbb{N}$. Additionally, let $\lambda_{0}\geq 0$ be the supremum of the numbers $\lambda>0$ such that

[TABLE]

*with the convention that $\lambda_{0}=0$ if no such $\lambda$ exists.

Then the integration problem in the class $\mathcal{H}_{d,\alpha,{\boldsymbol{\gamma}}}$ is strongly tractable in the randomized setting, with the exponent of strong tractability lying in the interval*

[TABLE]

In particular, if $\lambda_{0}\geq\alpha$, then the exponent of strong tractability is $1/(\alpha+1/2)$.

Proof.

The upper bound on the exponent of strong tractability for $\alpha>0$ and $\lambda_{0}>0$ follows from Theorem 11 and the discussion above, noting that we require $\lambda<\min(\lambda_{0},\alpha)$. For $\alpha=0$ it is well-known that the exponent of strong tractability in the randomized setting is $2$, which can be achieved, e.g., by the simple Monte Carlo method. For $\lambda_{0}=0$ we use the same method to obtain that the exponent of strong tractability is at most 2.

For the lower bound note that $n(\varepsilon,d)\geq n(\varepsilon,1)\geq c\cdot\varepsilon^{-1/(\alpha+1/2)}$ for some $c>0$, see, e.g., [16]. $\Box$

Remark 14**.**

In the same way as it was done in [26], we could also comment on other forms of tractability, like polynomial tractability, under modified summability assumptions on the weights. We omit the details.

For further information on tractability in various settings and an overview of various kinds of error criteria in the context of tractability, see the trilogy [17]–[19].

6 Conclusion

We showed in this paper that the randomized lattice algorithm $M_{n}$ achieves a randomized integration error with a convergence order arbitrarily close to ${\mathcal{O}}(n^{-\alpha-1/2})$ in $\mathcal{H}_{d,\alpha,{\boldsymbol{\gamma}}}$ with $\alpha>1/2$, where the implied constant is independent of $d$ as long as $\sum_{j=1}^{\infty}\gamma_{j}^{1/\alpha}<\infty$. By additionally making use of a random shift, this result can be extended to all $\alpha>0$.

If the weights only satisfy a weaker summability condition $\sum_{j=1}^{\infty}\gamma_{j}^{1/\lambda}<\infty$ for some $\lambda<\alpha$, then we can still obtain error bounds that are independent of $d$ but the convergence rate is reduced correspondingly to be close to ${\mathcal{O}}(n^{-\lambda-1/2})$.

The question whether component-by-component constructions can be used in the context of this setting remains open for future research.

Acknowledgement

P. Kritzer is supported by the Austrian Science Fund (FWF) Project F5506-N26, which is part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”. F.Y. Kuo acknowledges the financial support from the Australian Research Council (FT130100655 and DP150101770). D. Nuyens acknowledges the financial support from the KU Leuven research fund (OT:3E130287 and C3:3E150478).

P. Kritzer and M. Ullrich furthermore gratefully acknowledge the partial support of the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) in Vienna under the thematic programme “Tractability of High Dimensional Problems and Discrepancy”.

P. Kritzer, F.Y. Kuo, and D. Nuyens acknowledge the supports from the Taiwanese National Center for Theoretical Sciences (NCTS) – Mathematics Division, and the National Science Foundation Grant DMS-1638521 to the Statistical and Applied Mathematical Sciences Institute (SAMSI) under its 2017 year-long program on “Quasi-Monte Carlo and High-Dimensional Sampling Methods for Applied Mathematics”.