  We propose a sparse grid stochastic collocation method for long-time
simulations of stochastic differential equations (SDEs) driven by white noise.
The method uses pre-determined sparse quadrature rules for the forcing term and
constructs evolving set of sparse quadrature rules for the solution variables
in time. We carry out a restarting scheme to keep the dimension of random
variables for the forcing term, therefore also the number of quadrature points,
independent of time. At each restart, a sparse quadrature rule for the current
solution variables is constructed based on the knowledge of moments and the
previous quadrature rules via a minimization procedure. In this way, the method
allows us to capture the long-time solutions accurately using small degrees of
freedom. We apply the algorithm to low-dimensional nonlinear SDEs and
demonstrate its capability in long-time simulations numerically.
