A scalable parametric approximation to multi-normal probability integrals

The evaluation of probability integrals remains a central technical issue in reliability assessment of engineered systems. Recently, a newclass of approaches has been proposed by Bucher (2009) and Naess et al. (2009). The underlying idea is to formulate a scalable parameterization of the integrals in such a way that the scale factor is uniquely related to the probability. By calculating the integrals for a range of scale factors chosen such that e.g. Monte Carlo sampling is efficient for their solution (for large probabilities), the parameters in the parameterization may be readily estimated. Thereafter, the parameterization may be applied for the scale factors corresponding to the integral of interest which typically corresponds to a small probability. The performance of such approaches thus depends on the choice of parameterization of the probability integral and the functional form utilized for the extrapolation. The present paper proposes a scheme for the parameterization and the corresponding functional forms for the extrapolation, considering multi-normal probability integrals. The proposed scheme is illustrated on four principal examples including series and parallel systems and a case with multiple design points. Through the examples the performance of the proposed scheme is investigated, and the advantages are emphasized.