We introduce a novel random integration algorithm that exhibits a high convergence order for functions characterized by sparse frequencies or rapidly decaying Fourier coefficients. Specifically, for integration in periodic isotropic Sobolev spaces and isotropic Sobolev spaces with compact support, our approach achieves a nearly optimal root mean square error bound. In contrast to previous nearly optimal algorithms, our method exhibits polynomial tractability. Our integration algorithm also enjoys nearly optimal bound for weighted Sobolev space. By incorporating the trick of change of variable, our algorithm is proven to achieve the semi-exponential convergence order for the integration of analytic functions, which marks a significant improvement over the previously obtained super-polynomial convergence order. Furthermore, for integration involving Wiener-type functions, the sample complexity of our algorithm remains independent of the decay rate of the Fourier coefficients.
We study a Monte Carlo algorithm that is based on a specific (randomly shifted and dilated) lattice point set. The main result of this paper is that the mean squared error for a given compactly supported, square-integrable function is bounded by
n−1/2 times the
L2-norm of the Fourier transform outside a region around the origin, where
n is the expected number of function evaluations. As corollaries we obtain the order of convergence for the Sobolev spaces
Hps with isotropic, anisotropic or mixed smoothness for all values of the parameters. This proves, in particular, that the optimal order of convergence in the latter case is
n−s−1/2 for
p≥2, which is, in contrast to the case of deterministic algorithms, independent of the dimension. This shows that Monte Carlo algorithms can improve the order by more than
n−1/2 for a whole class of practically important function classes. All results carry over to functions defined on the unit cube without boundary conditions.
In this paper we study multivariate integration for a weighted Korobov space for which the Fourier coefficients of the functions decay exponentially fast. This implies that the functions of this space are infinitely times differentiable. Weights of the Korobov space monitor the influence of each variable and each group of variables. We show that there are numerical integration rules which achieve an exponential convergence of the worst-case integration error. We also investigate the dependence of the worst-case error on the number of variables s s , and show various tractability results under certain conditions on weights of the Korobov space. Tractability means that the dependence on s s is never exponential, and sometimes the dependence on s s is polynomial or there is no dependence on s s at all.
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 numerical integration for a weighted Korobov space of analytic periodic functions for which the Fourier coefficients decay exponentially fast. In particular, we are interested in how the error depends on the dimension d. Many recent papers deal with this problem or similar problems and provide matching necessary and sufficient conditions for various notions of tractability. In most cases even simple algorithms are known which allow to achieve these notions of tractability. However, there is a gap in the literature: while for the notion of exponential-weak tractability one knows matching necessary and sufficient conditions, so far no explicit algorithm has been known which yields the desired result. In this paper we close this gap and prove that Korobov lattice rules are suitable algorithms in order to achieve exponential-weak tractability for integration in weighted Korobov spaces of analytic periodic functions.
In this talk we study multivariate integration over R^s for weighted function space of infinitely many times differentiable functions for which the Fourier transform decays exponentially fast. The weights allows us to define the different influence of each variables. We prove that we can achieve the exponential convergence rate using a classical quasi-Monte Carlo method. More specific, we prove two convergence rates of
O(e−ND+B1) and
O(e−lnD1NNB1), where D and B are respectively defined by the exponential decay of the Fourier coefficients and of the integrand, for two different function spaces. We also show some numerical results that prove our theorems.
We study the numerical approximation of high-dimensional integrals using randomized quasi-Monte Carlo (RQMC) methods, with a focus on scrambled Sobol' sequences. While asymptotically faster than Monte Carlo, classical RQMC suffers from error bounds that grow exponentially in the dimension
s, and its root mean squared error (RMSE) convergence rate is generally no better than
O(N−3/2). To overcome these limitations, we combine two recent developments: Suzuki's coarse scrambling and the median trick. Coarse scrambling randomizes Sobol' sequences according to their generating base polynomials and significantly reduces the maximal gain coefficient. By appropriately choosing the base polynomials, we show that the coefficient can be made uniformly bounded in
s, yielding an
O(N−1/2) RMSE for
L2 integrands with a dimension-independent constant. The median trick then enables near-optimal convergence for function classes beyond
L2: for
Lp integrands with
p∈(1,2), the median of independent coarsely scrambled estimates achieves an error of
O(N−1+1/p) with high probability; for integrands in the Haar wavelet space
Hwav,α,s,p,q with
αp:=α−(1/p−1/2)+>0, we prove a high-probability error bound of
O(N−αp−1/2+ε) for any
ε>0; the bound improves to
O(N−r−αp−1/2+ε) when the integrand has dominating mixed derivatives of order
r≥1 that belong to
Hwav,α,s,p,q. The latter two bounds are uniform in
s under suitable conditions on the ANOVA components of the integrands. Numerical experiments confirm the predicted convergence rates and demonstrate the robustness of the proposed approach.