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.
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.
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).
Shifted rank-1 lattice rules, a special class of quasi-Monte Carlo methods, have recently been proposed by the present authors for the integration of functions belonging to certain "weighted" Sobolev spaces. The shifts in these rules were generated in a deterministic manner. In contrast, in this paper we generate these shifts randomly. This allows probabilistic estimates for the error in a given integral. It also reduces the number of operations required to find the generating vectors for the underlying lattice rules component-by-component. The rules thus constructed achieve a worst-case strong tractability error bound in an average or probabilistic sense.
This paper studies the error bounds of multivariate integration in weighted function spaces using lattice rules of the Korobov form, in which the generating vector for an n-point rule with n prime has the form (1,a,...,a d -1 ) (mod,n). With the parameter a chosen optimally, we establish new error bounds for Korobov lattice rules in weighted Korobov spaces. In particular, we prove that if the weights decay sufficiently fast, the optimal Korobov lattice rule has an error bound of order O(n^{-\alpha/2 +\delta})$ (for arbitrary δ>0), with the implied constant depending at worst polynomially on the dimension. Here α>1 is the smoothness parameter of the weighted Korobov spaces. We generalize the construction to the case where n is a product of arbitrary distinct prime numbers, with the purpose of reducing the construction cost without sacrificing much of the quality of the lattice rules. A corresponding result is deduced for weighted Sobolev spaces of nonperiodic functions, using randomly shifted optimal Korobov lattice rules. A comparison of the worst-case errors for Korobov lattice rules and the recent component-by-component constructions is presented. The investigation establishes the usefulness of (shifted) optimal Korobov lattice rules for integration, even in high dimensions, if the weights which characterize the weighted spaces are suitably chosen.
We study the worst-case approximation of multivariate periodic functions from weighted Korobov spaces H d , α , γ with smoothness α > 1 / 2 and weights γ in L p ( [ 0 , 1 ] d ) , 1 ≤ p ≤ ∞ . We analyze a median lattice algorithm that reconstructs a truncated Fourier series by estimating coefficients on a hyperbolic-cross-type index set using R rank-1 lattice sampling rules with independent randomly chosen generating vectors, and then aggregating the coefficient estimators via the componentwise median. For odd R and odd prime lattice size N , where R depends logarithmically on N , we prove high-probability error bounds in both L ∞ and L 2 . Interpolation then yields the result for all p ∈ [ 1 , ∞ ] . In particular, with high probability, the worst-case L p -error is bounded, for any β > 0 , by C d , α , β , γ , τ , p R 1 2 + ( 1 2 − 1 p ) + N − α + ( 1 2 − 1 p ) + + β , where ( x ) + = max { x , 0 } , and the constant C d , α , β , γ , τ , p is independent of N and R . For p = ∞ , the constant is dimension-independent under the summability condition ∑ j = 1 ∞ γ j 1 / ( 2 α ) < ∞ .
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.