Rigorous error estimates for the heuristic Monte Carlo integration method

Develop precise error estimates for the heuristic Monte Carlo integration scheme that computes interface areas and cell volumes via ray-based Monte Carlo sampling and integrates functions by averaging values at Voronoi vertices and centroids, quantifying its approximation error in a principled way.

Background

The heuristic Monte Carlo (HMC) method is proposed to combine strengths of the exact polygonal method (accurate but expensive in high dimensions) and the plain Monte Carlo method (scales well but may require many function evaluations). HMC estimates areas/volumes via Monte Carlo rays and then averages function values at vertices and centroids to approximate integrals.

While computationally appealing for medium dimensions, the authors emphasize that the approach is heuristic. They explicitly acknowledge the lack of rigorous error estimates and call attention to the need for theoretical guarantees.

References

However we have to admit that the averaging is merely heuristically motivated and we don't know of any precise error estimates.

— Voronoi Graph -- Improved raycasting and integration schemes for high dimensional Voronoi diagrams (2405.10050 - Sikorski et al., 16 May 2024) in Section “Comparison of the suggested methods”