Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1409.5894 · Springer proceedings in mathematics & statistics

Optimal Point Sets for Quasi-Monte Carlo Integration of Bivariate Periodic Functions with Bounded Mixed Derivatives

Aicke Hinrichs, Jens Oettershagen

Published Jan 1, 2016• 26 citations
arXivHTMLPDF
DOI
10.1007/978-3-319-33507-0_19

Related papers

Error bounds for quasi-Monte Carlo integration with uniform point sets

Harald Niederreiter

Journal of Computational and Applied Mathematics•Dec 3, 2002•27 citations

A New Quasi-Monte Carlo Algorithm for Numerical Integration of Smooth Functions

Emanouil I. Atanassov, Ivan Dimov, Mariya K. Durchova

Lecture notes in computer science•Jan 1, 2004•3 citations

Construction of Quasi-Monte Carlo Rules for Multivariate Integration in Spaces of Permutation-Invariant Functions

Dirk Nuyens, Gowri Suryanarayana, Markus Weimar

arXiv:1507.08084•Constructive Approximation•Jan 9, 2017•4 citations

Quasi-Monte Carlo integration of functions of unbounded variation

Ronald F. A. Cools, Tim Pillards

qMC integration of functions of unbounded variation Quasi-Monte Carlo Functions of unbounded variation Conclusion

•Sep 1, 2006•0 citations

Lattice rules and beyond..

Dirk Nuyens

In this talk I will show the beautiful simplicity of approximating high-dimensional integrals using lattice rules. In current engineering applications it is relatively easy to come up with very high and ultra high dimensional integrals, which might be expectations, that depend on hundreds, thousands, or even an infinite number of variables. As the minimal non-trivial tensor product rule, constructed from a 2-point rule, in 100 dimensions already uses 2^{100} = 1267650600228229401496703205376 ≈ 10^{30} function evaluations—which would take 10^{24} seconds at 1 function evaluation per microsecond and we remember the age of the universe to be only approximately 10^{17} seconds—we all know why to use Monte Carlo or quasi-Monte Carlo methods where we can freely choose the number of function evaluations independent of the number of dimensions. While plain vanilla Monte Carlo methods are very robust and easy to apply, in the case of quasi-Monte Carlo methods a lot more details can pop up. I will first introduce the most elegant—in my biased opinion, which I will defend—of all quasi-Monte Carlo methods: the rank-1 lattice rule. This method is defined by one single integer generating vector together with the total number of points in the rule. Approximating a multivariate integral by this method can be done in one line of Matlab code! As one is often interested in successive approximations I will next introduce a rank-1 lattice sequence. Maybe surprisingly, also here, one simple line of Matlab code is sufficient to calculate a segment of a lattice sequence (with a special property). While for Monte Carlo integration the usual assumption is L2 integrability, leading to an error bound of O(N^{−1/2}). For quasi-Monte Carlo methods we generally work with more involved function space settings, leading to error bounds of the form O(N^{−α}) with α ≥ 1/2. I will show the standard function space settings and dive into the weighted function spaces which are being used in current theory to be able to make the error bounds independent of the number of dimensions. We will gloss over the Korobov space of periodic smooth functions, the unanchored Sobolev space with first mixed derivatives, and the cosine space which is a special space of smooth non-periodic functions. All of which are spaces of so-called mixed dominating smoothness, which in terms of mixed derivatives would mean ∂^{k_1+···+k_s} /(∂x_1^{k_1} · · · ∂x_s^{k_s}) exists for all (k_1, . . . , k_s) ≤ (α, . . . , α). For non-periodic function spaces lattice rules and sequences can only reach O(N^{−1}) convergence. To arrive at higher-order convergence of O(N^{−α}) for non-periodic functions we dive into higher-order Sobolev spaces (of mixed dominating smoothness) in combination with digital nets, but here disguised as a lattice rule, namely polynomial lattice rules. Here, all arithmetic is over a finite field (we will choose the one of order 2) and our lattice rules now take the form of a lattice of polynomials which we map back to points in the unit cube. Higher order can now be obtained by means of the interlacing method from Josef Dick. Although more complicated, again, given the generating matrices of the digital net (as a bunch of integers), and two lines for a magical de Bruijn sequence, we can approximate an integral by such a digital sequence with a single line of (complicated) Matlab code. While Matlab/Octave one-liners might be fun, and somehow instructive, I will also point you to my online resources where code is available to (a) construct good lattice rules, lattice sequences and polynomial lattice rules (for product, order-dependent, product-and-order-dependent aka POD, and smoothness-driven product-and-order dependent aka SPOD) weighted spaces, and (b) generate points given the generating vector or matrices. Armed with this software you will be able to attack your own problems in no time. This talk is based upon the results of collaborations with many colleagues.

•Aug 1, 2016•0 citations

Quasi-Monte Carlo methods for mixture distributions and approximated distributions via piecewise linear interpolation

Tiangang Cui, Josef Dick, Friedrich Pillichshammer

Abstract We study numerical integration over bounded regions in $$\mathbb {R}^s$$ R s , $$s \ge 1$$ s ≥ 1 , with respect to some probability measure. We replace random sampling with quasi-Monte Carlo methods, where the underlying point set is derived from deterministic constructions which aim to fill the space more evenly than random points. Ordinarily, such quasi-Monte Carlo point sets are designed for the uniform measure, and the theory only works for product measures when a coordinate-wise transformation is applied. Going beyond this setting, we first consider the case where the target density is a mixture distribution where each term in the mixture comes from a product distribution. Next, we consider target densities which can be approximated with such mixture distributions. In order to be able to use an approximation of the target density, we require the approximation to be a sum of coordinate-wise products and that the approximation is positive everywhere (so that they can be re-scaled to probability density functions). We use tensor product hat function approximations for this purpose here, since a hat function approximation of a positive function is itself positive. We also study more complex algorithms, where we first approximate the target density with a general Gaussian mixture distribution and approximate this mixture distribution with an adaptive hat function approximation on rotated intervals. The Gaussian mixture approximation allows us (at least to some degree) to locate the essential parts of the target density, whereas the adaptive hat function approximation allows us to approximate the finer structure of the target density. We prove convergence rates for each of the integration techniques based on quasi-Monte Carlo sampling for integrands with bounded partial mixed derivatives. The employed algorithms are based on digital (t, s)-sequences over the finite field $$\mathbb {F}_2$$ F 2 and an inversion method. Numerical examples illustrate the performance of the algorithms for some target densities and integrands.

arXiv:2304.14786•Advances in Computational Mathematics•Feb 1, 2025•3 citations
In this talk I will show the beautiful simplicity of approximating high-dimensional integrals using lattice rules. In current engineering applications it is relatively easy to come up with very high and ultra high dimensional integrals, which might be expectations, that depend on hundreds, thousands, or even an infinite number of variables. As the minimal non-trivial tensor product rule, constructed from a 2-point rule, in 100 dimensions already uses 2^{100} = 1267650600228229401496703205376 ≈ 10^{30} function evaluations—which would take 10^{24} seconds at 1 function evaluation per microsecond and we remember the age of the universe to be only approximately 10^{17} seconds—we all know why to use Monte Carlo or quasi-Monte Carlo methods where we can freely choose the number of function evaluations independent of the number of dimensions. While plain vanilla Monte Carlo methods are very robust and easy to apply, in the case of quasi-Monte Carlo methods a lot more details can pop up. I will first introduce the most elegant—in my biased opinion, which I will defend—of all quasi-Monte Carlo methods: the rank-1 lattice rule. This method is defined by one single integer generating vector together with the total number of points in the rule. Approximating a multivariate integral by this method can be done in one line of Matlab code! As one is often interested in successive approximations I will next introduce a rank-1 lattice sequence. Maybe surprisingly, also here, one simple line of Matlab code is sufficient to calculate a segment of a lattice sequence (with a special property). While for Monte Carlo integration the usual assumption is L2 integrability, leading to an error bound of O(N^{−1/2}). For quasi-Monte Carlo methods we generally work with more involved function space settings, leading to error bounds of the form O(N^{−α}) with α ≥ 1/2. I will show the standard function space settings and dive into the weighted function spaces which are being used in current theory to be able to make the error bounds independent of the number of dimensions. We will gloss over the Korobov space of periodic smooth functions, the unanchored Sobolev space with first mixed derivatives, and the cosine space which is a special space of smooth non-periodic functions. All of which are spaces of so-called mixed dominating smoothness, which in terms of mixed derivatives would mean ∂^{k_1+···+k_s} /(∂x_1^{k_1} · · · ∂x_s^{k_s}) exists for all (k_1, . . . , k_s) ≤ (α, . . . , α). For non-periodic function spaces lattice rules and sequences can only reach O(N^{−1}) convergence. To arrive at higher-order convergence of O(N^{−α}) for non-periodic functions we dive into higher-order Sobolev spaces (of mixed dominating smoothness) in combination with digital nets, but here disguised as a lattice rule, namely polynomial lattice rules. Here, all arithmetic is over a finite field (we will choose the one of order 2) and our lattice rules now take the form of a lattice of polynomials which we map back to points in the unit cube. Higher order can now be obtained by means of the interlacing method from Josef Dick. Although more complicated, again, given the generating matrices of the digital net (as a bunch of integers), and two lines for a magical de Bruijn sequence, we can approximate an integral by such a digital sequence with a single line of (complicated) Matlab code. While Matlab/Octave one-liners might be fun, and somehow instructive, I will also point you to my online resources where code is available to (a) construct good lattice rules, lattice sequences and polynomial lattice rules (for product, order-dependent, product-and-order-dependent aka POD, and smoothness-driven product-and-order dependent aka SPOD) weighted spaces, and (b) generate points given the generating vector or matrices. Armed with this software you will be able to attack your own problems in no time. This talk is based upon the results of collaborations with many colleagues.
Abstract We study numerical integration over bounded regions in Rs\mathbb {R}^sRs R s , s≥1s \ge 1s≥1 s ≥ 1 , with respect to some probability measure. We replace random sampling with quasi-Monte Carlo methods, where the underlying point set is derived from deterministic constructions which aim to fill the space more evenly than random points. Ordinarily, such quasi-Monte Carlo point sets are designed for the uniform measure, and the theory only works for product measures when a coordinate-wise transformation is applied. Going beyond this setting, we first consider the case where the target density is a mixture distribution where each term in the mixture comes from a product distribution. Next, we consider target densities which can be approximated with such mixture distributions. In order to be able to use an approximation of the target density, we require the approximation to be a sum of coordinate-wise products and that the approximation is positive everywhere (so that they can be re-scaled to probability density functions). We use tensor product hat function approximations for this purpose here, since a hat function approximation of a positive function is itself positive. We also study more complex algorithms, where we first approximate the target density with a general Gaussian mixture distribution and approximate this mixture distribution with an adaptive hat function approximation on rotated intervals. The Gaussian mixture approximation allows us (at least to some degree) to locate the essential parts of the target density, whereas the adaptive hat function approximation allows us to approximate the finer structure of the target density. We prove convergence rates for each of the integration techniques based on quasi-Monte Carlo sampling for integrands with bounded partial mixed derivatives. The employed algorithms are based on digital (t, s)-sequences over the finite field F2\mathbb {F}_2F2​ F 2 and an inversion method. Numerical examples illustrate the performance of the algorithms for some target densities and integrands.