Related papers: Fast construction of higher order digital nets for…
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 investigate multivariate integration in weighted unanchored Sobolev spaces of smoothness of arbitrarily high order. As quadrature points we employ higher order polynomial lattice point sets over $\mathbb{F}_{2}$ which are…
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…
We study numerical integration of smooth functions defined over the $s$-dimensional unit cube. A recent work by Dick et al. (2019) has introduced so-called extrapolated polynomial lattice rules, which achieve the almost optimal rate of…
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…
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…
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…
Lattice rules are among the most prominently studied quasi-Monte Carlo methods to approximate multivariate integrals. A rank-1 lattice rule to approximate an $s$-dimensional integral is fully specified by its generating vector $\mathbf{z}…
We investigate quasi-Monte Carlo integration using higher order digital nets in weighted Sobolev spaces of arbitrary fixed smoothness $\alpha \in \mathbb{N}$, $\alpha \ge 2$, defined over the $s$-dimensional unit cube. We prove that…
In this paper we study quasi-Monte Carlo integration of smooth functions using digital nets. We fold digital nets over $\mathbb{Z}_{b}$ by means of the $b$-adic tent transformation, which has recently been introduced by the authors, and…
Lattice rules and polynomial lattice rules are quadrature rules for approximating integrals over the $s$-dimensional unit cube. Since no explicit constructions of such quadrature methods are known for dimensions $s > 2$, one usually has to…
Quasi-Monte Carlo (QMC) quadrature rules using higher order digital nets and sequences have been shown to achieve the almost optimal rate of convergence of the worst-case error in Sobolev spaces of arbitrary fixed smoothness $\alpha\in…
We investigate quasi-Monte Carlo rules for the numerical integration of multivariate periodic functions from Besov spaces $S^r_{p,q}B(\mathbb{T}^d)$ with dominating mixed smoothness $1/p<r<2$. We show that order 2 digital nets achieve the…
We consider the efficient construction of polynomial lattice rules, which are special cases of so-called quasi-Monte Carlo (QMC) rules. These are of particular interest for the approximate computation of multivariate integrals where the…
In this paper we give explicit constructions of point sets in the $s$ dimensional unit cube yielding quasi-Monte Carlo algorithms which achieve the optimal rate of convergence of the worst-case error for numerically integrating high…
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…
Lattice rules are among the most prominently studied quasi-Monte Carlo methods to approximate multivariate integrals. A rank-$1$ lattice rule to approximate an $s$-dimensional integral is fully specified by its \emph{generating vector}…
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…
Frolov's cubature formula on the unit hypercube has been considered important since it attains an optimal rate of convergence for various function spaces. Its integration nodes are given by shrinking a suitable full rank…
The (fast) component-by-component (CBC) algorithm is an efficient tool for the construction of generating vectors for quasi-Monte Carlo rank-1 lattice rules in weighted reproducing kernel Hilbert spaces. We consider product weights, which…