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$L_2$-approximation using median lattice algorithms

Zexin Pan, Peter Kritzer, Takashi Goda

TL;DR

This work tackles multivariate $L_2$-approximation in weighted Korobov spaces $\mathcal{H}_{d,\alpha,\boldsymbol{\gamma}}$ by introducing a median lattice-based algorithm that aggregates $R$ independent randomized rank-1 lattice rules to estimate Fourier coefficients over a truncated index set $\mathcal{A}_d(N_*)$. The median of the $R$ estimates yields high-probability error bounds that decay nearly as $M^{-\alpha}$ (up to $\varepsilon$) with the total cost $M=RN$, and the dimension dependence can be controlled or eliminated under summability assumptions on the product weights $\boldsymbol{\gamma}$. The authors establish that, for individual functions, the randomized $L_2$-approximation error can be as close to optimal as possible, up to logarithmic factors, and provide tractability results showing dimension-free or polynomial-in-$d$ behavior under certain weight conditions. Numerical experiments in $d=2$ corroborate the practical performance, illustrating faster-than-$L_2$-worst-case rates compared to single-lattice methods and highlighting the influence of truncation and sampling on observed convergence. Overall, the median lattice approach contributes a robust, information-based complexity perspective to randomized lattice-based approximation, achieving near-optimal rates for high-dimensional function classes while offering parallelizable computation via independent lattice trials.

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.

$L_2$-approximation using median lattice algorithms

TL;DR

This work tackles multivariate -approximation in weighted Korobov spaces by introducing a median lattice-based algorithm that aggregates independent randomized rank-1 lattice rules to estimate Fourier coefficients over a truncated index set . The median of the estimates yields high-probability error bounds that decay nearly as (up to ) with the total cost , and the dimension dependence can be controlled or eliminated under summability assumptions on the product weights . The authors establish that, for individual functions, the randomized -approximation error can be as close to optimal as possible, up to logarithmic factors, and provide tractability results showing dimension-free or polynomial-in- behavior under certain weight conditions. Numerical experiments in corroborate the practical performance, illustrating faster-than--worst-case rates compared to single-lattice methods and highlighting the influence of truncation and sampling on observed convergence. Overall, the median lattice approach contributes a robust, information-based complexity perspective to randomized lattice-based approximation, achieving near-optimal rates for high-dimensional function classes while offering parallelizable computation via independent lattice trials.

Abstract

In this paper, we study the problem of multivariate -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 -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.
Paper Structure (11 sections, 7 theorems, 90 equations, 3 figures, 1 algorithm)

This paper contains 11 sections, 7 theorems, 90 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Weighted Korobov spaces
  4. $L_2$-approximation
  5. Rank-1 lattice rules
  6. The median algorithm and its error
  7. Median lattice-based algorithm
  8. Error analysis
  9. Dependence of the error bound on the dimension
  10. Numerical experiments
  11. Discussion

Key Result

Lemma 5

For $L\geq 1$, we have where $\zeta(q):=\sum_{n=1}^\infty n^{-q}$ is the Riemann zeta function. We further have, for any $\tau>0$, where $P_{L}(\tau,d,\boldsymbol{\gamma})$ is defined as in eqn:PN, but with $N\in\mathbb{N}$ replaced by $L$.

Figures (3)

  • Figure 1: Results for the test functions $f_1$ (left) and $f_2$ (right): $L_2$-approximation error versus the total number of function evaluations $M$.
  • Figure 2: Results for the test functions $f_1$ (left) and $f_2$ (right): $L_2$-approximation error versus $N_*$.
  • Figure 3: $N_*$ versus the total number of function evaluations $M$.

Theorems & Definitions (21)

  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 5
  • proof
  • Corollary 6
  • proof
  • Remark 7
  • Lemma 8
  • proof
  • ...and 11 more