We two distinct practical problems in high-dimensional approximation. The first concerns the approximation of integrals using a particular randomised lattice rule, while the second focuses on function approximation with a generalisation of the rank-1 lattice known as the generated set. These algorithms are applied to functions in reproducing kernel Hilbert spaces defined on the torus, which are contained within the Wiener algebra. Our randomised lattice algorithm extends previous approaches by reducing the randomness, allowing for greater control over the structure of the sample points. The generating vector of the lattice is predetermined before the algorithm runs, with the only random element being the selection of the number of sample points. We demonstrate that there exist generating vectors for which this algorithm achieves a near-optimal rate of randomised error convergence. Furthermore, we produce a method by which such a generating vector can be constructed component by component. This method can be implemented efficiently using a fast algorithm, where the time required to produce a suitable generating vector exceeds that of computing the randomised error by only a logarithmic factor. We also introduce a random shift to this randomised algorithm in order to produce results for a larger function space where pointwise evaluation is not well defined and provide lower bounds on the error which can be achieved by these fixed vector algorithms. Generated sets are an extension of rank-1 lattice rules which allow the generator to have real, rather than strictly integer, components. While rank-1 lattices can be shown to only achieve at best half the optimal order of error convergence for function approximation in the worst-case L²-setting, we show that this restriction does not apply to generated sets, which are capable of achieving the optimal order. In a function space that generalises the Korobov space, we will demonstrate the existence of generators for which a least squares approximation algorithm is able to produce an error rate very close to the optimal bound. Additionally, we show that rational generating vectors with sufficiently large common denominators can achieve a comparable rate. Finally, we specialise these results to the weighted Korobov space, where we provide a more precise error bound and prove a tractability result, subject to a common condition on the weights of the space.
We two distinct practical problems in high-dimensional approximation. The first concerns the approximation of integrals using a particular randomised lattice rule, while the second focuses on function approximation with a generalisation of the rank-1 lattice known as the generated set. These algorithms are applied to functions in reproducing kernel Hilbert spaces defined on the torus, which are contained within the Wiener algebra. Our randomised lattice algorithm extends previous approaches by reducing the randomness, allowing for greater control over the structure of the sample points. The generating vector of the lattice is predetermined before the algorithm runs, with the only random element being the selection of the number of sample points. We demonstrate that there exist generating vectors for which this algorithm achieves a near-optimal rate of randomised error convergence. Furthermore, we produce a method by which such a generating vector can be constructed component by component. This method can be implemented efficiently using a fast algorithm, where the time required to produce a suitable generating vector exceeds that of computing the randomised error by only a logarithmic factor. We also introduce a random shift to this randomised algorithm in order to produce results for a larger function space where pointwise evaluation is not well defined and provide lower bounds on the error which can be achieved by these fixed vector algorithms. Generated sets are an extension of rank-1 lattice rules which allow the generator to have real, rather than strictly integer, components. While rank-1 lattices can be shown to only achieve at best half the optimal order of error convergence for function approximation in the worst-case L²-setting, we show that this restriction does not apply to generated sets, which are capable of achieving the optimal order. In a function space that generalises the Korobov space, we will demonstrate the existence of generators for which a least squares approximation algorithm is able to produce an error rate very close to the optimal bound. Additionally, we show that rational generating vectors with sufficiently large common denominators can achieve a comparable rate. Finally, we specialise these results to the weighted Korobov space, where we provide a more precise error bound and prove a tractability result, subject to a common condition on the weights of the space.
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, γ j \gamma _j , satisfy the summability condition ∑ j = 1 ∞ γ j 1 / α > ∞ \sum _{j=1}^{\infty }\gamma _j^{1/\alpha }>\infty , where α \alpha is 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 ∞ γ j 1 / α > ∞ \sum _{j=1}^{\infty }\gamma _j^{1/\alpha }>\infty holds. 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.
A randomised lattice rule algorithm with pre-determined generating vector and random number of points for Korobov spaces with 0<α≤1/2
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−α−1/2+ε), ε>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 α>1/2. Compared to the optimal deterministic rate of O(n−α+ε), ε>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<α≤1/2. Also here we obtain the near optimal convergence of O(n−α−1/2+ε), ε>0; or in more detail, we obtain O(rn−α−1/2+1/(2r)+ε′) which holds for any choices of ε′>0 and r∈N with r>1/(2α).
This paper studies the multivariate approximation of functions in weighted Korobov spaces using multiple rank-1 lattice rules. It has been shown by Kämmerer and Volkmer (2019) that algorithms based on multiple rank-1 lattices achieve the optimal convergence rate for the L∞ error in Wiener-type spaces, up to logarithmic factors. While this result was translated to weighted Korobov spaces in the recent monograph by Dick, Kritzer, and Pillichshammer (2022), the analysis requires the smoothness parameter α to be greater than 1 and is restricted to product weights. In this paper, we extend this result for multiple rank-1 lattice-based algorithms to the case where 1/2<α≤1 and for general weights, covering a broader range of periodic functions with low smoothness and general relative importance of variables. We also provide a summability condition on the weights to ensure strong polynomial tractability for any α>1/2. Furthermore, by incorporating random shifts into multiple rank-1 lattice-based algorithms, we prove that the resulting randomized algorithm achieves a nearly optimal convergence rate in terms of the worst-case root mean squared L2 error, while retaining the same tractability property.
We study the multivariate integration problem for periodic functions from the weighted Korobov space in the randomized setting. We introduce a new randomized rank-1 lattice rule with a randomly chosen number of points, which avoids the need for component-by-component construction in the search for good generating vectors while still achieving nearly the optimal rate of the randomized error. Our idea is to exploit the fact that at least half of the possible generating vectors yield nearly the optimal rate of the worst-case error in the deterministic setting. By randomly choosing generating vectors r r times and comparing their corresponding worst-case errors, one can find one generating vector with a desired worst-case error bound with a very high probability, and the (small) failure probability can be controlled by increasing r r logarithmically as a function of the number of points. Numerical experiments are conducted to support our theoretical findings.
L2-Approximation Using Median Lattice Algorithms
Abstract In this paper, we study the problem of multivariate L2 L 2 -approximation of functions belonging to a weighted Korobov space. We propose and analyze a median lattice-based algorithm, inspired by median integration rules, which have attracted significant attention in the theory of quasi-Monte Carlo methods. Our algorithm approximates the Fourier coefficients associated with a suitably chosen frequency index set, where each coefficient is estimated by taking the median over approximations from randomly shifted rank-1 lattice rules with independently chosen generating vectors. We prove that the algorithm achieves, with high probability, a convergence rate of the L2 L 2 -approximation error that is arbitrarily close to optimal with respect to the number of function evaluations. Furthermore, we show that the error bound depends only polynomially on the dimension, or is even independent of the dimension, under certain summability conditions on the weights. Numerical experiments illustrate the performance of the proposed median lattice-based algorithm.
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.