We investigate the periodic L_2-discrepancy of infinite sequences §_d in [0,1)^d and its analytic counterpart, the diaphony. We prove that infinite order-2 digital sequences over 𝔽_2 attain the optimal order L_2,N^ per(§_d) ≤ C_d (log N)^d/2/N for all N ∈ℕ∖{1}, matching known lower bounds for infinitely many N ∈ℕ. This confirms the conjectured optimality of order-2 constructions. By this result, we improve upon previously known constructions using order-5 digital sequences, and reduce the underlying dimension for the interlacing construction from 5d to 2d, significantly improving practicality. We establish our bounds within a broader framework of quasi-Monte Carlo integration for periodic Besov spaces S_p,q^rB(𝕋^d) with dominating mixed smoothness r ∈ (1/p,2), where p,q∈ [1,∞]. Rules based on infinite order-2 digital sequences yield worst-case errors of order (log N)^(d-1)(1-1/q) / N^min(r,1) for r ≠1, and (log N)^d(1-1/q)/N for r=1, for all N ∈ℕ∖{1}, while preserving extensibility in N.

In this paper, we study the problem of multivariate L_2-approximation of functions belonging to a weighted Korobov space. We propose and analyze a median lattice-based algorithm, inspired by median integration rules, which have attracted significant attention in the theory of quasi-Monte Carlo methods. Our algorithm approximates the Fourier coefficients associated with a suitably chosen frequency index set, where each coefficient is estimated by taking the median over approximations from randomly shifted rank-1 lattice rules with independently chosen generating vectors. We prove that the algorithm achieves, with high probability, a convergence rate of the L_2-approximation error that is arbitrarily close to optimal with respect to the number of function evaluations. Furthermore, we show that the error bound depends only polynomially on the dimension, or is even independent of the dimension, under certain summability conditions on the weights. Numerical experiments illustrate the performance of the proposed median lattice-based algorithm.

In the recent papers “The fast reduced QMC matrix-vector product” (Dick, Ebert, Herrmann, Kritzer, Longo; J. Comput. Appl. Math., 2024) and “Column reduced digital nets” (Anupindi, Kritzer; Numer. Algorithms, 2025), it was proposed to use QMC rules based on reduced digital nets which provide a speed-up in the computation of QMC vector-matrix products that may occur in practical applications. In this paper, we provide upper bounds on the quality parameter of row reduced and column-row reduced digital nets, which are helpful for the error analysis of using reduced point sets as integration nodes in a QMC rule. We also give remarks on further aspects and comparisons of row reduced, column reduced, and column-row reduced digital nets.

We study the worst-case approximation of multivariate periodic functions from the weighted Korobov space H_d,α,γ with smoothness α>1/2 in the Lebesgue norm L_p([0,1]^d) for 1≤ p≤∞. We analyze a median lattice algorithm that reconstructs a truncated Fourier series by approximating the coefficients on a hyperbolic-cross-type index set using R rank-1 lattice sampling rules with independent randomly chosen generating vectors, and then aggregating the resulting coefficient estimators via the componentwise median. For an odd number of repetitions R>1 and an odd prime lattice size N, we prove high-probability error bounds in both L_∞ and L_2. Interpolation then yields the result for all 1 ≤ p≤∞. In particular, with a high probability, the algorithm satisfies err(H_d,α,γ,L_p,A) ≤ C_d,α,β,,p N^- α+ (1/2 - 1/p)_+ + β, 1 ≤ p≤∞, β>0, where (x)_+ = max{x, 0}, N is the number of function evaluations, and the weights and the constant C_d,α,β,,p are independent of N. For p=∞, C_d,α,β,,∞ is dimension-independent under the summability condition ∑_j=1^∞ γ_j^1/(2α)<∞. These results extend recent analyses of median-based lattice approximation in L_2 and complement related multiple-shift lattice approaches, showing that median aggregation yields nearly optimal L_p-approximation rates (up to logarithmic factors and an arbitrarily small loss) in weighted Korobov spaces.

We study embeddings between reproducing kernel Hilbert spaces H(K) of functions of d ∈ℕ∪{∞} variables. The kernels K are superpositions of weighted finite tensor products of a fixed univariate kernel. The basic idea for the embeddings is to compensate a change of the univariate kernel by a suitable transformation of the weights. For the proofs we employ (d ∈ℕ) and develop (d = ∞) a discrete calculus on the cone of all weights, where completely monotone weights play a particular role. We sketch how to apply the embedding results to computational problems, as, e.g., numerical integration or function recovery.

We study the problem of multivariate L_2-approximation of functions in a weighted Korobov space using a median lattice-based algorithm recently proposed by the authors. In the original work, the algorithm requires knowledge of the smoothness and weights of the Korobov space to construct the hyperbolic cross index set, where each coefficient is estimated via the median of approximations obtained from randomly shifted, randomly chosen rank-1 lattice rules. In this paper, we introduce a universal median lattice-based algorithm, which eliminates the need for any prior information on smoothness and weights. Although the tractability property of the algorithm slightly deteriorates, we prove that, for individual functions in the Korobov space with arbitrary smoothness and (downward-closed) weights, it achieves an L_2-approximation error arbitrarily close to the optimal rate with respect to the number of function evaluations.

Digital nets provide an efficient way to generate integration nodes of quasi-Monte Carlo (QMC) rules. For certain applications, as e.g. in uncertainty quantification, we are interested in obtaining a speed-up in computing products of a matrix with the vectors corresponding to the nodes of a QMC rule. In the recent paper The fast reduced QMC matrix-vector product (Dick et al. J. Comput. Appl. Math. 440, 115642 2024), a speed up was obtained by using so-called reduced lattices and row reduced digital nets. In this work, we propose a different multiplication algorithm where we exploit the repetitive structure of column reduced digital nets instead of row reduced digital nets. This method has advantages over the previous one, as it facilitates the error analysis when using the integration nodes in a QMC rule. We also provide an upper bound for the quality parameter of column reduced digital nets, and numerical tests to illustrate the efficiency of the new algorithm.

We consider linear problems in the worst case setting. That is, given a linear operator and a pool of admissible linear measurements, we want to approximate the values of the operator uniformly on a convex and balanced set by means of algorithms that use at most $n$ such measurements. It is known that, in general, linear algorithms do not yield an optimal approximation. However, as we show in this paper, an optimal approximation can always be obtained with a homogeneous algorithm. This is of interest to us for two reasons. First, the homogeneity allows us to extend any error bound on the unit ball to the full input space. Second, homogeneous algorithms are better suited to tackle problems on cones, a scenario that is far less understood than the classical situation of balls. We use the optimality of homogeneous algorithms to prove solvability for a family of problems defined on cones. We illustrate our results by several examples.

We study the approximation of integrals $\int_D f(\boldsymbol{x}^\top A) \mathrm{d} \mu(\boldsymbol{x})$, where $A$ is a matrix, by quasi-Monte Carlo (QMC) rules $N^{-1} \sum_{k=0}^{N-1} f(\boldsymbol{x}_k^\top A)$. We are interested in cases where the main cost arises from calculating the products $\boldsymbol{x}_k^\top A$. We design QMC rules for which the computation of $\boldsymbol{x}_k^\top A$, $k = 0, 1, \ldots, N-1$, can be done fast, and for which the error of the QMC rule is similar to the standard QMC error. We do not require that $A$ has any particular structure. For instance, this approach can be used when approximating the expected value of a function with a multivariate normal random variable with a given covariance matrix, or when approximating the expected value of the solution of a PDE with random coefficients. The speed-up of the computation time is sometimes better and sometimes worse than the fast QMC matrix-vector product from [Dick, Kuo, Le Gia, and Schwab, Fast QMC Matrix-Vector Multiplication, SIAM J. Sci. Comput. 37 (2015)]. As in that paper, our approach applies to (polynomial) lattice point sets, but also to digital nets (we are currently not aware of any approach which allows one to apply the fast method from the aforementioned paper of Dick, Kuo, Le Gia, and Schwab to digital nets). Our method does not use FFT, instead we use repeated values in the quadrature points to derive a reduction in the computation time. This arises from the reduced CBC construction of lattice rules and polynomial lattice rules. The reduced CBC construction has been shown to reduce the computation time for the CBC construction. Here we show that it can also be used to also reduce the computation time of the QMC rule.

A large literature specifies conditions under which the information complexity for a sequence of numerical problems defined for dimensions 1,2,… grows at a moderate rate, i.e., the sequence of problems is tractable. Here, we focus on the situation where the space of available information consists of all linear functionals, and the problems are defined as linear operator mappings between Hilbert spaces. We unify the proofs of known tractability results and generalize a number of existing results. These generalizations are expressed as five theorems that provide equivalent conditions for (strong) tractability in terms of sums of functions of the singular values of the solution operators.

We give an overview of certain aspects of tractability analysis of multivariate problems. This paper is not intended to give a complete account of the subject, but provides an insight into how the theory works for particular types of problems. We mainly focus on linear problems on Hilbert spaces, and mostly allow arbitrary linear information. In such cases, tractability analysis is closely linked to an analysis of the singular values of the operator under consideration. We also highlight the more recent developments regarding exponential and generalized tractability. The theoretical results are illustrated by several examples throughout the article.

Let $f:[0,1]^d\to\mathbb{R}$ be a completely monotone integrand as defined by Aistleitner and Dick (2015) and let points $\boldsymbol{x}_0,\dots,\boldsymbol{x}_{n-1}\in[0,1]^d$ have a non-negative local discrepancy (NNLD) everywhere in $[0,1]^d$. We show how to use these properties to get a non-asymptotic and computable upper bound for the integral of $f$ over $[0,1]^d$. An analogous non-positive local discrepancy (NPLD) property provides a computable lower bound. It has been known since Gabai (1967) that the two dimensional Hammersley points in any base $b\ge2$ have non-negative local discrepancy. Using the probabilistic notion of associated random variables, we generalize Gabai's finding to digital nets in any base $b\ge2$ and any dimension $d\ge1$ when the generator matrices are permutation matrices. We show that permutation matrices cannot attain the best values of the digital net quality parameter when $d\ge3$. As a consequence the computable absolutely sure bounds we provide come with less accurate estimates than the usual digital net estimates do in high dimensions. We are also able to construct high dimensional rank one lattice rules that are NNLD. We show that those lattices do not have good discrepancy properties: any lattice rule with the NNLD property in dimension $d\ge2$ either fails to be projection regular or has all its points on the main diagonal.

Lattice rules are among the most prominently studied quasi-Monte Carlo methods to approximate multivariate integrals. A rank-1 lattice rule for an s-dimensional integral is specified by its generating vector z∈Zs and its number of points N. While there are many results on the existence of “good” rank-1 lattice rules, there are no explicit constructions of good generating vectors for dimensions s≥3. Therefore one resorts to computer search algorithms. In a recent paper by Ebert et al. in the Journal of Complexity, we showed a component-by-component digit-by-digit (CBC-DBD) construction for good generating vectors for integration of functions in weighted Korobov classes equipped with product weights. Here, we generalize this result to arbitrary positive weights, answering an open question from the paper of Ebert et al. We include a section on how the algorithm can be implemented in the case of POD weights, implying that the CBC-DBD construction is competitive with the classical CBC construction.

In Chapter 2 we introduced the Korobov spaces $${\mathcal{H}}_{{{\text{kor}},d,\alpha ,\gamma }}$$ and studied numerical integration using lattice rules for these spaces. The parameter $$\alpha$$ is related to the smoothness of the elements of $${\mathcal{H}}_{{{\text{kor}},d,\alpha ,\gamma }}$$ (see Propositions 2.2 and 2.4) and is called the smoothness parameter of these spaces. For an optimal integration rule the rate of convergence of the worst-case error is essentially of order $$O(N^{ - \alpha } )$$ and therefore reflects the smoothness of the reference space. In the cases considered so far, the smoothness parameter $$\alpha$$ has always been finite and hence we have observed convergence rates of polynomial order.

The so-called p-sets go back to definitions due to Korobov in the 1950s and Hua and Wang in the 1970s. Since then, these sets have been largely ignored because a number of other constructions have been discovered which yield a better convergence rate in terms of the cardinality of the point sets in integration rules.

The discrepancy of the quadrature points is an important quality criterion for QMC integration rules, as we have seen in Section 1.6, where we have studied discrepancy and the Koksma–Hlawka inequality. This of course also applies to lattice rules and the underlying lattice point sets. In this chapter we study the (extreme) discrepancy of lattice point sets and show its relation to the figure of merit R. Based on the latter we obtain efficient constructions of lattice point sets with the almost optimal order of magnitude of discrepancy.

In this chapter, we discuss a randomization method for rank-1 lattice rules that is an adaption of a technique introduced by Bakhvalov in [7] in 1961. We present a randomized algorithm $$A_{N,d}^{{{\text{ran}}}}$$ for numerical integration of elements of the weighted Korobov space $$H_{{{\text{kol}},d,\alpha ,\gamma }}$$ that uses at most N integration nodes and that is based on rank-1 lattice rules as building blocks.

In the preceding chapters, we have seen how (rank-1) lattice rules can be used for numerical integration, in particular in Korobov and Sobolev spaces. Moreover, we have outlined how we can efficiently construct the generating vectors of such (good) lattice rules, e.g., by the CBC algorithm.

In this chapter we discuss the approximation of integrals of the form.