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. Copyright ASCE 2007.