Abstract In this paper, we study the problem of multivariate $$L_2$$ 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 $$L_2$$ 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.
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.
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 ) < ∞ .
Abstract. We study quasi-Monte Carlo (QMC) integration over the multidimensional unit cube in several weighted function spaces with different smoothness classes. We consider approximating the integral of a function by the median of several integral estimates under independent and random choices of the underlying QMC point sets (either linearly scrambled digital nets or infinite-precision polynomial lattice point sets). Even though our approach does not require any information on the smoothness and weights of a target function space as an input, we can prove a probabilistic upper bound on the worst-case error for the respective weighted function space, where the failure probability converges to 0 exponentially fast as the number of estimates increases. Our obtained rates of convergence are nearly optimal for function spaces with finite smoothness, and we can attain a dimension-independent super-polynomial convergence for a class of infinitely differentiable functions. This implies that our median-based QMC rule is universal in the sense that it does not need to be adjusted to the smoothness and the weights of the function spaces and yet exhibits the nearly optimal rate of convergence. Numerical experiments support our theoretical results.
We study quasi-Monte Carlo (QMC) integration of smooth functions defined over the multidimensional unit cube. Inspired by a recent work of Pan and Owen, we study a new construction-free median QMC rule which can exploit the smoothness and the weights of function spaces adaptively. For weighted Korobov spaces, we draw a sample of r independent generating vectors of rank-1 lattice rules, compute the integral estimate for each, and approximate the true integral by the median of these r estimates. For weighted Sobolev spaces, we use the same approach but with the rank-1 lattice rules replaced by high-order polynomial lattice rules. A major advantage over the existing approaches is that we do not need to construct good generating vectors by a computer search algorithm, while our median QMC rule achieves almost the optimal worst-case error rate for the respective function space with any smoothness and weights, with a probability that converges to 1 exponentially fast as r increases. Numerical experiments illustrate and support our theoretical findings.
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 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.
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.