We study numerical integration of multivariate functions defined over the -dimensional unit cube . For an integrable function we denote the integral of by
For an -element set , we consider approximating by
For a Banach space with norm , the worst-case error by the algorithm is defined by
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
where denotes the fractional part of a non-negative real numbers . The resulting QMC algorithm is called a lattice rule with generating vector
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
where we write
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 , 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
where the dot denotes the usual inner product of two vectors and denotes the -th Fourier coefficient defined by
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
Then the weighted Korobov space with smoothness , denoted by , is a reproducing kernel Hilbert space with the reproducing kernel 
where we write and . The inner product of the space is given by
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
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
Define the dual lattice for by
Then the following property is well known.
For and , we have
For a non-empty subset , define
Then the squared worst-case error is given by
There is a concise computable form of the criterion when is a natural number
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.
Let and .
Compute for all and let
If , let and go to Step 2.
for , the set of non-negative real numbers, we only need arithmetic operations with memory. In the case of POD weights, i.e.,
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.
Let , and be given. The generating vector constructed by Algorithm 1 satisfies
for any , where denotes the Riemann zeta function and denotes the Euler totient function defined by
For any , let us write . Then it follows from Jensen’s inequality that
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.,
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
This criterion turns out to be very useful in our context (it has for instance recently also been used in ).
Let and . For any and such that whenever , we have
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
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.
Let , and such that whenever . Let be constructed by Algorithm 1 based on the criterion . Then the following holds true:
For general weights and , assume that there exist such that , ,
Then the worst-case error depends only polynomially on and is bounded by
for some constant which is independent of and . If the above conditions hold for , the worst-case error is bounded independently of .
In particular, in the case of product weights and , assume that there exists such that
Then the worst-case error is independent of and bounded by
for arbitrarily small .
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.
For any real and any , we have
Proof of Theorem 1.
Recalling the definition of , we have
Let us define , for which we have . Moreover, let
Then it is straightforward to see that
Hence it holds for any non-empty subset that
Moreover, by assuming whenever , we obtain a lower bound
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
For the first term in the parenthesis, we have
For the second term in the parenthesis, applying Lemma 2 with and gives
Using these bounds, we obtain
Note that this bound on the inner sum on the expression of also applies to the case .
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
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
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
Moreover, as proven in [1, Theorem 5.35], we have
for any non-empty . By using the lower bound (2) on , the following result holds true.
Let and . For any and such that whenever , we have
Applying the result from Proposition 1, we can prove the following tractability properties. For general weights we use
for any non-empty . We assume that the sum in the case of product weights to ensure a dimension independent bound on the sum