Search

In previous work (Kuo, Nuyens, Wilkes, 2023), we showed that a lattice rule with a pre-determined generating vector but random number of points can achieve the near optimal convergence of $O(n^{-\alpha-1/2+\epsilon})$, $\epsilon > 0$, for the worst case expected error, commonly referred to as the randomised error, for numerical integration of high-dimensional functions in the Korobov space with smoothness $\alpha > 1/2$. Compared to the optimal deterministic rate of $O(n^{-\alpha+\epsilon})$, $\epsilon > 0$, such a randomised algorithm is capable of an extra half in the rate of convergence. In this paper, we show that a pre-determined generating vector also exists in the case of $0 < \alpha \le 1/2$. Also here we obtain the near optimal convergence of $O(n^{-\alpha-1/2+\epsilon})$, $\epsilon > 0$; or in more detail, we obtain $O(\sqrt{r} \, n^{-\alpha-1/2+1/(2r)+\epsilon'})$ which holds for any choices of $\epsilon' > 0$ and $r \in \mathbb{N}$ with $r > 1/(2\alpha)$.

We show that a very simple randomised algorithm for numerical integration can produce a near optimal rate of convergence for integrals of functions in the $d$-dimensional weighted Korobov space. This algorithm uses a lattice rule with a fixed generating vector and the only random element is the choice of the number of function evaluations. For a given computational budget $n$ of a maximum allowed number of function evaluations, we uniformly pick a prime $p$ in the range $n/2 < p \le n$. We show error bounds for the randomised error, which is defined as the worst case expected error, of the form $O(n^{-\alpha - 1/2 + \delta})$, with $\delta > 0$, for a Korobov space with smoothness $\alpha > 1/2$ and general weights. The implied constant in the bound is dimension-independent given the usual conditions on the weights. We present an algorithm that can construct suitable generating vectors \emph{offline} ahead of time at cost $O(d n^4 / \ln n)$ when the weight parameters defining the Korobov spaces are so-called product weights. For this case, numerical experiments confirm our theory that the new randomised algorithm achieves the near optimal rate of the randomised error., Comment: 29 pages, 2 figures

The kernel interpolant in a reproducing kernel Hilbert space is optimal in the worst-case sense among all approximations of a function using the same set of function values. In this paper, we compare two search criteria to construct lattice point sets for use in lattice-based kernel approximation. The first candidate, $\calP_n^*$, is based on the power function that appears in machine learning literature. The second, $\calS_n^*$, is a search criterion used for generating lattices for approximation using truncated Fourier series. We find that the empirical difference in error between the lattices constructed using $\calP_n^*$ and $\calS_n^*$ is marginal. The criterion $\calS_n^*$ is preferred as it is computationally more efficient and has a proven error bound., Comment: 16 pages, 3 figures

We approximate $d$-variate periodic functions in weighted Korobov spaces with general weight parameters using $n$ function values at lattice points. We do not limit $n$ to be a prime number, as in currently available literature, but allow any number of points, including powers of $2$, thus providing the fundamental theory for construction of embedded lattice sequences. Our results are constructive in that we provide a component-by-component algorithm which constructs a suitable generating vector for a given number of points or even a range of numbers of points. It does so without needing to construct the index set on which the functions will be represented. The resulting generating vector can then be used to approximate functions in the underlying weighted Korobov space. We analyse the approximation error in the worst-case setting under both the $L_2$ and $L_{\infty}$ norms. Our component-by-component construction under the $L_2$ norm achieves the best possible rate of convergence for lattice-based algorithms, and the theory can be applied to lattice-based kernel methods and splines. Depending on the value of the smoothness parameter $\alpha$, we propose two variants of the search criterion in the construction under the $L_{\infty}$ norm, extending previous results which hold only for product-type weight parameters and prime $n$. We also provide a theoretical upper bound showing that embedded lattice sequences are essentially as good as lattice rules with a fixed value of $n$. Under some standard assumptions on the weight parameters, the worst-case error bound is independent of $d$., Comment: 27 pages, 4 figures

We investigate the application of efficient recursive numerical integration strategies to models in lattice gauge theory from quantum field theory. Given the coupling structure of the physics problems and the group structure within lattice cubature rules for numerical integration, we show how to approach these problems efficiently by means of Fast Fourier Transform techniques. In particular, we consider applications to the quantum mechanical rotor and compact U(1) lattice gauge theory, where the physical dimensions are two and three. This proceedings article reviews our results presented in J. Comput. Phys 443 (2021) 110527., Comment: arXiv admin note: text overlap with arXiv:2011.05451

We introduce a new method to approximate integrals $\int_{\mathbb{R}^d} f(\boldsymbol{x}) \, \mathrm{d} \boldsymbol{x}$ which simply scales lattice rules from the unit cube $[0,1]^d$ to properly sized boxes on $\mathbb{R}^d$, hereby achieving higher-order convergence that matches the smoothness of the integrand function $f$ in a certain Sobolev space of dominating mixed smoothness. Our method only assumes that we can evaluate the integrand function $f$ and does not assume a particular density nor the ability to sample from it. In particular, for the theoretical analysis we show a new result that the method of adding Bernoulli polynomials to a function to make it "periodic" on a box without changing its integral value over the box, is equivalent to an orthogonal projection from a well chosen Sobolev space of dominating mixed smoothness to an associated periodic Sobolev space of the same dominating mixed smoothness, which we call a Korobov space. We note that the Bernoulli polynomial method is often not used because of its excessive computational complexity and also here we only make use of it in our theoretical analysis. We show that our new method of applying scaled lattice rules to increasing boxes can be interpreted as orthogonal projections with decreasing projection error. Such a method would not work on the unit cube since then the committed error caused by non-periodicity of the integrand would be constant, but for integration on the Euclidean space we can use the certain decay towards zero when the boxes grow. Hence we can bound the truncation error as well as the projection error and show higher-order convergence in applying scaled lattice rules for integration on Euclidean space. We illustrate our theoretical analysis by numerical experiments which confirm our findings., Comment: 40 pages, 5 figures

High dimensional integrals are abundant in many fields of research including quantum physics. The aim of this paper is to develop efficient recursive strategies to tackle a class of high dimensional integrals having a special product structure with low order couplings, motivated by models in lattice gauge theory from quantum field theory. A novel element of this work is the potential benefit in using lattice cubature rules. The group structure within lattice rules combined with the special structure in the physics integrands may allow efficient computations based on Fast Fourier Transforms. Applications to the quantum mechanical rotor and compact $U(1)$ lattice gauge theory in two and three dimensions are considered.

Lattice rules are among the most prominently studied quasi-Monte Carlo methods to approximate multivariate integrals. A rank-1 lattice rule to approximate an $s$-dimensional integral is fully specified by its generating vector $\mathbf{z} \in \mathbb{Z}^s$ and its number of points $N$. While there are many results on the existence of "good" rank-1 lattice rules, there are no explicit constructions for good generating vectors for dimensions $s \ge 3$. This is why one usually resorts to computer search algorithms. Motivated by earlier work of Korobov from 1963 and 1982, we present two variants of search algorithms for good lattice rules and show that the resulting rules exhibit a convergence rate in weighted function spaces that can be arbitrarily close to the optimal rate. Moreover, contrary to most other algorithms, we do not need to know the smoothness of our integrands in advance, the generating vector will still recover the convergence rate associated with the smoothness of the particular integrand, and, under appropriate conditions on the weights, the error bounds can be stated without dependence on $s$. The search algorithms presented in this paper are two variants of the well-known component-by-component (CBC) construction, one of which is combined with a digit-by-digit (DBD) construction. We present numerical results for both algorithms using fast construction algorithms in the case of product weights. They confirm our theoretical findings., Comment: 38 pages, 3 figures

In a recent paper by the same authors, we provided a theoretical foundation for the component-by-component (CBC) construction of lattice algorithms for multivariate $L_2$ approximation in the worst case setting, for functions in a periodic space with general weight parameters. The construction led to an error bound that achieves the best possible rate of convergence for lattice algorithms. Previously available literature covered only weights of a simple form commonly known as product weights. In this paper we address the computational aspect of the construction. We develop fast CBC construction of lattice algorithms for special forms of weight parameters, including the so-called POD weights and SPOD weights which arise from PDE applications, making the lattice algorithms truly applicable in practice. With $d$ denoting the dimension and $n$ the number of lattice points, we show that the construction cost is $\mathcal{O}(d\,n\log(n) + d^2\log(d)\,n)$ for POD weights, and $\mathcal{O}(d\,n\log(n) + d^3\sigma^2\,n)$ for SPOD weights of degree $\sigma\ge 2$. The resulting lattice generating vectors can be used in other lattice-based approximation algorithms, including kernel methods or splines.

This paper provides the theoretical foundation for the construction of lattice algorithms for multivariate $L_2$ approximation in the worst case setting, for functions in a periodic space with general weight parameters. Our construction leads to an error bound that achieves the best possible rate of convergence for lattice algorithms. This work is motivated by PDE applications in which bounds on the norm of the functions to be approximated require special forms of weight parameters (so-called POD weights or SPOD weights), as opposed to the simple product weights covered by the existing literature. Our result can be applied to other lattice-based approximation algorithms, including kernel methods or splines.

We consider rank-1 lattices for integration and reconstruction of functions with series expansion supported on a finite index set. We explore the connection between the periodic Fourier space and the non-periodic cosine space and Chebyshev space, via tent transform and then cosine transform, to transfer known results from the periodic setting into new insights for the non-periodic settings. Fast discrete cosine transform can be applied for the reconstruction phase. To reduce the size of the auxiliary index set in the associated component-by-component (CBC) construction for the lattice generating vectors, we work with a bi-orthonormal set of basis functions, leading to three methods for function reconstruction in the non-periodic settings. We provide new theory and efficient algorithmic strategies for the CBC construction. We also interpret our results in the context of general function approximation and discrete least-squares approximation.

In many models used in engineering and science, material properties are uncertain or spatially varying. For example, in geophysics, and porous media flow in particular, the uncertain permeability of the material is modelled as a random field. These random fields can be highly anisotropic. Efficient solvers, such as the Multiple Semi-Coarsened Multigrid (MSG) method, see [15, 16, 17], are required to compute solutions for various realisations of the uncertain material. The MSG method is an extension of the classic Multigrid method, that uses additional coarse grids that are coarsened in only a single coordinate direction. In this sense, it closely resembles the extension of Multilevel Monte Carlo (MLMC) [6] to Multi-Index Monte Carlo (MIMC) [9]. We present an unbiased MIMC method that reuses the MSG coarse solutions, similar to the work in [11]. Our formulation of the estimator can be interpreted as the problem of learning the unknown distribution of the number of samples across all indices, and unifies the previous work on adaptive MIMC [19] and unbiased estimation [18]. We analyse the cost of this new estimator theoretically and present numerical experiments with various anisotropic random fields, where the unknown coefficients in the covariance model are considered as hyperparameters. We illustrate its robustness and superiority over unbiased MIMC without sample reuse.

In this paper, we propose a numerical method to approximate the solution of the time-dependent Schr\"odinger equation with periodic boundary condition in a high-dimensional setting. We discretize space by using the Fourier pseudo-spectral method on rank-$1$ lattice points, and then discretize time by using a higher-order exponential operator splitting method. In this scheme the convergence rate of the time discretization depends on properties of the spatial discretization. We prove that the proposed method, using rank-$1$ lattice points in space, allows to obtain higher-order time convergence, and, additionally, that the necessary condition on the space discretization can be independent of the problem dimension $d$. We illustrate our method by numerical results from 2 to 8 dimensions which show that such higher-order convergence can really be obtained in practice., Comment: 15 pages, 2 figures

We introduce the multivariate decomposition finite element method for elliptic PDEs with lognormal diffusion coefficient $a=\exp(Z)$ where $Z$ is a Gaussian random field defined by an infinite series expansion $Z(\boldsymbol{y}) = \sum_{j\ge1} y_j\,\phi_j$ with $y_j\sim\mathcal{N}(0,1)$ and a given sequence of functions $\{\phi_j\}_{j\ge1}$. 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, taking into account the truncation error. Under appropriate conditions, the MDFEM achieves higher-order convergence rates in term of error versus cost, i.e., to achieve an accuracy of $O(\epsilon)$ the computational cost is $O(\epsilon^{-1/\lambda-d'/\lambda}) = O(\epsilon^{-(p^*+d'/\tau)/(1-p^*)})$ where $\epsilon^{-1/\lambda}$ and $\epsilon^{-d'/\lambda}$ are respectively the cost of the quasi-Monte Carlo cubature and the finite element approximations, with $d' = d \, (1+\delta')$ for some $\delta' \ge 0$ and $d$ the physical dimension, and $0 < p^* \le (2+d'/\tau)^{-1}$ is a parameter representing the sparsity of $\{\phi_j\}_{j\ge1}$., Comment: 49 pages

In the analysis of using quasi-Monte Carlo (QMC) methods to approximate expectations of a linear functional of the solution of an elliptic PDE with random diffusion coefficient the sensitivity w.r.t. the parameters is often stated in terms of product-and-order-dependent (POD) weights. The (offline) fast component-by-component (CBC) construction of an $N$-point QMC method making use of these POD weights leads to a cost of $\mathcal{O}(s N \log(N) + s^2 N)$ with $s$ the parameter truncation dimension. When $s$ is large this cost is prohibitive. As an alternative Herrmann and Schwab introduced an analysis resulting in product weights to reduce the construction cost to $\mathcal{O}(s N \log(N))$. We here show how the reduced CBC method can be used for POD weights to reduce the cost to $\mathcal{O}(\sum_{j=1}^{\min\{s,s^*\}} (m-w_j+j) \, b^{m-w_j})$, where $N=b^m$ with prime $b$, $w_1 \le \cdots \le w_s$ are nonnegative integers and $s^*$ can be chosen much smaller than $s$ depending on the regularity of the random field expansion as such making it possible to use the POD weights directly. We show a total error estimate for using randomly shifted lattice rules constructed through the reduced CBC construction., Comment: 18 pages

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 (MDM), to compute infinite dimensional integrals, with the finite element method (FEM), 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 log-factor which typically appears in error bounds for multivariate quadrature, i.e., cubature, methods; and we take care of the fact that the number of points $n$ 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., Comment: 31 pages

We approximate the solution for the time dependent Schr\"odinger equation (TDSE) in two steps. We first use a pseudo-spectral collocation method that uses samples of functions on rank-1 or rank-r lattice points with unitary Fourier transforms. We then get a system of ordinary differential equations in time, which we solve approximately by stepping in time using the Strang splitting method. We prove that the numerical scheme proposed converges quadratically with respect to the time step size, given that the potential is in a Korobov space with the smoothness parameter greater than $9/2$. Particularly, we prove that the required degree of smoothness is independent of the dimension of the problem. We demonstrate our new method by comparing with results using sparse grids from [12], with several numerical examples showing large advantage for our new method and pushing the examples to higher dimensionality. The proposed method has two distinctive features from a numerical perspective: (i) numerical results show the error convergence of time discretization is consistent even for higher-dimensional problems; (ii) by using the rank-$1$ lattice points, the solution can be efficiently computed (and further time stepped) using only $1$-dimensional Fast Fourier Transforms., Comment: Modified. 40pages, 5 figures. The proof of Lemma 1 is updated after the paper is published

The Multilevel Monte Carlo method is an efficient variance reduction technique. It uses a sequence of coarse approximations to reduce the computational cost in uncertainty quantification applications. The method is nowadays often considered to be the method of choice for solving PDEs with random coefficients when many uncertainties are involved. When using Full Multigrid to solve the deterministic problem, coarse solutions obtained by the solver can be recycled as samples in the Multilevel Monte Carlo method, as was pointed out by Kumar, Oosterlee and Dwight [Int. J. Uncertain. Quantif., 7 (2017), pp. 57--81]. In this article, an alternative approach is considered, using Quasi-Monte Carlo points, to speed up convergence. Additionally, our method comes with an improved variance estimate which is also valid in case of the Monte Carlo based approach. The new method is illustrated on the example of an elliptic PDE with lognormal diffusion coefficient. Numerical results for a variety of random fields with different smoothness parameters in the Mat\'ern covariance function show that sample recycling is more efficient when the input random field is nonsmooth.

In this paper we focus on efficient implementations of the Multivariate Decomposition Method (MDM) for approximating integrals of $\infty$-variate functions. Such $\infty$-variate integrals occur for example as expectations in uncertainty quantification. Starting with the anchored decomposition $f = \sum_{\mathfrak{u}\subset\mathbb{N}} f_\mathfrak{u}$, where the sum is over all finite subsets of $\mathbb{N}$ and each $f_\mathfrak{u}$ depends only on the variables $x_j$ with $j\in\mathfrak{u}$, our MDM algorithm approximates the integral of $f$ by first truncating the sum to some `active set' and then approximating the integral of the remaining functions $f_\mathfrak{u}$ term-by-term using Smolyak or (randomized) quasi-Monte Carlo (QMC) quadratures. The anchored decomposition allows us to compute $f_\mathfrak{u}$ explicitly by function evaluations of $f$. Given the specification of the active set and theoretically derived parameters of the quadrature rules, we exploit structures in both the formula for computing $f_\mathfrak{u}$ and the quadrature rules to develop computationally efficient strategies to implement the MDM in various scenarios. In particular, we avoid repeated function evaluations at the same point. We provide numerical results for a test function to demonstrate the effectiveness of the algorithm.

This article provides a high-level overview of some recent works on the application of quasi-Monte Carlo (QMC) methods to PDEs with random coefficients. It is based on an in-depth survey of a similar title by the same authors, with an accompanying software package which is also briefly discussed here. Embedded in this article is a step-by-step tutorial of the required analysis for the setting known as the uniform case with first order QMC rules. The aim of this article is to provide an easy entry point for QMC experts wanting to start research in this direction and for PDE analysts and practitioners wanting to tap into contemporary QMC theory and methods., Comment: arXiv admin note: text overlap with arXiv:1606.06613

This article provides an overview of some interfaces between the theory of quasi-Monte Carlo (QMC) methods and applications. We summarize three QMC theoretical settings: first order QMC methods in the unit cube $[0,1]^s$ and in $\mathbb{R}^s$, and higher order QMC methods in the unit cube. One important feature is that their error bounds can be independent of the dimension $s$ under appropriate conditions on the function spaces. Another important feature is that good parameters for these QMC methods can be obtained by fast efficient algorithms even when $s$ is large. We outline three different applications and explain how they can tap into the different QMC theory. We also discuss three cost saving strategies that can be combined with QMC in these applications. Many of these recent QMC theory and methods are developed not in isolation, but in close connection with applications.

In a previous paper (J. Comp. Phys. 230 (2011), 3668--3694), the authors proposed a new practical method for computing expected values of functionals of solutions for certain classes of elliptic partial differential equations with random coefficients. This method was based on combining quasi-Monte Carlo (QMC) methods for computing the expected values with circulant embedding methods for sampling the random field on a regular grid. It was found capable of handling fluid flow problems in random heterogeneous media with high stochastic dimension, but a convergence theory was missing. This paper provides a convergence analysis for the method in the case when the QMC method is a specially designed randomly shifted lattice rule. The convergence result depends on the eigenvalues of the underlying nested block circulant matrix and can be independent of the number of stochastic variables under certain assumptions. In fact the QMC analysis applies to general factorisations of the covariance matrix to sample the random field. The error analysis for the underlying fully discrete finite element method allows for locally refined meshes (via interpolation from a regular sampling grid of the random field). Numerical results on a non-regular domain with corner singularities in two spatial dimensions and on a regular domain in three spatial dimensions are included.

In this paper we prove, under mild conditions, that the positive definiteness of the circulant matrix appearing in the circulant embedding method is always guaranteed, provided the enclosing cube is sufficiently large. We examine in detail the case of the Mat\'ern covariance, and prove (for fixed correlation length) that, as $h_0\rightarrow 0$, positive definiteness is guaranteed when the random field is sampled on a cube of size order $(1 + \nu^{1/2} \log h_0^{-1})$ times larger than the size of the physical domain. (Here $h_0$ is the mesh spacing of the regular grid and $\nu$ the Mat\'ern smoothness parameter.) We show that the sampling cube can become smaller as the correlation length decreases when $h_0$ and $\nu$ are fixed. Our results are confirmed by numerical experiments. We prove several results about the decay of the eigenvalues of the circulant matrix. These lead to the conjecture, verified by numerical experiment, that they decay with the same rate as the Karhunen--Lo\`{e}ve eigenvalues of the covariance operator.

In a series of papers, in 1993, 1994 & 1996, Sloan & Niederreiter introduced a modification of lattice rules for non-periodic functions, called "vertex modified lattice rules"', and a particular breed called "optimal vertex modified lattice rules". In the 1994 paper, Niederreiter & Sloan concentrate explicitly on Fibonacci lattice rules, which are a particular good choice of 2-dimensional lattice rules. Error bounds in this series of papers were given related to the star discrepancy. In this paper we pose the problem in terms of the so-called unanchored Sobolev space, which is a reproducing kernel Hilbert space often studied nowadays in which functions have $L_2$-integrable mixed first derivatives. It is known constructively that randomly shifted lattice rules, as well as deterministic tent-transformed lattice rules and deterministic fully symmetrized lattice rules can achieve close to $O(N^{-1})$ convergence in this space, see Sloan, Kuo & Joe (2002) and Dick, Nuyens & Pillichshammer (2014) respectively. We derive a break down of the worst-case error of vertex modified lattice rules in the unanchored Sobolev space in terms of the worst-case error in a Korobov space, a multilinear space and some additional "mixture term". For the 1-dimensional case this worst-case error is obvious and gives an explicit expression for the trapezoidal rule. In the 2-dimensional case this mixture term also takes on an explicit form for which we derive upper and lower bounds. For this case we prove that there exist lattice rules with a nice worst-case error bound with the additional mixture term of the form $N^{-1} \log^2(N)$., Comment: The data from the table and the figure can be downloaded as the file "wce.csv" (comma separated values)

We present an adaptive version of the Multi-Index Monte Carlo method, introduced by Haji-Ali, Nobile and Tempone (2016), for simulating PDEs with coefficients that are random fields. A classical technique for sampling from these random fields is the Karhunen-Lo\`eve expansion. Our adaptive algorithm is based on the adaptive algorithm used in sparse grid cubature as introduced by Gerstner and Griebel (2003), and automatically chooses the number of terms needed in this expansion, as well as the required spatial discretizations of the PDE model. We apply the method to a simplified model of a heat exchanger with random insulator material, where the stochastic characteristics are modeled as a lognormal random field, and we show consistent computational savings.

We analyze a new random algorithm for numerical integration of $d$-variate functions over $[0,1]^d$ from a weighted Sobolev space with dominating mixed smoothness $\alpha\ge 0$ and product weights $1\ge\gamma_1\ge\gamma_2\ge\cdots>0$, where the functions are continuous and periodic when $\alpha>1/2$. The algorithm is based on rank-$1$ lattice rules with a random number of points~$n$. For the case $\alpha>1/2$, we prove that the algorithm achieves almost the optimal order of convergence of $\mathcal{O}(n^{-\alpha-1/2})$, where the implied constant is independent of the dimension~$d$ if the weights satisfy $\sum_{j=1}^\infty \gamma_j^{1/\alpha}<\infty$. The same rate of convergence holds for the more general case $\alpha>0$ by adding a random shift to the lattice rule with random $n$. This shows, in particular, that the exponent of strong tractability in the randomized setting equals $1/(\alpha+1/2)$, if the weights decay fast enough. We obtain a lower bound to indicate that our results are essentially optimal. This paper is a significant advancement over previous related works with respect to the potential for implementation and the independence of error bounds on the problem dimension. Other known algorithms which achieve the optimal error bounds, such as those based on Frolov's method, are very difficult to implement especially in high dimensions. Here we adapt a lesser-known randomization technique introduced by Bakhvalov in 1961. This algorithm is based on rank-$1$ lattice rules which are very easy to implement given the integer generating vectors. A simple probabilistic approach can be used to obtain suitable generating vectors., Comment: 17 pages

The (fast) component-by-component (CBC) algorithm is an efficient tool for the construction of generating vectors for quasi-Monte Carlo rank-1 lattice rules in weighted reproducing kernel Hilbert spaces. We consider product weights, which assigns a weight to each dimension. These weights encode the effect a certain variable (or a group of variables by the product of the individual weights) has. Smaller weights indicate less importance. Kuo (2003) proved that the CBC algorithm achieves the optimal rate of convergence in the respective function spaces, but this does not imply the algorithm will find the generating vector with the smallest worst-case error. In fact it does not. We investigate a generalization of the component-by-component construction that allows for a general successive coordinate search (SCS), based on an initial generating vector, and with the aim of getting closer to the smallest worst-case error. The proposed method admits the same type of worst-case error bounds as the CBC algorithm, independent of the choice of the initial vector. Under the same summability conditions on the weights as in [Kuo,2003] the error bound of the algorithm can be made independent of the dimension $d$ and we achieve the same optimal order of convergence for the function spaces from [Kuo,2003]. Moreover, a fast version of our method, based on the fast CBC algorithm by Nuyens and Cools, is available, reducing the computational cost of the algorithm to $O(d \, n \log(n))$ operations, where $n$ denotes the number of function evaluations. Numerical experiments seeded by a Korobov-type generating vector show that the new SCS algorithm will find better choices than the CBC algorithm and the effect is better when the weights decay slower., Comment: 13 pages, 1 figure, MCQMC2016 conference (Stanford)

In this paper we analyze the approximation of multivariate integrals over the Euclidean plane for functions which are analytic. We show explicit upper bounds which attain the exponential rate of convergence. We use an infinite grid with different mesh sizes and lengths in each direction to sample the function, and then truncate it. In our analysis, the mesh sizes and the truncated domain are chosen by optimally balancing the truncation error and the discretization error. This paper derives results in comparable function space settings, extended to $\R^s$, as which were recently obtained in the unit cube by Dick, Larcher, Pillichshammer and Wo{\'z}niakowski (2011). They showed that both lattice rules and regular grids, with different mesh sizes in each direction, attain exponential rates, hence motivating us to analyze only cubature formula based on regular meshes. We further also amend the analysis of older publications, e.g., Sloan and Osborn (1987) and Sugihara (1987), using lattice rules on $\R^s$ by taking the truncation error into account and extending them to take the anisotropy of the function space into account., Comment: 23 pages, 3 figures

We present a Multi-Index Quasi-Monte Carlo method for the solution of elliptic partial differential equations with random coefficients. By combining the multi-index sampling idea with randomly shifted rank-1 lattice rules, the algorithm constructs an estimator for the expected value of some functional of the solution. The efficiency of this new method is illustrated on a three-dimensional subsurface flow problem with lognormal diffusion coefficient with underlying Mat\'ern covariance function. This example is particularly challenging because of the small correlation length considered, and thus the large number of uncertainties that must be included. We show numerical evidence that it is possible to achieve a cost inversely proportional to the requested tolerance on the root-mean-square error, for problems with a smoothly varying random field

This article provides a survey of recent research efforts on the application of quasi-Monte Carlo (QMC) methods to elliptic partial differential equations (PDEs) with random diffusion coefficients. It considers, and contrasts, the uniform case versus the lognormal case, single-level algorithms versus multi-level algorithms, first order QMC rules versus higher order QMC rules, and deterministic QMC methods versus randomized QMC methods. It gives a summary of the error analysis and proof techniques in a unified view, and provides a practical guide to the software for constructing and generating QMC points tailored to the PDE problems. The analysis for the uniform case can be generalized to cover a range of affine parametric operator equations.

We develop algorithms for multivariate integration and approximation in the weighted half-period cosine space of smooth non-periodic functions. We use specially constructed tent-transformed rank-1 lattice points as cubature nodes for integration and as sampling points for approximation. For both integration and approximation, we study the connection between the worst-case errors of our algorithms in the cosine space and the worst-case errors of some related algorithms in the well-known weighted Korobov space of smooth periodic functions. By exploiting this connection, we are able to obtain constructive worst-case error bounds with good convergence rates for the cosine space., Comment: Journal of Complexity, Available online 26 May 2016

We study multivariate integration of functions that are invariant under the permutation (of a subset) of their arguments. Recently, in Nuyens, Suryanarayana, and Weimar (Adv. Comput. Math. (2016), 42(1):55--84), the authors derived an upper estimate for the $n$th minimal worst case error for such problems, and showed that under certain conditions this upper bound only weakly depends on the dimension. We extend these results by proposing two (semi-) explicit construction schemes. We develop a component-by-component algorithm to find the generating vector for a shifted rank-$1$ lattice rule that obtains a rate of convergence arbitrarily close to $\mathcal{O}(n^{-\alpha})$, where $\alpha>1/2$ denotes the smoothness of our function space and $n$ is the number of cubature nodes. Further, we develop a semi-constructive algorithm that builds on point sets which can be used to approximate the integrands of interest with a small error; the cubature error is then bounded by the error of approximation. Here the same rate of convergence is achieved while the dependence of the error bounds on the dimension $d$ is significantly improved.

We further develop the \emph{Multivariate Decomposition Method} (MDM) for the Lebesgue integration of functions of infinitely many variables $x_1,x_2,x_3,\ldots$ with respect to a corresponding product of a one dimensional probability measure. Although a number of concepts of infinite-dimensional integrals have been used in the literature, questions of uniqueness and compatibility have mostly not been studied. We show that, under appropriate convergence conditions, the Lebesgue integral equals the `anchored' integral, independently of the anchor. The MDM assumes that point values of $f_{\mathfrak{u}}$ are available for important subsets ${\mathfrak{u}}$, at some known cost. In this paper we introduce a new setting, in which it is assumed that each $f_{\mathfrak{u}}$ belongs to a normed space $F_{\mathfrak{u}}$, and that bounds $B_{\mathfrak{u}}$ on $\|f_{\mathfrak{u}}\|_{F_{\mathfrak{u}}}$ are known. This contrasts with the assumption in many papers that weights $\gamma_{\mathfrak{u}}$, appearing in the norm of the infinite-dimensional function space, are somehow known. Often such weights $\gamma_{\mathfrak{u}}$ were determined by minimizing an error bound depending on the $B_{\mathfrak{u}}$, the $\gamma_{\mathfrak{u}}$ \emph{and} the chosen algorithm, resulting in weights that depend on the algorithm. In contrast, in this paper only the bounds $B_{\mathfrak{u}}$ are assumed known. We give two examples in which we specialize the MDM: in the first case $F_{\mathfrak{u}}$ is the $|{\mathfrak{u}}|$-fold tensor product of an anchored reproducing kernel Hilbert space, and in the second case it is a particular non-Hilbert space for integration over an unbounded domain.

We study multivariate integration of functions that are invariant under permutations (of subsets) of their arguments. We find an upper bound for the $n$th minimal worst case error and show that under certain conditions, it can be bounded independent of the number of dimensions. In particular, we study the application of unshifted and randomly shifted rank-$1$ lattice rules in such a problem setting. We derive conditions under which multivariate integration is polynomially or strongly polynomially tractable with the Monte Carlo rate of convergence $O(n^{-1/2})$. Furthermore, we prove that those tractability results can be achieved with shifted lattice rules and that the shifts are indeed necessary. Finally, we show the existence of rank-$1$ lattice rules whose worst case error on the permutation- and shift-invariant spaces converge with (almost) optimal rate. That is, we derive error bounds of the form $O(n^{-\lambda/2})$ for all $1 \leq \lambda < 2 \alpha$, where $\alpha$ denotes the smoothness of the spaces. Keywords: Numerical integration, Quadrature, Cubature, Quasi-Monte Carlo methods, Rank-1 lattice rules., Comment: 26 pages; minor changes due to reviewer's comments; the final publication is available at link.springer.com