  We present a reduced basis stochastic Galerkin method for partial
differential equations with random inputs. In this method, the reduced basis
methodology is integrated into the stochastic Galerkin method, resulting in a
significant reduction in the cost of solving the Galerkin system. To reduce the
main cost of matrix-vector manipulation involved in our reduced basis
stochastic Galerkin approach, the secant method is applied to identify the
number of reduced basis functions. We present a general mathematical framework
of the methodology, validate its accuracy and demonstrate its efficiency with
numerical experiments.
