Sequential Monte Carlo Sampling of Feasible Regions Represented by R-Functions

Abstract

Feasible-region sampling and volume estimation become increasingly inefficient when constraints define only a small fraction of a convenient bounding domain. We combine R-functions, which encode compound feasibility as a continuous signed field, with sequential Monte Carlo (SMC) samplers. The proposed construction exploits not only the sign of the R-function, which determines exact feasibility, but also its continuous negative values outside the feasible set, which provide graded information on constraint violation. Logistic intermediate distributions use this information to guide particles progressively towards feasibility, while the final target distribution remains uniform on the feasible region. Experiments on intersections and unions, including disconnected cases of hyperballs, show substantial variance reductions relative to direct rejection sampling, with the statistical advantage generally increasing as the feasible fraction decreases. For sufficiently small feasible regions, these advantages persist even after accounting for computational cost. To isolate the benefit of continuous R-function information, we also compare the proposed method with an SMC sampler with a region-only (RO) construction that uses the same sequential particle machinery but retains only binary feasible/infeasible information. We demonstrate that the R-function-informed SMC method concentrates particles in the feasible region earlier and produces lower-variance volume estimates than RO.