Lattice rules with random nn achieve nearly the optimal O(n−α−1/2)\mathcal{O}(n^{-\alpha-1/2}) error independently of the dimension

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 page

Rank-1 lattice rules for multivariate integration in spaces of permutation-invariant functions: Error bounds and tractability

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.co

Good intermediate-rank lattice rules based on the weighted star discrepancy

We study the problem of constructing good intermediate-rank lattice rules in the sense of having a low weighted star discrepancy. The intermediate-rank rules considered here are obtained by “copying” rank-1 lattice rules. We show that such rules can be constructed using a component-by-component technique and prove that the bound for the weighted star discrepancy achieves the optimal convergence rate

Good lattice rules with a composite number of points based on the product weighted star discrepancy

Rank-1 lattice rules based on a weighted star discrepancy with weights of a product form have been previously constructed under the assumption that the number of points is prime. Here, we extend these results to the non-prime case. We show that if the weights are summable, there exist lattice rules whose weighted star discrepancy is O(n−1+δ), for any δ > 0, with the implied constant independent of the dimension and the number of lattice points, but dependent on δ and the weights. Then we show that the generating vector of such a rule can be constructed using a component-by-component (CBC) technique. The cost of the CBC construction is analysed in the final part of the paper

Construction of quasi-Monte Carlo rules for multivariate integration in spaces of permutation-invariant functions

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