The relationship between Galerkin and collocation methods in statistical transmission line analysis

This paper discusses the relationship between two standard methods for the stochastic analysis of linear circuits, namely the stochastic Galerkin method (SGM) and the stochastic collocation method (SCM), based on a multidimensional Gaussian quadrature. It is established that the SCM corresponds to an approximate factorization of the SGM, involving matrix polynomials sharing the same coefficients as the pertinent polynomial chaos basis functions. Under certain assumptions, the two methods coincide. These findings are illustrated by means of a frequency-domain simulation of a transmission line circuit