Uncertainty Quantification Using Multi-Dimensional Hermite Polynomials

Abstract

The general stochastic problem involves the propagation of input uncertainties through a computation model to arrive at a random output vector. This paper presents the application of the multi-dimensional Hermite polynomials to reduce an unknown random output vector into a significantly simpler unknown vector of numbers. The unknown numbers are evaluated using a collocation method because it has the important practical advantage of allowing existing deterministic numerical codes to be used as “black boxes”. A simple laterally loaded pile example involving two input random variables demonstrated that a third- or fourth-order Hermite expansion is adequate to reproduce probabilities of failure between 10 -3 and 10 -4 . A simple and efficient 2-term recurrence method for obtaining Hermite polynomials of any order in the case of two random dimensions is proposed. To our knowledge, this proposal appears to be original.