An efficient implementation algorithm for quasi-Monte Carlo approximations of high-dimensional integrals

TL;DR

This work targets high-dimensional numerical integration by improving quasi-Monte Carlo lattice rules through an affine-coordinate transformation that reveals a tensor-product structure, enabling efficient evaluation via multilevel dimension iteration (MDI). The proposed MDI-LR algorithm combines the transformed tensor-product view with MDI to compute multi-dimensional sums with reduced time and memory requirements, achieving a practical polynomial complexity of about $O(N^2 d^3)$ in many regimes. Extensive numerical experiments compare MDI-LR to standard lattice rules and improved variants, demonstrating substantial speedups in medium and high dimensions while preserving or enhancing accuracy. The results indicate that MDI-LR revitalizes QMC lattice rules for high-dimensional problems and provides a concrete pathway toward scalable, high-dimensional integration in applications, with avenues for further extension to related Monte Carlo frameworks.

Abstract

In this paper, we develop and test a fast numerical algorithm, called MDI-LR, for efficient implementation of quasi-Monte Carlo lattice rules for computing $d$-dimensional integrals of a given function. It is based on the idea of converting and improving the underlying lattice rule into a tensor product rule by an affine transformation and adopting the multilevel dimension iteration approach which computes the function evaluations (at the integration points) in the tensor product multi-summation in cluster and iterates along each (transformed) coordinate direction so that a lot of computations can be reused. The proposed algorithm also eliminates the need for storing integration points and computing function values independently at each point. Extensive numerical experiments are presented to gauge the performance of the algorithm MDI-LR and to compare it with standard implementation of quasi-Monte Carlo lattice rules. It is also showed numerically that the algorithm MDI-LR can achieve a computational complexity of order $O(N^2d^3)$ or better, where $N$ represents the number of points in each (transformed) coordinate direction and $d$ standard for the dimension. Thus, the algorithm MDI-LR effectively overcomes the curse of dimensionality and revitalizes QMC lattice rules for high-dimensional integration.

Figures (12)

• Figure 1: $81$-point lattice with generating vectors $(1, 2), (1, 4)$, and $(1, 7)$.
• Figure 1: Left: 81-point lattice with the generating vector $(1, 4)$. Right: transformed lattice after affine coordinate transformation.
• Figure 1: CUP time comparison of SLR and MDI-LR simulations: the number of lattice points increases in dimension (left), the number of lattice points increases slowly (right).
• Figure 1: Performance comparison of algorithm MDI-LR with $n=1+10^{d}$ and $a=4,6,8,10,12,14,16$ for computing $I_d(f)$, $I_d(\widehat{f})$ and $I_d(\widetilde{f})$. Top left: $d=5$, CPU time comparison. Top right: $d=10$, CPU time comparison. Bottom left: $d=5$, comparison of relative errors. Bottom right: $d=10$, comparison of relative errors
• Figure 1: The relationship between the CPU time and parameter $N$ when $d=5$: $I_d(f)$ (left), $I_d(\widehat{f})$ (middle), $I_d(\widetilde{f})$ (right).

Theorems & Definitions (14)

• Definition 1: rank-one lattice rule
• Theorem 2
• Theorem 3
• Theorem 4
• Remark 2.1
• Example 1: Fibonacci lattice
• Example 2: Korobov lattice
• Example 3: CBC lattice
• Theorem 1
• Proof 1