  We extend stochastic basis adaptation and spatial domain decomposition
methods to solve time varying stochastic partial differential equations (SPDEs)
with a large number of input random parameters. Stochastic basis adaptation
allows the determination of a low dimensional stochastic basis representation
of a quantity of interest (QoI). Extending basis adaptation to time-dependent
problems is challenging because small errors introduced in the previous time
steps of the low dimensional approximate solution accumulate over time and
cause divergence from the true solution. To address this issue we have
introduced an approach where the basis adaptation varies at every time step so
that the low dimensional basis is adapted to the QoI at that time step. We have
coupled the time-dependent basis adaptation with domain decomposition to
further increase the accuracy in the representation of the QoI. To illustrate
the construction, we present numerical results for one-dimensional time varying
linear and nonlinear diffusion equations with random space-dependent diffusion
coefficients. Stochastic dimension reduction techniques proposed in the
literature have mainly focused on quantifying the uncertainty in time
independent and scalar QoI. To the best of our knowledge, this is the first
attempt to extend dimensional reduction techniques to time varying and
spatially dependent quantities such as the solution of SPDEs.
