Polynomial chaos expansions of response quantities have been widely used in Computational Stochastic Mechanics and are well documented. Introduced in conjunction with a truncated Karhunen-Loeve-representation of the input random field, they represent global approximations in the Hilbert space of functions of (usually standard Gaussian) random variables. However, the global approximation character may lead to inefficient convergence behavior for higher order response moments or small response probabilities.
Therefore, after multiplicative decomposition in a deterministic and a random part, local polynomial expansions of the solution are introduced by partitioning the domain of random variables and the physical domain. By carefully choosing the local basis, the problem decouples in the random domain. The expansion coefficients can then be determined independently by parallel processing. Moreover, local expansions allow to construct new hybrid simulation schemes, that is, combinations of analytical and simulation based techniques.
For reliability estimation, the expansion can be interpreted as a local response surface. Starting from the global approximation, a local response surface can be constructed by computing the design point and sensitivities. After that, suitable local approximations can be introduced by decomposing the region of most probable failure.