On interpolating between probability distributions

Abstract

Uncertainty in the simulation of physical systems is common and may be due to parameter variations or noise. When the uncertainty is random in nature, probability distributions for the quantities of interest are obtained. Sometimes, knowing only the mean and variance is sufficient; at other times, safety and reliability considerations require knowledge of the complete distribution. Computing these distributions is often a time-consuming task and needs to be repeated when some system parameters are changed. In this paper, formulas for interpolating between probability density and mass functions in spaces of arbitrary dimensionality are presented. It is found that these formulas give accurate results even when the functions one is interpolating between are not that “close”. As the mesh used in interpolation is refined, the accuracy of the interpolated quantities increases; accordingly, in addition to the more complicated and robust interpolation formulas meant for the case of a coarse mesh, simplified versions that result in good accuracy when the mesh is fine are also given. Savings in computational effort up to a factor of one hundred are common even when the more complicated interpolation formulas are used. This means that interpolation is a lucrative alternative to Monte Carlo simulation and even to the Generalized Cell Mapping (GCM) method when complete probability distributions, as opposed to only the low-order statistics, are needed. It is expected that this technique will relieve much of the burden of repeated, time-consuming simulations as certain relevant parameters are varied. In addition, the given formulas may be interpreted as general-purpose algorithms for the blending of shapes, thereby leading to applications beyond what is considered in this paper.