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A Monte Carlo method for integration of multivariate smooth functions I: Sobolev spaces

Mario Ullrich

Published Apr 20, 2016• 1 citation
OpenAlex

Abstract

We study a Monte Carlo algorithm that is based on a specific (randomly shifted and dilated) lattice point set. The main result of this paper is that the mean squared error for a given compactly supported, square-integrable function is bounded by $n^{-1/2}$ times the $L_2$-norm of the Fourier transform outside a region around the origin, where $n$ is the expected number of function evaluations. As corollaries we obtain the order of convergence for the Sobolev spaces $H^s_p$ with isotropic, anisotropic or mixed smoothness for all values of the parameters. This proves, in particular, that the optimal order of convergence in the latter case is $n^{-s-1/2}$ for $p\ge2$, which is, in contrast to the case of deterministic algorithms, independent of the dimension. This shows that Monte Carlo algorithms can improve the order by more than $n^{-1/2}$ for a whole class of practically important function classes. All results carry over to functions defined on the unit cube without boundary conditions.

Related papers

On the Randomization of Frolov's Algorithm for Multivariate Integration

David Krieg

We are concerned with the numerical integration of functions from the Sobolev space $H^{r,\text{mix}}([0,1]^d)$ of dominating mixed smoothness $r\in\mathbb{N}$ over the $d$-dimensional unit cube. In 1976, K. K. Frolov introduced a deterministic quadrature rule whose worst case error has the order $n^{-r} \, (\log n)^{(d-1)/2}$ with respect to the number $n$ of function evaluations. This is known to be optimal. 39 years later, Erich Novak and me introduced a randomized version of this algorithm using $d$ random dilations. We showed that its error is bounded above by a constant multiple of $n^{-r-1/2} \, (\log n)^{(d-1)/2}$ in expectation and by $n^{-r} \, (\log n)^{(d-1)/2}$ almost surely. The main term $n^{-r-1/2}$ is again optimal and it turns out that the very same algorithm is also optimal for the isotropic Sobolev space $H^s([0,1]^d)$ of smoothness $s>d/2$. We also added a random shift to this algorithm to make it unbiased. Just recently, Mario Ullrich proved that the expected error of the resulting algorithm on $H^{r,\text{mix}}([0,1]^d)$ is even bounded above by $n^{-r-1/2}$. This thesis is a review of the mentioned upper bounds and their proofs.

arXiv:1603.04637•Mar 15, 2016•0 citations

Richardson Extrapolation of Polynomial Lattice Rules

Josef Dick, Takashi Goda, Takehito Yoshiki

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 rule, named the extrapolated polynomial lattice rule, which achieves the almost optimal rate of convergence. Extrapolated polynomial lattice rules are constructed in two steps: (i) construction of classical polynomial lattice rules over $\Bbb{F}_b$ with $\alpha$ consecutive sizes of nodes, $b^{m-\alpha+1},\ldots,b^{m}$, and (ii) recursive application of Richardson extrapolation to a chain of $\alpha$ approximate values of the integral obtained by consecutive polynomial lattice rules. We prove the existence of good extrapolated polynomial lattice rules achieving the almost optimal order of convergence of the worst-case error in Sobolev spaces with general weights. Then, by restricting to product weights, we show that such good extrapolated polynomial lattice rules can be constructed by the fast component-by-component algorithm under a computable quality criterion. The required total construction cost is of order $(s+\alpha)N\log N$, which improves the currently known result for the interlaced polynomial lattice rule, which is of order $s\alpha N\log N$. We also study the dependence of the worst-case error bound on the dimension. A big advantage of our method compared to interlaced polynomial lattice rules is that the fast QMC matrix vector method can be used in this setting, while still achieving the same rate of convergence. Such a method was previously not known. Numerical experiments for test integrands support our theoretical result.

arXiv:1707.03989•SIAM Journal on Numerical Analysis•Jan 1, 2019•11 citations

A note on unshifted lattice rules for high-dimensional integration in weighted unanchored Sobolev spaces

Takashi Goda

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 that achieve a worst-case error of O ( n − 1 / 4 ( log n ) 1 / 2 ) , with the implied constant independent of the dimension, under certain summability conditions on the weights. Although this convergence rate is inferior to the one achievable for the shifted-averaged root mean squared worst-case error, the result does not rely on random shifting or transformation and holds unconditionally without any conjecture, in contrast to the result of Kazashi and Sloan.

arXiv:2504.14768•Applied Mathematics Letters•May 28, 2025•1 citation

A random integration algorithm for high-dimensional function spaces

Liang Chen, Minqiang Xu, Haizhang Zhang

We introduce a novel random integration algorithm that exhibits a high convergence order for functions characterized by sparse frequencies or rapidly decaying Fourier coefficients. Specifically, for integration in periodic isotropic Sobolev spaces and isotropic Sobolev spaces with compact support, our approach achieves a nearly optimal root mean square error bound. In contrast to previous nearly optimal algorithms, our method exhibits polynomial tractability. Our integration algorithm also enjoys nearly optimal bound for weighted Sobolev space. By incorporating the trick of change of variable, our algorithm is proven to achieve the semi-exponential convergence order for the integration of analytic functions, which marks a significant improvement over the previously obtained super-polynomial convergence order. Furthermore, for integration involving Wiener-type functions, the sample complexity of our algorithm remains independent of the decay rate of the Fourier coefficients.

arXiv:2406.16627•Mathematics of Computation•Dec 30, 2025•2 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

A Universal Median Quasi-Monte Carlo Integration

Takashi Goda, Kosuke Suzuki, Makoto Matsumoto

Abstract. We study quasi-Monte Carlo (QMC) integration over the multidimensional unit cube in several weighted function spaces with different smoothness classes. We consider approximating the integral of a function by the median of several integral estimates under independent and random choices of the underlying QMC point sets (either linearly scrambled digital nets or infinite-precision polynomial lattice point sets). Even though our approach does not require any information on the smoothness and weights of a target function space as an input, we can prove a probabilistic upper bound on the worst-case error for the respective weighted function space, where the failure probability converges to 0 exponentially fast as the number of estimates increases. Our obtained rates of convergence are nearly optimal for function spaces with finite smoothness, and we can attain a dimension-independent super-polynomial convergence for a class of infinitely differentiable functions. This implies that our median-based QMC rule is universal in the sense that it does not need to be adjusted to the smoothness and the weights of the function spaces and yet exhibits the nearly optimal rate of convergence. Numerical experiments support our theoretical results.

arXiv:2209.13186•SIAM Journal on Numerical Analysis•Feb 16, 2024•12 citations
We study a Monte Carlo algorithm that is based on a specific (randomly shifted and dilated) lattice point set. The main result of this paper is that the mean squared error for a given compactly supported, square-integrable function is bounded by n−1/2n^{-1/2}n−1/2 times the L2L_2L2​-norm of the Fourier transform outside a region around the origin, where nnn is the expected number of function evaluations. As corollaries we obtain the order of convergence for the Sobolev spaces HpsH^s_pHps​ with isotropic, anisotropic or mixed smoothness for all values of the parameters. This proves, in particular, that the optimal order of convergence in the latter case is n−s−1/2n^{-s-1/2}n−s−1/2 for p≥2p\ge2p≥2, which is, in contrast to the case of deterministic algorithms, independent of the dimension. This shows that Monte Carlo algorithms can improve the order by more than n−1/2n^{-1/2}n−1/2 for a whole class of practically important function classes. All results carry over to functions defined on the unit cube without boundary conditions.
We are concerned with the numerical integration of functions from the Sobolev space Hr,mix([0,1]d)H^{r,\text{mix}}([0,1]^d)Hr,mix([0,1]d) of dominating mixed smoothness r∈Nr\in\mathbb{N}r∈N over the ddd-dimensional unit cube. In 1976, K. K. Frolov introduced a deterministic quadrature rule whose worst case error has the order n−r (log⁡n)(d−1)/2n^{-r} \, (\log n)^{(d-1)/2}n−r(logn)(d−1)/2 with respect to the number nnn of function evaluations. This is known to be optimal. 39 years later, Erich Novak and me introduced a randomized version of this algorithm using ddd random dilations. We showed that its error is bounded above by a constant multiple of n−r−1/2 (log⁡n)(d−1)/2n^{-r-1/2} \, (\log n)^{(d-1)/2}n−r−1/2(logn)(d−1)/2 in expectation and by n−r (log⁡n)(d−1)/2n^{-r} \, (\log n)^{(d-1)/2}n−r(logn)(d−1)/2 almost surely. The main term n−r−1/2n^{-r-1/2}n−r−1/2 is again optimal and it turns out that the very same algorithm is also optimal for the isotropic Sobolev space Hs([0,1]d)H^s([0,1]^d)Hs([0,1]d) of smoothness s>d/2s>d/2s>d/2. We also added a random shift to this algorithm to make it unbiased. Just recently, Mario Ullrich proved that the expected error of the resulting algorithm on Hr,mix([0,1]d)H^{r,\text{mix}}([0,1]^d)Hr,mix([0,1]d) is even bounded above by n−r−1/2n^{-r-1/2}n−r−1/2. This thesis is a review of the mentioned upper bounds and their proofs.
We study multivariate numerical integration of smooth functions in weighted Sobolev spaces with dominating mixed smoothness α≥2\alpha\geq 2α≥2 defined over the sss-dimensional unit cube. We propose a new quasi-Monte Carlo (QMC)-based quadrature rule, named the extrapolated polynomial lattice rule, which achieves the almost optimal rate of convergence. Extrapolated polynomial lattice rules are constructed in two steps: (i) construction of classical polynomial lattice rules over Fb\Bbb{F}_bFb​ with α\alphaα consecutive sizes of nodes, bm−α+1,…,bmb^{m-\alpha+1},\ldots,b^{m}bm−α+1,…,bm, and (ii) recursive application of Richardson extrapolation to a chain of α\alphaα approximate values of the integral obtained by consecutive polynomial lattice rules. We prove the existence of good extrapolated polynomial lattice rules achieving the almost optimal order of convergence of the worst-case error in Sobolev spaces with general weights. Then, by restricting to product weights, we show that such good extrapolated polynomial lattice rules can be constructed by the fast component-by-component algorithm under a computable quality criterion. The required total construction cost is of order (s+α)Nlog⁡N(s+\alpha)N\log N(s+α)NlogN, which improves the currently known result for the interlaced polynomial lattice rule, which is of order sαNlog⁡Ns\alpha N\log NsαNlogN. We also study the dependence of the worst-case error bound on the dimension. A big advantage of our method compared to interlaced polynomial lattice rules is that the fast QMC matrix vector method can be used in this setting, while still achieving the same rate of convergence. Such a method was previously not known. Numerical experiments for test integrands support our theoretical result.
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 that achieve a worst-case error of O ( n − 1 / 4 ( log n ) 1 / 2 ) , with the implied constant independent of the dimension, under certain summability conditions on the weights. Although this convergence rate is inferior to the one achievable for the shifted-averaged root mean squared worst-case error, the result does not rely on random shifting or transformation and holds unconditionally without any conjecture, in contrast to the result of Kazashi and Sloan.
We introduce a novel random integration algorithm that exhibits a high convergence order for functions characterized by sparse frequencies or rapidly decaying Fourier coefficients. Specifically, for integration in periodic isotropic Sobolev spaces and isotropic Sobolev spaces with compact support, our approach achieves a nearly optimal root mean square error bound. In contrast to previous nearly optimal algorithms, our method exhibits polynomial tractability. Our integration algorithm also enjoys nearly optimal bound for weighted Sobolev space. By incorporating the trick of change of variable, our algorithm is proven to achieve the semi-exponential convergence order for the integration of analytic functions, which marks a significant improvement over the previously obtained super-polynomial convergence order. Furthermore, for integration involving Wiener-type functions, the sample complexity of our algorithm remains independent of the decay rate of the Fourier coefficients.
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 quasi-Monte Carlo (QMC) integration over the multidimensional unit cube in several weighted function spaces with different smoothness classes. We consider approximating the integral of a function by the median of several integral estimates under independent and random choices of the underlying QMC point sets (either linearly scrambled digital nets or infinite-precision polynomial lattice point sets). Even though our approach does not require any information on the smoothness and weights of a target function space as an input, we can prove a probabilistic upper bound on the worst-case error for the respective weighted function space, where the failure probability converges to 0 exponentially fast as the number of estimates increases. Our obtained rates of convergence are nearly optimal for function spaces with finite smoothness, and we can attain a dimension-independent super-polynomial convergence for a class of infinitely differentiable functions. This implies that our median-based QMC rule is universal in the sense that it does not need to be adjusted to the smoothness and the weights of the function spaces and yet exhibits the nearly optimal rate of convergence. Numerical experiments support our theoretical results.