$L_2$-approximation using median lattice algorithms

TL;DR

This paper introduces a median lattice algorithm for multivariate $L_2$-approximation in weighted Korobov spaces, achieving near-optimal convergence with dimension-independent error bounds under certain conditions.

Contribution

The paper proposes a novel median lattice-based method for $L_2$-approximation that leverages median estimates from randomized lattice rules, improving convergence and dimension dependence.

Findings

Achieves near-optimal convergence rates with high probability.

Error bounds depend polynomially or are independent of dimension.

Numerical experiments confirm the effectiveness of the proposed method.

Abstract

In this paper, we study the problem of multivariate $L_2$-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 $L_2$-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.