Mark Campanelli

Many problems involve the computation of the numeric output of a mathematical function of one or more numeric input variables. Typically, these input variables have uncertainty/variability, and can be modeled mathematically as random variables with a joint probability distribution. In this setting, the output from the function is itself a random variable having some probability distribution. Furthermore, the output distribution of a given function may be used as an input distribution to another function to produce yet another output distribution. This process, termed the propagation of distributions, can be computationally challenging.

The propagation of distributions has many applications in the realm of uncertainty quantification, including: evaluating/improving measurement precision, tolerancing for product design and manufacture, and risk management. Important issues in the propagation of distributions through a single function include the number and probabilistic dependence of the inputs, as well as the nonlinearity and computational expense of the function. A popular and generally applicable computational approach to the propagation of distributions is by Monte Carlo methods (MCMs). Using a random number generator, a typical MCM generates a numerical representation of the output distribution by repeatedly evaluating the given function at independently sampled values from the joint input distribution.

In this work, a sampling-free approach to the propagation of distributions is presented. The computational technique behind this alternative numerical approach, termed the Sandbox Algorithm, is described. A comparison to existing approaches is made, and some applications of the algorithm, implemented in MATLAB, are explored.