Error Estimation of Polynomial Chaos Approximations in Transient Structural Dynamics

Abstract

Usually, within stochastic framework, a testing dataset is used to evaluate the approximation error between a surrogate model (e.g., a polynomial chaos expansion) and the exact model. We propose here another method to estimate the quality of an approximated solution of a stochastic process, within the context of structural dynamics. We demonstrate that the approximation error is governed by an equation based on the residue of the approximate solution. This problem can be solved numerically using an approximated solution, here a coarse Monte Carlo simulation. The developed estimate is compared to a reference solution on a simple case. The study of this comparison makes it possible to validate the efficiency of the proposed method. This validation has been observed using different sets of simulations. To illustrate the applicability of the proposed approach to a more challenging problem, we also present a problem with a large number of random parameters. This illustration shows the interest of the method compared to classical estimates.

Verification of polynomial chaos surrogates in the framework of structural vibrations with uncertainties

“…An alternative approach, commonly used in statistical learning, is to measure the mean square error on an empirical test dataset. In Dao et al [27], the authors proposed such an approach, but the error was derived from a residual based problem. However, using a small dataset leads to a non-robust estimate of the mean square error, while employing a large size of testing datasets leads to prohibitive computation costs.…”

“…While Dao et al explored transient dynamics in their study [27], our focus in this paper is on structural vibrations. This type of problem is quite common, as uncertainties often manifest themselves in the mid-frequency range.…”