Stochastic collocation methods facilitate the numerical solution of partial differential equations (PDEs) with random data and give rise to long sequences of similar linear systems. When elliptic PDEs with random diffusion coefficients are discretized with mixed finite element methods in the physical domain we obtain saddle point systems. These are trivial to solve when considered individually; the challenge lies in exploiting their similarities to recycle information and minimize the cost of solving the entire sequence. We apply stochastic collocation to a model stochastic elliptic problem and discretize in physical space using Raviart-Thomas elements. We propose an efficient solution strategy for the resulting linear systems that is more robust than any other in the literature. In particular, we show that it is feasible to use finely-tuned algebraic multigrid preconditioning if key set-up information is reused. The proposed solver is robust with respect to variations in the discretization and statistical parameters for stochastically linear and nonlinear data. © 2010 The author. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.