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A simple universal algorithm for high-dimensional integration

Takashi Goda, David Krieg

TL;DR

The paper tackles efficient high-dimensional integration over $[0,1]^d$ for functions in weighted Korobov spaces $\mathcal{H}_{d,\alpha,\gamma}$ by proposing a simple universal randomized lattice-quadrature algorithm. It randomizes over primes $p$ and generating vectors $z$, and uses the median of independent lattice estimates to achieve near-optimal convergence: the randomized error satisfies $e^{\rm ran}_{d,\alpha,\gamma} \le C\,n^{-\alpha-1/2+\varepsilon}$ and the deterministic error satisfies $e^{\rm det}_{d,\alpha,\gamma} \le C\,n^{-\alpha+\varepsilon}$ (with high probability). The analysis leverages the Korobov kernel, the Fourier-weight sum $V_d(\alpha,\gamma)$, and a median-amplification technique, showing dimension-free guarantees when $\gamma \in \ell_{1/\alpha}$, and extends to nonperiodic functions via tent- or smooth-transforms. Numerical experiments with periodic and nonperiodic test functions corroborate the theoretical rates, demonstrating universality and robustness across smoothness levels and dimensions. The result is a simple, implementable, and near-universal approach for high-dimensional quadrature that adaptively attains near-optimal rates without detailed prior knowledge of the integrand.

Abstract

We present a simple universal algorithm for high-dimensional integration which has the optimal error rate (independent of the dimension) in all weighted Korobov classes both in the randomized and the deterministic setting. Our theoretical findings are complemented by numerical tests.

A simple universal algorithm for high-dimensional integration

TL;DR

The paper tackles efficient high-dimensional integration over for functions in weighted Korobov spaces by proposing a simple universal randomized lattice-quadrature algorithm. It randomizes over primes and generating vectors , and uses the median of independent lattice estimates to achieve near-optimal convergence: the randomized error satisfies and the deterministic error satisfies (with high probability). The analysis leverages the Korobov kernel, the Fourier-weight sum , and a median-amplification technique, showing dimension-free guarantees when , and extends to nonperiodic functions via tent- or smooth-transforms. Numerical experiments with periodic and nonperiodic test functions corroborate the theoretical rates, demonstrating universality and robustness across smoothness levels and dimensions. The result is a simple, implementable, and near-universal approach for high-dimensional quadrature that adaptively attains near-optimal rates without detailed prior knowledge of the integrand.

Abstract

We present a simple universal algorithm for high-dimensional integration which has the optimal error rate (independent of the dimension) in all weighted Korobov classes both in the randomized and the deterministic setting. Our theoretical findings are complemented by numerical tests.

Paper Structure

This paper contains 4 sections, 7 theorems, 52 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Preliminaries
  2. Analysis of the randomized error
  3. Analysis of the deterministic error
  4. Numerical examples

Key Result

Theorem 1

Algorithm alg:median has the following properties. For any $\alpha>1/2$ and $\gamma\in (0,1]^\mathbb{N}$, and any $\varepsilon >0$, there are positive constants $n_0=n_0(\alpha,\varepsilon,\gamma,d)$ and $C=C(\alpha,\varepsilon,\gamma,d)$ such that the following holds. Here, both $n_0$ and $C$ are independent of the dimension $d$ if $\gamma \in \ell_{1/\alpha}$.

Figures (3)

  • Figure 1: Results for the test functions $f_1$ and $f_2$ with $d=20$.
  • Figure 2: Results for the test function $f_{a,c}$ with $d=50$ and $c=2a+1$. Each subfigure corresponds to a different value of $a$.
  • Figure 3: Results for the nonperiodic test function with $d=10$.

Theorems & Definitions (17)

  • Theorem 1
  • Remark 2: Non-periodic functions
  • Remark 3
  • Lemma 4
  • proof
  • Remark 5
  • Lemma 6
  • Proposition 7
  • proof
  • Corollary 8
  • ...and 7 more