First-order second-moment method
In probability theory, the first-order second-moment (FOSM) method, also referenced as mean value first-order second-moment (MVFOSM) method, is a probabilistic method to determine the stochastic moments of a function with random input variables. The name is based on the derivation, which uses a "first-order" Taylor series and the first and "second moments" of the input variables. Approximation. Consider the objective function formula_1, where the input vector formula_2 is a realization of the random vector formula_3 with probability density function formula_4. Because formula_3 is randomly distributed, formula_6 is also randomly distributed . Following the FOSM method, the mean value of formula_6 is approximated by The variance of formula_6 is approximated by where formula_11 is the length/dimension of formula_2 and formula_13 is the partial derivative of formula_6 at the mean vector formula_15 with respect to the "i"-th entry of formula_2. More accurate, second-order second-moment approximations are also available Derivation. The objective function is approximated by a Taylor series at the mean vector formula_15. The mean value of formula_6 is given by the integral Inserting the first-order Taylor series yields The variance of formula_6 is given by the integral According  to the computational formula for the variance, this can be written as Inserting the Taylor series yields Higher-order approaches. The following abbreviations are introduced. In the following, the entries of the random vector formula_3 are assumed to be independent. Considering also the second-order terms of the Taylor expansion, the approximation of the mean value is given by The second-order approximation of the variance is given by The skewness of formula_6 can be determined from the third central moment formula_31. When considering only linear terms of the Taylor series, but higher-order moments, the third central moment is  approximated by For the second-order approximations of the third central moment as well as for the derivation of all higher-order approximations see Appendix D of Ref. Taking into account the quadratic terms of the Taylor series and the third moments of the input variables is referred to as second-order third-moment method. However, the full second-order approach of the variance (given above) also includes fourth-order moments of input parameters, the full second-order approach of the skewness 6th-order moments, and the full second-order approach of the kurtosis up to 8th-order moments. Practical  application. There are several examples in the literature where the FOSM method is employed to estimate the stochastic distribution of the buckling load of axially compressed structures (see e.g. Ref.). For structures which are very sensitive to deviations from the ideal structure (like cylindrical shells) it has been proposed to use the FOSM method as a design approach. Often the applicability is checked by comparison with a Monte Carlo simulation. Two comprehensive application examples of the full second-order method specifically oriented towards the fatigue crack growth in a metal railway axle are discussed and checked by comparison with a  Monte Carlo simulation in Ref. In engineering practice, the objective function often is not given as analytic expression, but for instance as a result of a finite-element simulation. Then the derivatives of the objective function need to be estimated by the central differences method. The number of evaluations of the objective function equals formula_33. Depending on the number of random variables this still can mean a significantly smaller number of evaluations than performing a Monte Carlo simulation. However, when using the FOSM method as a design procedure, a lower bound shall be estimated, which is actually not given by  the FOSM approach. Therefore, a type of distribution needs to be assumed for the distribution of the objective function, taking into account the approximated mean value and standard deviation.