Polynomial Lattice Rules Research Articles

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 of periodic functions defined over the high-dimensional unit cube. Recent work by Kritzer, Kuo, Nuyens and Ullrich [J. Approx. Theory 240 (2019), pp. 96–113] shows that rank-1 lattice rules with a randomly chosen number of points and good generating vector achieve almost the optimal order of the randomized error in weighted Korobov spaces, and moreover, that the error is bounded independently of the dimension if the weight parameters,γj\gamma _j, satisfy the summability condition∑j=1∞γj1/α>∞\sum _{j=1}^{\infty }\gamma _j^{1/\alpha }>\infty, whereα\alphais a smoothness parameter. The argument is based on the existence result that at least half of the possible generating vectors yield almost the optimal order of the worst-case error in the same function spaces.In this paper we provide a component-by-component construction algorithm of such randomized rank-1 lattice rules, without any need to check whether the constructed generating vectors satisfy a desired worst-case error bound. Similarly to the above-mentioned work, we prove that our algorithm achieves almost the optimal order of the randomized error and that the error bound is independent of the dimension if the same condition∑j=1∞γj1/α>∞\sum _{j=1}^{\infty }\gamma _j^{1/\alpha }>\inftyholds. We also provide analogous results for tent-transformed lattice rules for weighted half-period cosine spaces and for polynomial lattice rules in weighted Walsh spaces, respectively.

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 over the multi-dimensional unit cube. Inspired by a recent work of Pan and Owen, we study a new construction-free median QMC rule which can exploit the smoothness and the weights of function spaces adaptively. For weighted Korobov spaces, we draw a sample of $r$ independent generating vectors of rank-1 lattice rules, compute the integral estimate for each, and approximate the true integral by the median of these $r$ estimates. For weighted Sobolev spaces, we use the same approach but with the rank-1 lattice rules replaced by high-order polynomial lattice rules. A major advantage over the existing approaches is that we do not need to construct good generating vectors by a computer search algorithm, while our median QMC rule achieves almost the optimal worst-case error rate for the respective function space with any smoothness and weights, with a probability that converges to 1 exponentially fast as $r$ increases. Numerical experiments illustrate and support our theoretical findings.

Multilevel quadrature for elliptic problems on random domains by the coupling of FEM and BEM

Elliptic boundary value problems which are posed on a random domain can be mapped to a fixed, nominal domain. The randomness is thus transferred to the diffusion matrix and the loading. While this domain mapping method is quite efficient for theory and practice, since only a single domain discretisation is needed, it also requires the knowledge of the domain mapping. However, in certain applications, the random domain is only described by its random boundary, while the quantity of interest is defined on a fixed, deterministic subdomain. In this setting, it thus becomes necessary to compute a random domain mapping on the whole domain, such that the domain mapping is the identity on the fixed subdomain and maps the boundary of the chosen fixed, nominal domain on to the random boundary. To overcome the necessity of computing such a mapping, we therefore couple the finite element method on the fixed subdomain with the boundary element method on the random boundary. We verify on one hand the regularity of the solution with respect to the random domain mapping required for many multilevel quadrature methods, such as the multilevel quasi-Monte Carlo quadrature using Halton points, the multilevel sparse anisotropic Gauss–Legendre and Clenshaw–Curtis quadratures and multilevel interlaced polynomial lattice rules. On the other hand, we derive the coupling formulation and show by numerical results that the approach is feasible.

Construction of good polynomial lattice rules in weighted Walsh spaces by an alternative component-by-component construction

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 function spaces under consideration and prove that the obtained error bounds can, under certain conditions on the involved weights, be made independent of the dimension. We also demonstrate that our alternative component-by-component construction, which is independent of the underlying smoothness of the function space, can be implemented relatively easily in a fast manner. Numerical experiments confirm our theoretical findings.

Component-by-Component Digit-by-Digit Construction of Good Polynomial Lattice Rules in Weighted Walsh Spaces

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 achieves excellent convergence order, and, under suitable conditions, we can vanquish the curse of dimensionality by considering function spaces equipped with coordinate weights. The construction algorithm is based on a quality measure that is independent of the underlying smoothness of the function space and can be implemented in a fast manner (without the use of fast Fourier transformations). Furthermore, we illustrate our findings with extensive numerical results.

MDFEM: Multivariate decomposition finite element method for elliptic PDEs with uniform random diffusion coefficients using higher-order QMC and FEM

We introduce the multivariate decomposition finite element method (MDFEM) for solving elliptic PDEs with uniform random diffusion coefficients. We show that the MDFEM can be used to reduce the computational complexity of estimating the expected value of a linear functional of the solution of the PDE. The proposed algorithm combines the multivariate decomposition method, to compute infinite-dimensional integrals, with the finite element method, to solve different instances of the PDE. The strategy of the MDFEM is to decompose the infinite-dimensional problem into multiple finite-dimensional ones which lends itself to easier parallelization than to solve a single large dimensional problem. Our first result adjusts the analysis of the multivariate decomposition method to incorporate the \((\ln (n))^d\)-factor which typically appears in error bounds for d-dimensional n-point cubature formulae and we take care of the fact that n needs to come, e.g., in powers of 2 for higher order approximations. For the further analysis we specialize the cubature methods to be two types of quasi-Monte Carlo (QMC) rules, being digitally shifted polynomial lattice rules and interlaced polynomial lattice rules. The second and main contribution then presents a bound on the error of the MDFEM and shows higher-order convergence w.r.t. the total computational cost in case of the interlaced polynomial lattice rules in combination with a higher-order finite element method. We show that the cost to achieve an error \(\epsilon \) is of order \(\epsilon ^{-a_{\mathrm {MDFEM}}}\) with \(a_{\mathrm {MDFEM}} = 1/\lambda + d'/\tau \) if the QMC cubature errors can be bounded by \(n^{-\lambda }\) and the FE approximations converge like \(h^\tau \) with cost \(h^{d'}\), where \(\lambda = \tau (1-p^*) / (p^* (1+d'/\tau ))\) and \(p^*\) is a parameter representing the “sparsity” of the random field expansion. A comparison with a dimension truncation algorithm shows that the MDFEM will perform better than the truncation algorithm if \(p^*\) is sufficiently small, i.e., the representation of the random field is sufficiently sparse.

MDFEM: Multivariate decomposition finite element method for elliptic PDEs with lognormal diffusion coefficients using higher-order QMC and FEM

We introduce the multivariate decomposition finite element method (MDFEM) for elliptic PDEs with lognormal diffusion coefficients, that is, when the diffusion coefficient has the form a = exp(Z) where Z is a Gaussian random field defined by an infinite series expansion Z(y) = ∑j≥1yjϕj with yj ~ N(0,1) and a given sequence of functions {ϕj}j≥1. We use the MDFEM to approximate the expected value of a linear functional of the solution of the PDE which is an infinite-dimensional integral over the parameter space. The proposed algorithm uses the multivariate decomposition method (MDM) to compute the infinite-dimensional integral by a decomposition into finite-dimensional integrals, which we resolve using quasi-Monte Carlo (QMC) methods, and for which we use the finite element method (FEM) to solve different instances of the PDE. We develop higher-order quasi-Monte Carlo rules for integration over the finite-dimensional Euclidean space with respect to the Gaussian distribution by use of a truncation strategy. By linear transformations of interlaced polynomial lattice rules from the unit cube to a multivariate box of the Euclidean space we achieve higher-order convergence rates for functions belonging to a class of anchored Gaussian Sobolev spaces while taking into account the truncation error. These cubature rules are then used in the MDFEM algorithm. Under appropriate conditions, the MDFEM achieves higher-order convergence rates in terms of error versus cost, i.e., to achieve an accuracy of O(ε) the computational cost is O(ε-1/λ-d′/λ)=O(ε-(p*+d′/τ)/(1-p*)) where ε−1/λ and ε−d′/λ) are respectively the cost of the quasi-Monte Carlo cubature and the finite element approximations, with d′= d (1+δ′) for some δ′ ≥ 0 and d the physical dimension, and 0<p*≤(2+d′/τ)-1 is a parameter representing the sparsity of {ϕj}j≥1.

Weighted integration over a hyperrectangle based on digital nets and sequences

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 hyperrectangle. 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 hyperrectangle but is otherwise independent of the product measure.

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 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 used for function classes with different smoothness and/or weights.In this paper we prove that a lattice rule constructed by the CBC algorithm for the weighted Korobov space with given smoothness and weights achieves the almost optimal rate of convergence with good tractability properties for general classes of smoothness and weights which satisfy some summability conditions. Such a stability result also can be shown for polynomial lattice rules in weighted Walsh spaces. We further give bounds on the weighted star discrepancy and discuss the tractability properties for these QMC rules. The results are comparable to those obtained for Halton, Sobol and Niederreiter sequences.

Richardson extrapolation allows truncation of higher-order digital nets and sequences

Abstract We study numerical integration of smooth functions defined over the $s$-dimensional unit cube. A recent work by Dick et al. (2019, Richardson extrapolation of polynomial lattice rules. SIAM J. Numer. Anal., 57, 44–69) has introduced so-called extrapolated polynomial lattice rules, which achieve the almost optimal rate of convergence for numerical integration, and can be constructed by the fast component-by-component search algorithm with smaller computational costs as compared to interlaced polynomial lattice rules. In this paper we prove that, instead of polynomial lattice point sets, truncated higher-order digital nets and sequences can be used within the same algorithmic framework to explicitly construct good quadrature rules achieving the almost optimal rate of convergence. The major advantage of our new approach compared to original higher-order digital nets is that we can significantly reduce the precision of points, i.e., the number of digits necessary to describe each quadrature node. This finding has a practically useful implication when either the number of points or the smoothness parameter is so large that original higher-order digital nets require more than the available finite-precision floating-point representations.

Richardson Extrapolation of Polynomial Lattice Rules

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 \emph{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 $\mathbb{F}_b$ 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 interlaced polynomial lattice rule, that 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.

Quasi--Monte Carlo Integration for Affine-Parametric, Elliptic PDEs: Local Supports and Product Weights

We analyze convergence rates of first-order quasi--Monte Carlo (QMC) integration with randomly shifted lattice rules and for higher-order, interlaced polynomial lattice rules for a class of countably parametric integrands that result from linear functionals of solutions of linear, elliptic diffusion equations with affine-parametric, uncertain coefficient function $a(x,{y}) = \bar{a}(x) + \sum_{j\geq 1} y_j \psi_j(x)$ in a bounded domain $D\subset \mathbb{R}^d$. Extending the result in [F. Y. Kuo, C. Schwab, and I. H. Sloan, SIAM J. Numer. Anal., 50 (2012), pp. 3351--3374], where $\psi_j$ was assumed to have global support in the domain $D$, we assume in the present paper that ${supp}(\psi_j)$ is localized in $D$ and that we have control on the overlaps of these supports. Under these conditions we prove dimension-independent convergence rates in [1/2,1) of randomly shifted lattice rules with product weights and corresponding higher-order convergence rates by higher-order, interlaced polynomial lattice rules with product weights. The product structure of the QMC weights facilitates work bounds for the fast, component-by-component constructions of [D. Nuyens and R. Cools, Math. Comp., 75 (2006), pp. 903--920] which scale linearly with respect to the parameter dimension $s$. The dimension-independent convergence rates are only limited by the degree of digit interlacing used in the construction of the higher-order QMC quadrature rule and, for locally supported coefficient functions, by the summability of the locally supported coefficient sequence in the affine-parametric coefficient.

The Bb-adic Symmetrization of Digital Nets for Quasi-Monte Carlo Integration

AbstractThe notion of symmetrization, also known as Davenport’s reflection principle, is well known in the area of the discrepancy theory and quasi- Monte Carlo (QMC) integration. In this paper we consider applying a symmetrization technique to a certain class of QMC point sets called digital nets over ℤb. Although symmetrization has been recognized as a geometric technique in the multi-dimensional unit cube, we give another look at symmetrization as a geometric technique in a compact totally disconnected abelian group with dyadic arithmetic operations. Based on this observation we generalize the notion of symmetrization from base 2 to an arbitrary base b ∈ ℕ, b ≥ 2. Subsequently, we study the QMC integration error of symmetrized digital nets over ℤbin a reproducing kernel Hilbert space. The result can be applied to component-by-component construction or Korobov construction for finding good symmetrized (higher order) polynomial lattice rules which achieve high order convergence of the integration error for smooth integrands at the expense of an exponential growth of the number of points with the dimension. Moreover, we consider two-dimensional symmetrized Hammersley point sets in prime base b, and prove that the minimum Dick weight is large enough to achieve the best possible order of Lpdiscrepancy for all 1 ≤ p < ∞.

Construction of interlaced polynomial lattice rules for infinitely differentiable functions

We study multivariate integration over the $s$-dimensional unit cube in a weighted space of infinitely differentiable functions. It is known from a recent result by Suzuki that there exists a good quasi-Monte Carlo (QMC) rule which achieves a super-polynomial convergence of the worst-case error in this function space, and moreover, that this convergence behavior is independent of the dimension under a certain condition on the weights. In this paper we provide a constructive approach to finding a good QMC rule achieving such a dimension-independent super-polynomial convergence of the worst-case error. Specifically, we prove that interlaced polynomial lattice rules, with an interlacing factor chosen properly depending on the number of points $N$ and the weights, can be constructed using a fast component-by-component algorithm in at most $O(sN(\log N)^2)$ arithmetic operations to achieve a dimension-independent super-polynomial convergence. The key idea for the proof of the worst-case error bound is to use a variant of Jensen's inequality with a purposely-designed concave function.

Good interlaced polynomial lattice rules for numerical integration in weighted Walsh spaces

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 for smooth functions. A construction principle of higher order digital nets and sequences based on a digit interlacing function was introduced in Dick (2007), which interlaces classical digital nets or sequences whose number of components is a multiple of the dimension.In this paper, we study the use of polynomial lattice point sets for interlaced components. We call quadrature rules using such point sets interlaced polynomial lattice rules. We consider weighted Walsh spaces containing smooth functions and derive two upper bounds on the worst-case error for interlaced polynomial lattice rules, both of which can be employed as a quality criterion for the construction of interlaced polynomial lattice rules. We investigate the component-by-component construction and the Korobov construction as a means of explicit constructions of good interlaced polynomial lattice rules that achieve the optimal rate of the worst-case error. Through this approach we are able to obtain a good dependence of the worst-case error bounds on the dimension under certain conditions on the weights, while significantly reducing the construction cost as compared to higher order polynomial lattice rules.

Construction of Interlaced Scrambled Polynomial Lattice Rules of Arbitrary High Order

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 error for numerical integration of smooth functions defined on the $s$-dimensional unit cube. The key ingredient there is a digit interlacing function applied to the components of a randomly scrambled digital net whose number of components is $ds$, where the integer $d$ is the so-called interlacing factor. In this paper, we replace the randomly scrambled digital nets by randomly scrambled polynomial lattice point sets, which allows us to obtain a better dependence on the dimension while still achieving the optimal rate of convergence. Our results apply to Owen's full scrambling scheme as well as the simplifications studied by Hickernell, Matou\v{s}ek and Owen. We consider weighted function spaces with general weights, whose elements have square integrable partial mixed derivatives of order up to $\alpha\ge 1$, and derive an upper bound on the variance of the estimator for higher order scrambled polynomial lattice rules. Employing our obtained bound as a quality criterion, we prove that the component-by-component construction can be used to obtain explicit constructions of good polynomial lattice point sets. By first constructing classical polynomial lattice point sets in base $b$ and dimension $ds$, to which we then apply the interlacing scheme of order $d$, we obtain a construction cost of the algorithm of order $\mathcal{O}(dsmb^m)$ operations using $\mathcal{O}(b^m)$ memory in case of product weights, where $b^m$ is the number of points in the polynomial lattice point set.

Constructing good higher order polynomial lattice rules with modulus of reduced degree

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 F2 which are randomly digitally shifted and then folded using the tent transformation. We first prove the existence of good higher order polynomial lattice rules which achieve the optimal rate of the mean square worst-case error, while reducing the required degree of modulus by half as compared to higher order polynomial lattice rules whose quadrature points are randomly digitally shifted but not folded using the tent transformation. Thus we are able to restrict the search space of generating vectors significantly. We then study the component-by-component construction as an explicit means of obtaining good higher order polynomial lattice rules. In a way analogous to Baldeaux et al. (2012), we show how to calculate the quality criterion efficiently and how to obtain the fast component-by-component construction using the fast Fourier transform. Our result generalizes the previous result shown by Cristea et al. (2007), in which the degree of smoothness is fixed at 2 and classical polynomial lattice rules are considered.

Optimal randomized changing dimension algorithms for infinite-dimensional integration on function spaces with ANOVA-type decomposition

We study the numerical integration problem for functions with infinitely many variables. The function spaces of integrands we consider are weighted reproducing kernel Hilbert spaces with norms related to the ANOVA decomposition of the integrands. The weights model the relative importance of different groups of variables. We investigate randomized quadrature algorithms and measure their quality by estimating the randomized worst-case integration error.In particular, we provide lower error bounds for a very general class of randomized algorithms that includes non-linear and adaptive algorithms. Furthermore, we propose new randomized changing dimension algorithms (also called multivariate decomposition methods) and present favorable upper error bounds. For product weights and finite-intersection weights our lower and upper error bounds match and show that our changing dimension algorithms are optimal in the sense that they achieve convergence rates arbitrarily close to the best possible convergence rate. As more specific examples, we discuss unanchored Sobolev spaces of different degrees of smoothness and randomized changing dimension algorithms that use as building blocks interlaced scrambled polynomial lattice rules.

Infinite-Dimensional Integration in Weighted Hilbert Spaces: Anchored Decompositions, Optimal Deterministic Algorithms, and Higher-Order Convergence

We study numerical integration of functions depending on an infinite number of variables. We provide lower error bounds for general deterministic linear algorithms and provide matching upper error bounds with the help of suitable multilevel algorithms and changing dimension algorithms. More precisely, the spaces of integrands we consider are weighted reproducing kernel Hilbert spaces with norms induced by an underlying anchored function space decomposition. Here the weights model the relative importance of different groups of variables. The error criterion used is the deterministic worst case error. We study two cost models for function evaluation which depend on the number of active variables of the chosen sample points, and two classes of weights, namely product and order-dependent (POD) weights and the newly introduced weights with finite active dimension. We show for these classes of weights that multilevel algorithms achieve the optimal rate of convergence in the first cost model while changing dimension algorithms achieve the optimal convergence rate in the second model. As an illustrative example, we discuss the anchored Sobolev space with smoothness parameter $\alpha$ and provide new optimal quasi-Monte Carlo multilevel algorithms and quasi-Monte Carlo changing dimension algorithms based on higher-order polynomial lattice rules.

Construction of scrambled polynomial lattice rules over $\mathbb{F}_{2}$ with small mean square weighted $\mathcal{L}_{2}$ discrepancy

The \(\mathcal{L}_{2}\) discrepancy is one of several well-known quantitative measures for the equidistribution properties of point sets in the high-dimensional unit cube. The concept of weights was introduced by Sloan and Woźniakowski to take into account the relative importance of the discrepancy of lower dimensional projections. As known under the name of quasi-Monte Carlo methods, point sets with small weighted \(\mathcal{L}_{2}\) discrepancy are useful in numerical integration. This study investigates the component-by-component construction of polynomial lattice rules over the finite field \(\mathbb{F}_{2}\) whose scrambled point sets have small mean square weighted \(\mathcal{L}_{2}\) discrepancy. An upper bound on this discrepancy is proved, which converges at almost the best possible rate of N −2+δ for all δ>0, where N denotes the number of points. Numerical experiments confirm that the performance of our constructed polynomial lattice point sets is comparable or even superior to that of Sobol’ sequences.