Weighted integration over a cube based on digital nets and sequences

09/15/2020
by   Josef Dick, et al.
0

Quasi-Monte Carlo (QMC) methods are equal weight quadrature rules to approximate integrals over the unit cube with respect to the uniform measure. In this paper we discuss QMC integration with respect to general product measures defined on an arbitrary cube. We only require that the cumulative distribution function is invertible. We develop a worst-case error bound and study the dependence of the error on the number of points and the dimension for digital nets and sequences as well as polynomial lattice point sets, which are mapped to the domain using the inverse cumulative distribution function. We do not require any smoothness properties of the probability density function and the worst-case error does not depend on the particular choice of density function and its smoothness. The component-by-component construction of polynomial lattice rules is based on a criterion which depends only on the size of the cube but is otherwise independent of the product measure.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

12/23/2019

Stability of lattice rules and polynomial lattice rules constructed by the component-by-component algorithm

We study quasi-Monte Carlo (QMC) methods for numerical integration of mu...
01/24/2022

Construction-free median quasi-Monte Carlo rules for function spaces with unspecified smoothness and general weights

We study quasi-Monte Carlo (QMC) integration of smooth functions defined...
12/18/2020

A Tool for Custom Construction of QMC and RQMC Point Sets

We present LatNet Builder, a software tool to find good parameters for l...
01/16/2020

On Quasi-Monte Carlo Methods in Weighted ANOVA Spaces

In the present paper we study quasi-Monte Carlo rules for approximating ...
09/23/2021

Component-by-component construction of randomized rank-1 lattice rules achieving almost the optimal randomized error rate

We study a randomized quadrature algorithm to approximate the integral o...
10/05/2019

An algorithm to compute the t-value of a digital net and of its projections

Digital nets are among the most successful methods to construct low-disc...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.