Explain systematic bias of Monte Carlo vs. heuristic Monte Carlo integrals relative to polygonal integration

Determine the cause of the observed systematic bias in the integration results, specifically that the ray-based Monte Carlo integration method (Algorithm MonteCarlo using random rays and subsamples) systematically overestimates and the heuristic Monte Carlo integration method (which averages function values at Voronoi vertices and centroids while using Monte Carlo estimates for areas/volumes) systematically underestimates, relative to polygonal Leibnitz-based integration, for the test function f(x)=sin(x1^2) over Voronoi cells.

Background

The paper introduces three integration approaches over Voronoi cells: (i) a ray-based Monte Carlo method (MC) for area and volume integrals, (ii) an exact polygonal method based on recursive Leibnitz-type decompositions (P), and (iii) a heuristic Monte Carlo method (HMC) that uses Monte Carlo for areas/volumes and averages function values at vertices/centroids to mimic linear interpolation.

In numerical experiments (d=5, random nodes, function f(x)=sin(x1^2)), the authors compare MC and HMC against P using histograms of relative deviations. They report a consistent pattern: MC integrals overestimate while HMC integrals underestimate relative to P by around 2%. The authors explicitly state uncertainty about the reason for this behavior, suggesting it might be related to the function’s shape.