Independent Component Analysis to enhance performances of Karhunen-Loeve expansions for non-Gaussian stochastic processes: Application to uncertain systems

摘要

In the context of engineering systems, an essential step in uncertainty quantification is the development of accurate and efficient representation of the random input parameters. For such input parameters modeled as stochastic processes, Karhunen-Loeve expansion is a classical approach providing efficient representations using a set of uncorrelated, but generally statistically dependent random variables. The dependence structure among these random variables may be difficult to estimate statistically and is thus ignored in many practical applications. This simplifying assumption of independence may lead to considerable errors in estimating the variability in the system state, thus limiting the effectiveness of Karhunen-Loeve expansion in certain cases. In this paper, Independent Component Analysis is exploited to linearly transform the random variables used in Karhunen-Loeve expansion resulting into a set of random variables exhibiting higher order decorrelation. The stochastic wave equation is investigated for numerical illustration whereby the random stiffness coefficient is modeled as a non-Gaussian stochastic process. Under the assumption of independence among the random variables used in the Karhunen-Loeve expansion and Independent Component Analysis representations, the latter provides more accurate statistical characterization of the output process for the specific cases examined.