Imprecise random field analysis with parametrized kernel functions

Abstract

Abstract The application of isotropic random fields in engineering analysis requires the definition of their first two central moments, as well as their covariance function. In general, insufficient data are available to make a fully objective crisp estimate on these quantities, and hence, subjectivity enters implicitly into the analysis. The framework of imprecise probabilities is gaining popularity in this context, as it allows to explicitly separate the epistemic uncertainty, present due to data insufficiency, from the aleatory nature of the random parameters. However, an approach that is capable of handling imprecision in the complete definition of the imprecise random field is lacking to date. This paper proposes a framework for imprecise random field analysis with parametrized covariance functions. As such, the functional form of the covariance function is assumed to be known deterministically, whereas the governing parameters are subjected to imprecision. First, a comprehensive analysis of the effect of imprecise random fields, given imprecision on both the mean and auto-covariance structure, is presented. It is shown that the discretization of an imprecise random field, given interval-valued correlation lengths cannot be performed using interval arithmetical procedures as the resulting basis functions are in that case no longer a complete basis on L 2 . Therefore, an iterative procedure is proposed where the bounds on the imprecise random field basis are determined via an optimization procedure. This procedure provides exact bounds on the response of a system in case the system is monotonic with respect to the random field realizations. Two illustrative case studies on a mass-damper-spring system and a dynamic state-space model of a car suspension are included to illustrate the methodology. These case studies illustrate that indeed a separation between aleatory and epistemic uncertainty is possible in random field analysis, and hence, more objective results are obtained at only slightly increased computational cost.