Derandomised lattice rules for high dimensional integration

TL;DR

This paper develops a deterministic shifted lattice rule approach for high-dimensional integration, showing existence of near-optimal shifts and demonstrating promising numerical results with a new CBC-based shift selection algorithm.

Contribution

It introduces a deterministic shift method for lattice rules, proving the existence of near-optimal shifts and proposing a new CBC-based shift selection algorithm.

Findings

Existence of a half-shifted rule with squared worst-case error close to the shift-averaged error.

Numerical experiments show the effectiveness of the CBC for shift algorithm.

Deterministic shifts perform comparably to random shifts in high-dimensional integration.

Abstract

We seek shifted lattice rules that are good for high dimensional integration over the unit cube in the setting of an unanchored weighted Sobolev space of functions with square-integrable mixed first derivatives. Many existing studies rely on random shifting of the lattice, whereas here we work with lattice rules with a deterministic shift. Specifically, we consider "half-shifted" rules, in which each component of the shift is an odd multiple of $1/(2N)$, where $N$ is the number of points in the lattice. We show, by applying the principle that \emph{there is always at least one choice as good as the average}, that for a given generating vector there exists a half-shifted rule whose squared worst-case error differs from the shift-averaged squared worst-case error by a term of order only ${1/N^2}$. Numerical experiments, in which the generating vector is chosen component-by-component (CBC) as for randomly shifted lattices and then the shift by a new "CBC for shift" algorithm, yield encouraging results.

Tables

Table 1. Table 1: Shifts Δ s ∗ = ( 2 ​ m s ∗ − 1 ) / ( 2 ​ N ) superscript subscript Δ 𝑠 2 superscript subscript 𝑚 𝑠 1 2 𝑁 {\Delta_{s}^{*}=(2m_{s}^{*}-1)/(2N)} and κ ​ ( s , N ) 𝜅 𝑠 𝑁 \kappa(s,N) obtained by the CBC for shift algorithm for N = 2048 𝑁 2048 N=2048 , γ j = 1 / j 2 subscript 𝛾 𝑗 1 superscript 𝑗 2 \gamma_{j}=1/j^{2} ; and κ 0 ​ ( s , N ) subscript 𝜅 0 𝑠 𝑁 \kappa_{0}(s,N) , the values of κ ​ ( s , N ) 𝜅 𝑠 𝑁 \kappa(s,N) corresponding to zero shift, for s = 1 , … , 50 𝑠 1 … 50 s=1,\dots,50 . We see that κ ​ ( s , N ) 𝜅 𝑠 𝑁 \kappa(s,N) is less than 1 1 1 .

$s$	$m_s^{\ast }$	$\kappa (s,2048)$	${\kappa }_0(s,2048)$
1	1	0.708211	1.414765
2	227	0.774829	1.242600
3	17	0.804685	1.184076
4	1955	0.817572	1.159945
5	1273	0.827628	1.164227
6	1250	0.835811	1.153188
7	1698	0.841416	1.140436
8	1970	0.845575	1.135741
9	476	0.847988	1.134190
10	646	0.850682	1.130419
11	779	0.853486	1.129294
12	1093	0.855818	1.126409
13	1498	0.857239	1.123408
14	550	0.859090	1.122288
15	1218	0.860315	1.123017
16	1124	0.861422	1.121392
17	135	0.862420	1.120642
18	717	0.863531	1.120035
19	854	0.864463	1.119229
20	1634	0.865152	1.118282
21	1692	0.865776	1.117763
22	1002	0.866488	1.116437
23	1034	0.866953	1.117111
24	249	0.867514	1.117083
25	1477	0.868110	1.116343

Table 2. Table 2: Shifts Δ s ∗ = ( 2 ​ m s ∗ − 1 ) / ( 2 ​ N ) superscript subscript Δ 𝑠 2 superscript subscript 𝑚 𝑠 1 2 𝑁 {\Delta_{s}^{*}=(2m_{s}^{*}-1)/(2N)} and κ ​ ( s , N ) 𝜅 𝑠 𝑁 \kappa(s,N) obtained by the CBC for shift algorithm for N = 2048 𝑁 2048 N=2048 , γ j = 0.5 j subscript 𝛾 𝑗 superscript 0.5 𝑗 \gamma_{j}=0.5^{j} ; and κ 0 ​ ( s , N ) subscript 𝜅 0 𝑠 𝑁 \kappa_{0}(s,N) , the values of κ ​ ( s , N ) 𝜅 𝑠 𝑁 \kappa(s,N) corresponding to zero shift, for s = 1 , … , 50 𝑠 1 … 50 s=1,\dots,50 . We see that κ ​ ( s , N ) 𝜅 𝑠 𝑁 \kappa(s,N) is less than 1 1 1 .

s	$m_s^{\ast }$	$\kappa (s,2048)$	${\kappa }_0(s,2048)$
1	1	0.708211	1.414765
2	227	0.774829	1.242600
3	17	0.804685	1.184076
4	1955	0.817572	1.159945
5	422	0.829080	1.146448
6	1698	0.836308	1.130689
7	1917	0.841847	1.131901
8	2005	0.845626	1.127075
9	5	0.848370	1.121351
10	135	0.851825	1.116117
11	1139	0.853881	1.118105
12	1410	0.857111	1.111780
13	982	0.859260	1.109848
14	1151	0.860542	1.107647
15	751	0.862129	1.104858
16	1043	0.863598	1.102854
17	1083	0.864755	1.107635
18	412	0.866106	1.107113
19	211	0.867072	1.106381
20	854	0.867879	1.105469
21	418	0.868589	1.136709
22	849	0.869226	1.164846
23	13	0.876862	1.197944
24	1280	0.877108	1.197713
25	1229	0.882508	1.217421

Equations

$$I_s(f):={\int }_0^1\cdots {\int }_0^{1}f(x_1,\dots ,x_s)\mathrm{d}{x}_1\cdots \mathrm{d}{x}_s={\int }_{[0,1{]}^s}f(\mathbf{x})\mathrm{d}\mathbf{x}.$$

$$Q_{N,s}({\bm{z}},\bm{\Delta};f):=\frac{1}{N}\sum_{k=1}^{N}f\bigg{(}\bigg{\{}\frac{{k}{\bm{z}}}{N}+\bm{\Delta}\bigg{\}}\bigg{)},$$

$$e_{N,s}(\mathbf{z},\mathbf{\Delta }):=\underset{f\in H_s,\Vert f{\Vert }_{H_s}\le 1}{\sup }|Q_{N,s}(\mathbf{z},\mathbf{\Delta };f)-I_s(f)|.$$

$$\Vert f{\Vert }_{H_{s,\mathbf{\gamma }}}^2:=$$

$$e_{N,s}^{2}({\bm{z}},\bm{\Delta})={\frac{1}{N^{2}}\sum_{k=1}^{N}\sum_{k^{\prime}=1}^{N}\sum_{{\emptyset\neq}{\mathrm{\mathfrak{u}}}\subseteq\{1:s\}}\!\!\gamma_{{\mathrm{\mathfrak{u}}}}\prod_{j\in{\mathrm{\mathfrak{u}}}}\bigg{[}\frac{1}{2}B_{2}\bigg{(}\bigg{\{}\frac{(k-k^{\prime})z_{j}}{N}\bigg{\}}\bigg{)}+A_{k,k^{\prime},z_{j}}(\Delta_{j})\bigg{]},}$$

$$A_{k,k^{\prime},z}(\Delta):=\bigg{(}\bigg{\{}\frac{kz}{N}+\Delta\bigg{\}}-\frac{1}{2}\bigg{)}\bigg{(}\bigg{\{}\frac{k^{\prime}z}{N}+\Delta\bigg{\}}-\frac{1}{2}\bigg{)}.$$

$$\displaystyle{[e_{N,s}^{{\rm sh}}({\bm{z}})]^{2}}:=\,\int_{[0,1]^{s}}e_{N,s}^{2}({\bm{z}},\bm{\Delta})\,{\mathrm{d}}\bm{\Delta}=\,{\frac{1}{N}\sum_{k=1}^{N}\sum_{\emptyset\neq{\mathrm{\mathfrak{u}}}\subseteq\{1:s\}}\!\!\gamma_{{\mathrm{\mathfrak{u}}}}\prod_{j\in{\mathrm{\mathfrak{u}}}}B_{2}\bigg{(}\bigg{\{}\frac{kz_{j}}{N}\bigg{\}}\bigg{)}},$$

$$e_{N,s}^{{\rm sh}}({\bm{z}}^{*})\leq\Bigg{(}{\frac{1}{\varphi(N)}\sum_{\emptyset\neq{\mathrm{\mathfrak{u}}}\subseteq\{1:s\}}\gamma_{\mathrm{\mathfrak{u}}}^{\lambda}\left(\frac{2\zeta(2\lambda)}{(2\pi^{2})^{\lambda}}\right)^{|{\mathrm{\mathfrak{u}}}|}}\Bigg{)}^{1/(2\lambda)},$$

$$\sqrt{\mathbb{E}[|Q_{N,s}({\bm{z}}^{*}{,\cdot;}f)-I_{s}(f)|^{2}]}\leq\!\Bigg{(}{\frac{1}{\varphi(N)}\sum_{\emptyset\neq{\mathrm{\mathfrak{u}}}\subseteq\{1:s\}}\!\!\gamma_{\mathrm{\mathfrak{u}}}^{\lambda}\left(\frac{2\zeta(2\lambda)}{(2\pi^{2})^{\lambda}}\right)^{|{\mathrm{\mathfrak{u}}}|}}\Bigg{)}^{1/(2\lambda)}\!\!\|f\|_{H_{{s,{\bm{\gamma}}}}}.$$

$$\kappa (N,s):=\frac{e_{N,s}({\mathbf{z}}^{\ast },{\mathbf{\Delta }}^{\ast })}{e_{N,s}^{\mathrm{sh}}({\mathbf{z}}^{\ast })}.$$

$$|Q_{N,s}({\mathbf{z}}^{\ast },{\mathbf{\Delta }}^{\ast };f)-I_s(f)|\le \kappa (N,s)e_{N,s}^{\mathrm{sh}}({\mathbf{z}}^{\ast })\Vert f{\Vert }_{H_{s,\mathbf{\gamma }}}$$

$$\displaystyle\qquad\qquad\qquad\leq\kappa(N,s)\Bigg{(}{\frac{1}{\varphi(N)}\sum_{\emptyset\neq{\mathrm{\mathfrak{u}}}\subseteq\{1:s\}}\!\!\gamma_{\mathrm{\mathfrak{u}}}^{\lambda}\bigg{(}\frac{2\zeta(2\lambda)}{(2\pi^{2})^{\lambda}}\bigg{)}^{|{\mathrm{\mathfrak{u}}}|}}\Bigg{)}^{1/(2\lambda)}\!\!\|f\|_{H_{{s,{\bm{\gamma}}}}},$$

$$[e_{N,s}^{\frac{1}{2}\mathrm{sh}}(\mathbf{z}){]}^2:=\frac{1}{N^s}\sum _{\mathbf{\Delta }\in S_N^s}{e}_{N,s}^2(\mathbf{z};\mathbf{\Delta }),$$

$$\Big{|}[e_{N,s}^{\rm{sh}}(\bm{z})]^{2}-[e_{N,s}^{\frac{1}{2}\rm{sh}}(\bm{z})]^{2}\Big{|}\leq\frac{1}{4N^{2}}\sum_{\emptyset\neq{\mathrm{\mathfrak{u}}}\subseteq\{1:s\}}\gamma_{\mathrm{\mathfrak{u}}}\bigg{(}\frac{1}{3}\bigg{)}^{|\mathfrak{u}|}{|{\mathrm{\mathfrak{u}}}|}.$$

$$[e_{N,s}^{\mathrm{sh}}(\mathbf{z}){]}^2-[e_{N,s}^{\frac{1}{2}\mathrm{sh}}(\mathbf{z}){]}^2=\frac{1}{N^2}\sum _{k=1}^N\sum _{k^{\prime }=1}^N\sum _{\varnothing \ne \mathfrak{u}\subseteq \{1:s\}}{\gamma }_{\mathfrak{u}}\left(\prod _{j\in \mathfrak{u}}{a}_j^{k,k^{\prime }}-\prod _{j\in \mathfrak{u}}{b}_j^{k,k^{\prime }}\right),$$

$$a_j^{k,k^{\prime }}:=c_j^{k,k^{\prime }}+{\int }_0^1{A}_{k,k^{\prime },z_j}(\Delta )\mathrm{d}\Delta ,$$

$$\displaystyle c_{j}^{k,k^{\prime}}:=\frac{1}{2}B_{2}\bigg{(}\bigg{\{}\frac{(k-k^{\prime})z_{j}}{N}\bigg{\}}\bigg{)},$$

$$a_j^{k,k^{\prime }}-b_j^{k,k^{\prime }}$$

$$\displaystyle=\sum^{N}_{m=1}\bigg{(}\int^{\frac{m}{N}}_{\frac{m-1}{N}}A_{k,k^{\prime},z_{j}}(\Delta)\,{\mathrm{d}}\Delta-\frac{1}{N}A_{k,k^{\prime},z_{j}}\bigg{(}\frac{2m-1}{2N}\bigg{)}\bigg{)},$$

$$\kappa (N,s)=\frac{e_{N,s}((z_1^{\ast },\dots ,z_s^{\ast }),({\Delta }_1^{\ast },\dots ,{\Delta }_s^{\ast }))}{e_{N,s}^{\mathrm{sh}}(z_1^{\ast },\dots ,z_s^{\ast })},s=1,\dots ,s_{\max }.$$

$$\Delta^{*}_{1}\in\mathrm{argmin}\big{\{}e^{2}_{N,1}({z}^{*}_{1}{,}\Delta_{1})\mid\Delta_{1}\in{S_{N}}\big{\}},$$

$$\Delta^{*}_{s}\in\mathrm{argmin}\big{\{}e_{N,s}^{2}{(({z}^{*}_{1},\dotsc,{z}^{*}_{s}),(\Delta^{*}_{1},\dotsc,\Delta^{*}_{s-1},\Delta_{s}))}\mid\Delta_{s}\in{S_{N}}\big{\}},$$

Full text

1 Introduction

Lattice rules are often used for high dimensional integration over the unit cube, that is, for the numerical evaluation of the $s$-dimensional integral

[TABLE]

A shifted lattice rule for the approximation of the integral is an equal weight cubature rule of the form

[TABLE]

where ${\bm{z}}\in\{1,\ldots,N-1\}^{{s}}$ is the generating vector, $\bm{\Delta}\in[0,1]^{s}$ is the shift, while the braces around an $s$-vector indicate that each component of the vector is to be replaced by its fractional part in $[0,1)$. The special case $\bm{\Delta}=\bm{0}$ yields the unshifted lattice rule, which provably works well for periodic functions [7]. If the integrand is not periodic, the shift can play a useful role. The implementation of a shifted lattice rule is relatively easy once the vectors ${\bm{z}}$ and $\bm{\Delta}$ are prescribed, even when $s$ is very large, say, in the tens of thousands.

The central concern of this paper is the construction of a good shift vector $\bm{\Delta}$, given a specific choice of a good ${\bm{z}}$.

At the present time the overwhelmingly favored method for fixing the shifts in a non-periodic setting is to choose them randomly. In a randomly shifted lattice rule the shift $\bm{\Delta}$ is chosen from a uniform distribution on $[0,1]^{s}$, and the integral is approximated by an empirical estimate of the expected value, $\frac{1}{q}\sum_{i=1}^{q}Q_{N,s}({\bm{z}},\bm{\Delta}_{i};f)$, where $q$ is some fixed number, and $\bm{\Delta}_{1},\ldots,\bm{\Delta}_{q}\in[0,1]^{s}$ are $q$ independent samples from the uniform distribution on ${[0,1]^{s}}$. With the shift chosen randomly, all that remains in the randomly shifted case is to construct the integer vector ${\bm{z}}$, which can be done very effectively by using the component-by-component (CBC) construction to yield a vector ${\bm{z}}^{*}$ that gives a satisfactorily small value of the shift-averaged worst-case error.

In the present paper we construct a new kind of shifted lattice rule, which is derandomised in the sense that the generating vector is the same ${\bm{z}}^{*}$ determined by the CBC algorithm for the shift-averaged worst-case error, while the shift $\bm{\Delta}^{*}$ is determined by a new CBC construction, “CBC for shift”: the components of the shift vector is obtained one at a time, chosen from the odd multiples of $1/(2N)$.

We argue that there is a significant potential cost saving in this deterministic alternative, in that it becomes no longer necessary to compute an empirical average over shifts. In many applications there is not just a single integral to be evaluated, but rather many such integrals with different input parameters. In such a situation it may be seen as overkill to compute an error estimate for every single integral. If at any stage an error estimate is needed by a user of the present algorithm, then all that is needed is to replace the computed shift $\bm{\Delta}^{*}$ by $q$ randomly chosen shifts, in the knowledge that the resulting error estimate is just that of the CBC-constructed randomly shifted lattice rule.

Approaches to estimating the error for lattice rules for non-periodic functions without randomisation include [2, 3], where a mapping called the tent transform was applied to the lattice rule. In this paper, however, no transformation of the lattice points is considered.

1.1 Function spaces and worst-case errors

The central element in any CBC construction is the worst-case error, which for the case of the shifted lattice rule (1.1) and a Hilbert space $H_{s}$ may be defined by

[TABLE]

Here we consider a weighted unanchored Sobolev space of functions with square-integrable mixed first derivatives on $(0,1)^{s}$, with squared norm

[TABLE]

where $\{1:s\}=\{1,2,\ldots,s\}$, ${\bm{\gamma}}_{\mathrm{\mathfrak{u}}}$ is a positive number which is the “weight” corresponding to the subset ${\mathrm{\mathfrak{u}}}\subseteq\{1:s\}$, with ${\bm{\gamma}}_{\emptyset}=1$, and ${\bm{x}}_{\mathrm{\mathfrak{u}}}$ denotes the variables $x_{j}$ for $j\in{\mathrm{\mathfrak{u}}}$. It is well known that suitably decaying weights are essential if we are to have error bounds independent of dimension [8]. The squared worst-case error has an explicit formula (see e.g., [1])

[TABLE]

where

[TABLE]

For the randomly shifted lattice rule the relevant form of the worst-case error is the shift-averaged worst-case error $e_{N,s}^{{\rm sh}}({\bm{z}})$, defined by

[TABLE]

which is the expected value of the squared worst-case error, taken with respect to the random shift. Notice that the double sum over $k,k^{\prime}$ in (1.2) simplified to a single sum over $k$ in (1.3).

1.2 CBC constructions

The principle of a CBC construction is that, at stage $j$, one determines the $j$th component of the cubature points by seeking to minimise an error criterion for the $j$-dimensional problem; then with that component fixed one moves on to the next component, never going back.

In the case of randomly shifted lattice rules, $z_{1}^{*}$ is chosen to have the value $1$, and then, for $j=1,2,\ldots,s-1$, once $z_{1}^{*},z_{2}^{*},\ldots,z_{j}^{*}$ are fixed, $z_{j+1}$ is chosen to be the element from $\{1,\ldots,N-1\}$ that gives the smallest value of $[e_{N,j+1}^{{\rm sh}}(z_{1}^{*},\ldots,z_{j}^{*},z_{j+1})]^{2}$. The cost of the CBC algorithm for constructing ${\bm{z}}^{*}$ up to $s$ dimensions is of order $s\,N\log N$ using FFT [5], in the simplest case of “product weights”, in which there is only one sequence $\gamma_{1},\gamma_{2},\ldots,\gamma_{s}$ of weight parameters, and the value of $\gamma_{\mathrm{\mathfrak{u}}}$ is taken to be the product $\prod_{j\in{\mathrm{\mathfrak{u}}}}\gamma_{j}$. In this case the sum over ${\mathrm{\mathfrak{u}}}$ in (1.3) can be rewritten as a product of $s$ factors.

The proven quality of the CBC construction for randomly shifted lattice rules is very good, in the sense that, with $\zeta$ being the Riemann zeta function and $\varphi$ being the Euler totient function, for all $\lambda\in(\frac{1}{2},1]$,

[TABLE]

see e.g., [1]. It follows from the definition that for $f\in H_{{s,{\bm{\gamma}}}}$ one has as an error bound for the randomly shifted lattice rule constructed by CBC

[TABLE]

When $N$ is prime we have $\varphi(N)=N-1$. Thus the convergence rate is arbitrarily close to $1/N$ as $\lambda\to 1/2$, but with a constant that blows up as $\lambda\to 1/2$ because $\zeta(2\lambda)\to\infty$.

For our new derandomised lattice rule we take the components of the generating vector to be $z_{1}^{*},z_{2}^{*},\ldots,z_{s}^{*}$ as determined by the CBC algorithm for randomly shifted lattice rules. We then determine the components of the shift by a new CBC for shift algorithm, in which at stage $j$, with $\Delta_{1}^{*},\ldots,\Delta_{j}^{*}$ already fixed, we choose $\Delta_{j+1}$ by minimising the squared worst-case error $e_{N,j+1}^{2}((z_{1}^{*},\ldots,z_{j}^{*},z_{j+1}^{*}),(\Delta_{1}^{*},\ldots,\Delta_{j}^{*},\Delta_{j+1}))$. Of course it is not possible to check all real numbers in $[0,1)$ as values of $\Delta_{1},\ldots,{\Delta_{s}}$. We argue that it is sufficient to restrict the set of possible shift components to the odd multiples of $1/(2N)$, that is, to the $N$ values $S_{N}:=\{1/(2N),3/(2N),\ldots,(2N-1)/(2N)\}$.

Our argument for the sufficiency of restricting the search over shifts to the odd multiples of $1/(2N)$ is given in Theorem 2.1 below, in which we show that for any choice of generating vector ${\bm{z}}$, the average of the squared worst-case error over all shifts in $[0,1]^{s}$ differs from the average over the discrete set $S_{N}^{s}$ by a term of order only $1/N^{2}$.

The restriction from the continuous interval $[0,1]$ to the discrete set $S_{N}$ for the shift has previously been considered in [6] in a different CBC algorithm which constructs the components of ${\bm{z}}$ and $\bm{\Delta}$ simultaneously, in the order of $z_{1},\Delta_{1},z_{2},\Delta_{2},\ldots$.

Now we discuss the error with respect to the shift $\bm{\Delta}^{*}$ obtained by the present CBC for shift algorithm. Let us define the ratio:

[TABLE]

Then, from the definition of the worst-case error and using (1.4) we have the the following error bound for the present CBC algorithm:

[TABLE]

for all $\lambda\in(1/2,1]$, which is an explicit and deterministic error bound in which in any practical situation $\kappa(N,s)$ is a known constant. Numerical experiments in Section 3 suggest that $\kappa(N,s)$ can often be smaller than $1$, making the derandomised option attractive in practice.

It should be said the presented CBC for shift algorithm is expensive: the cost of a single evaluation of the worst-case error (1.2) is of order $sN^{2}$ in the simplest case of product weights, and therefore the cost of a search over $N$ values of the shift up to dimension $s$ is of order $sN^{3}$ (if we store the products during the search). But the cost is an off-line cost, since spare computing capacity can be used to complement existing CBC vectors ${\bm{z}}^{*}$ for randomly shifted lattice rules by deterministic shifts $\bm{\Delta}^{*}$ generated by the CBC for shift algorithm.

2 Error analysis

In this section, we show that for any choice of generating vector ${\bm{z}}$, the squared worst-case error with shift averaged over $S_{N}^{s}$, defined by

[TABLE]

differs from the average of the squared worst-case error over all shifts $[e^{\rm{sh}}_{N,s}(\bm{z})]^{2}$ by a term of order only ${1/N^{2}}$.

Theorem 2.1**.**

For arbitrary ${\bm{z}}\in\{1,\dotsc,N-1\}^{s}$, with $e_{N,s}^{\rm{sh}}(\bm{z})$ and $e_{N,s}^{\frac{1}{2}\rm{sh}}({\bm{z}})$ as defined in (1.3) and (2.1), respectively, we have

[TABLE]

Proof.

We see from (1.2) that

[TABLE]

where we write for $k,k^{\prime}=1,\ldots,N$, $j=1,\ldots,s$, $m=1,\ldots,N$,

[TABLE]

Since $|B_{2}(x)|\leq 1/6$ for all $x\in[0,1]$ and $|(x-1/2)(y-1/2)|\leq 1/4$ for all $x,y\in[0,1)$, we have trivially $|a_{j}^{k,k^{\prime}}|\leq 1/3$ and $|b_{j}^{k,k^{\prime}}|\leq 1/3$. It follows by induction that |\prod_{j\in{\mathrm{\mathfrak{u}}}}a_{j}^{k,k^{\prime}}-\prod_{j\in{\mathrm{\mathfrak{u}}}}b_{j}^{k,k^{\prime}}|\leq\big{(}\frac{1}{3}\big{)}^{|\mathfrak{u}|-1}\sum_{j\in\mathfrak{u}}|a_{j}^{k,k^{\prime}}-b_{j}^{k,k^{\prime}}|.

We therefore consider the difference

[TABLE]

which is precisely the error of a composite midpoint rule approximation to the integral of $A_{k,k^{\prime},z_{j}}(\Delta)=\{\frac{kz_{j}}{N}+\Delta\}\{\frac{k^{\prime}z_{j}}{N}+\Delta\}-\frac{1}{2}\{\frac{kz_{j}}{N}+\Delta\}-\frac{1}{2}\{\frac{k^{\prime}z_{j}}{N}+\Delta\}+\frac{1}{4}$.

Since $\frac{kz_{j}}{N}$ is a multiple of $\frac{1}{N}$, the function $\{\frac{kz_{j}}{N}+\Delta\}$ is linear in $\Delta$ on each subinterval $[\frac{m-1}{N},\frac{m}{N})$ of length $\frac{1}{N}$, and so the midpoint rule is exact on each subinterval. The same conclusion holds for $\{\frac{k^{\prime}z_{j}}{N}+\Delta\}$.

On the other hand, the function $\{\frac{kz_{j}}{N}+\Delta\}\{\frac{k^{\prime}z_{j}}{N}+\Delta\}$ is quadratic in $\Delta$ on each subinterval $[\frac{m-1}{N},\frac{m}{N})$, and its second derivative is the constant function $2$, which is uniformly continuous on $(\frac{m-1}{N},\frac{m}{N})$ and can be uniquely extended to $[\frac{m-1}{N},\frac{m}{N}]$. Therefore, the midpoint rule has error bounded by $\frac{2}{24}\frac{1}{N^{3}}$ on each subinterval, leading to the total error $|a_{j}^{k,k^{\prime}}-b_{j}^{k,k^{\prime}}|\leq\frac{1}{12N^{2}}$, and in turn yielding |\prod_{j\in{\mathrm{\mathfrak{u}}}}a_{j}^{k,k^{\prime}}-\prod_{j\in{\mathrm{\mathfrak{u}}}}b_{j}^{k,k^{\prime}}|\leq\big{(}\frac{1}{3}\big{)}^{|\mathfrak{u}|-1}\frac{|{\mathrm{\mathfrak{u}}}|}{12N^{2}}. This completes the proof. ∎

3 CBC for shift algorithm

Theorem 2.1 given in the previous section provides a good motivation to consider the following algorithm.

4 Numerical results

We ran the CBC for shift algorithm in weighted unanchored Sobolev spaces with product weights $\gamma_{j}=1/j^{2}$, $\gamma_{j}=0.9^{j}$, $\gamma_{j}=0.75^{j}$, and $\gamma_{j}=0.5^{j}$, with the number of points $N=1024$ and $2048$. We used the lattice generating vectors ${\bm{z}}^{*}$ from [4].

Table 1 shows the values of the indices $m_{s}^{*}$ for the components of the shifts $\Delta_{s}^{*}=(2m_{s}^{*}-1)/(2N)$ together with the values of $\kappa(s,N)$, for the case $N=2048$ and $\gamma_{j}=1/j^{2}$. As a comparison, we provide also the values of the ratio (1.5) with $\bm{\Delta}^{*}$ replaced by the zero shift vector, denoting the new ratio by $\kappa_{0}(N,s)$. We see that $\kappa(s,N)$ is less than $1$, whereas $\kappa_{0}(s,N)$ exceeds $1$.

Table 2 shows the same values for the case $\gamma_{j}=0.5^{j}$. Again, we see that $\kappa(s,N)$ is less than $1$, whereas $\kappa_{0}(s,N)$ exceeds $1$. The same observation holds for the other cases that we considered.

Acknowledgements

We gratefully acknowledge the financial support from the Australian Research Council (DP180101356).