Preintegration is not smoothing when monotonicity fails

TL;DR

Preintegration aims to simplify high-dimensional integrals with discontinuities by smoothing, but this paper shows that monotonicity is essential; without it, the resulting integrand often remains non-smooth with singularities.

Contribution

This paper demonstrates that the monotonicity condition in preintegration is necessary for ensuring smoothness of the resulting integrand, challenging previous assumptions.

Findings

Without monotonicity, the integrand has singularities.

Monotonicity ensures smoothness after preintegration.

Preintegration may not smooth the integrand if monotonicity fails.

Abstract

Preintegration is a technique for high-dimensional integration over $d$-dimensional Euclidean space, which is designed to reduce an integral whose integrand contains kinks or jumps to a $(d-1)$-dimensional integral of a smooth function. The resulting smoothness allows efficient evaluation of the $(d-1)$-dimensional integral by a Quasi-Monte Carlo or Sparse Grid method. The technique is similar to conditional sampling in statistical contexts, but the intention is different: in conditional sampling the aim is to reduce the variance, rather than to achieve smoothness. Preintegration involves an initial integration with respect to one well chosen real-valued variable. Griebel, Kuo, Sloan [Math. Comp. 82 (2013), 383--400] and Griewank, Kuo, Le\"ovey, Sloan [J. Comput. Appl. Maths. 344 (2018), 259--274] showed that the resulting $(d-1)$-dimensional integrand is indeed smooth under appropriate conditions, including a key assumption -- the integrand of the smooth function underlying the kink or jump is strictly monotone with respect to the chosen special variable when all other variables are held fixed. The question addressed in this paper is whether this monotonicity property with respect to one well chosen variable is necessary. We show here that the answer is essentially yes, in the sense that without this property the resulting $(d-1)$-dimensional integrand is generally not smooth, having square-root or other singularities.