Related papers: Component-by-component construction of randomized …
In this paper we present the first known deterministic algorithm for the construction of multiple rank-1 lattices for the approximation of periodic functions of many variables. The algorithm works by converting a potentially large…
In this paper we show error bounds for randomly subsampled rank-1 lattices. We pay particular attention to the ratio of the size of the subset to the size of the initial lattice, which is decisive for the computational complexity. In the…
Approximating multivariate periodic functions in weighted Korobov spaces via rank-1 lattices is fundamentally limited by frequency aliasing. Existing optimal-rate methods rely on randomized constructions or large pre-computations. We…
We show how to obtain a fast component-by-component construction algorithm for higher order polynomial lattice rules. Such rules are useful for multivariate quadrature of high-dimensional smooth functions over the unit cube as they achieve…
We study multivariate numerical integration of smooth functions in weighted Sobolev spaces with dominating mixed smoothness $\alpha\geq 2$ defined over the $s$-dimensional unit cube. We propose a new quasi-Monte Carlo (QMC)-based quadrature…
An existence result is presented for the worst-case error of lattice rules for high dimensional integration over the unit cube, in an unanchored weighted space of functions with square-integrable mixed first derivatives. Existing studies…
Higher order scrambled digital nets are randomized quasi-Monte Carlo rules which have recently been introduced in [J. Dick, Ann. Statist., 39 (2011), 1372--1398] and shown to achieve the optimal rate of convergence of the root mean square…
In this paper, we study an efficient algorithm for constructing point sets underlying quasi-Monte Carlo integration rules for weighted Korobov classes. The algorithm presented is a reduced fast component-by-component digit-by-digit…
We study multivariate integration over the $s$-dimensional unit cube in a weighted space of infinitely differentiable functions. It is known from a recent result by Suzuki that there exists a good quasi-Monte Carlo (QMC) rule which achieves…
In this work, we consider the approximate reconstruction of high-dimensional periodic functions based on sampling values. As sampling schemes, we utilize so-called reconstructing multiple rank-1 lattices, which combine several preferable…
We prove upper bounds on the order of convergence of lattice based algorithms for numerical integration in function spaces of dominating mixed smoothness on the unit cube with homogeneous boundary condition. More precisely, we study…
In a recent paper by the same authors, we provided a theoretical foundation for the component-by-component (CBC) construction of lattice algorithms for multivariate $L_2$ approximation in the worst case setting, for functions in a periodic…
In this paper, we propose a novel algorithm for function approximation in a weighted Korobov space based on shifted rank-1 lattice rules. To mitigate aliasing errors inherent in lattice-based Fourier coefficient estimation, we employ…
Rank-1 lattice rules are a class of equally weighted quasi-Monte Carlo methods that achieve essentially linear convergence rates for functions in a reproducing kernel Hilbert space (RKHS) characterized by square-integrable first-order mixed…
This paper investigates the construction of space-filling designs for computer experiments. The space-filling property is characterized by the covering and separation radii of a design, which are integrated through the unified criterion of…
Quadrature rules using higher order digital nets and sequences are known to exploit the smoothness of a function for numerical integration and to achieve an improved rate of convergence as compared to classical digital nets and sequences…
In this paper, we study an efficient algorithm for constructing node sets of high-quality quasi-Monte Carlo integration rules for weighted Korobov, Walsh, and Sobolev spaces. The algorithm presented is a reduced fast successive coordinate…
This study focuses on constructing efficient rank-1 lattices that enable the exact integration and reconstruction of functions within Chebyshev spaces, based on finite lower index sets. We establish the equivalence of different…
This short article studies a deterministic quasi-Monte Carlo lattice rule in weighted unanchored Sobolev spaces of smoothness $1$. Building on the error analysis by Kazashi and Sloan, we prove the existence of unshifted rank-1 lattice rules…
We study multivariate integration of functions that are invariant under permutations (of subsets) of their arguments. We find an upper bound for the $n$th minimal worst case error and show that under certain conditions, it can be bounded…