In this paper we construct polynomial lattice rules which have, in some sense, small gain coefficients using a component-by-component approach. The gain coefficients, as introduced by Owen, indicate to what degree the method improves upon Monte Carlo. We show that the variance of an estimator based on a scrambled polynomial lattice rule constructed component-by-component decays at a rate of $N^{-(2α+ 1) +δ}$, for all $δ>0$, assuming that the function under consideration has bounded fractional variation of order $α$ and where $N$ denotes the number of quadrature points. An analogous result is obtained for Korobov polynomial lattice rules. It is also established that these rules are almost optimal for the function space considered in this paper. Furthermore, we discuss the implementation of the component-by-component approach and show how to reduce the computational cost associated with it. Finally, we present numerical results comparing scrambled polynomial lattice rules and scrambled digital nets.
We study quasi-Monte Carlo (QMC) methods for numerical integration of multivariate functions defined over the high-dimensional unit cube. Lattice rules and polynomial lattice rules, which are special classes of QMC methods, have been intensively studied and the so-called component-by-component (CBC) algorithm has been well-established to construct rules which achieve the almost optimal rate of convergence with good tractability properties for given smoothness and set of weights. Since the CBC algorithm constructs rules for given smoothness and weights, not much is known when such rules are use…
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 dimension $d$ may be in the hundreds or thousands. We study a construction method that assembles the generating vector, which is in this case a vector of polynomials over a finite field, of the polynomial lattice rule in a digit-by-digit (or, equivalently, coefficient-by-coefficient) fashion. As we will show, the integration error of the corresponding QMC rules…
We study the efficient construction of good polynomial lattice rules, which are special instances of quasi-Monte Carlo (QMC) methods. The integration rules obtained are of particular interest for the approximation of multivariate integrals in weighted Walsh spaces. In particular, we study a construction algorithm which assembles the components of the generating vector, which is in this case a vector of polynomials over a finite field, of the polynomial lattice rule in a component-wise fashion. We show that the constructed QMC rules achieve the almost optimal error convergence order in the func…
Despite extensive research on symmetric polynomial quadrature rules for triangles, as well as approaches to their calculation, few studies have focused on non-polynomial functions, particularly on their integration using symmetric triangle rules. In this paper, we present two approaches to computing symmetric triangle rules for singular integrands by developing rules that can integrate arbitrary functions. The first approach is well suited for a moderate amount of points and retains much of the efficiency of polynomial quadrature rules. The second approach better addresses large amounts of poi…
Construction of interlaced scrambled polynomial lattice rules of arbitrary high order
Graduate School of Engineering, The University of Tokyo（东京大学工学研究科）；School of Mathematics and Statistics, The University of New South Wales（新南威尔士大学数学与统计学院）
On the role of weak Marcinkiewicz-Zygmund constants in polynomial approximation by orthogonal bases
School of Mathematics and Statistics, Guizhou University（贵州大学数学与统计学院）；Department of Mathematics "Tullio Levi-Civita", University of Padova（帕多瓦大学蒂利奥·莱维-奇维塔数学系）
Generalized eigenvalue methods for Gaussian quadrature rules
Georgia Institute of Technology（佐治亚理工学院）；Technische Universität Berlin（柏林工业大学）；Uit – The Arctic University of Norway（挪威北极大学）；Universität Konstanz（康斯坦茨大学）；North Carolina State University（北卡罗来纳州立大学）
