ANOVA Decomposition of Convex Piecewise Linear Functions

Abstract

Piecewise linear convex functions arise as integrands in stochastic programs. They are Lipschitz continuous on their domain, but do not belong to tensor product Sobolev spaces. Motivated by applying Quasi-Monte Carlo methods we show that all terms of their ANOVA decomposition, except the one of highest order, are smooth if the underlying densities are smooth and a certain geometric condition is satisfied. The latter condition is generically satisfied in the normal case. © Springer-Verlag Berlin Heidelberg 2013.