Preintegration is Not Smoothing When Monotonicity Fails

Abstract

Preintegration is a technique for high-dimensional integration over the 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 usually 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, Leövey, 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—that 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. The square-root singularity is generically enough to prevent the preintegrated function from belonging to the mixed derivative spaces typically used in Quasi-Monte Carlo or Sparse Grid analysis.