  In this paper, we propose a dynamically low-dimensional approximation method
to solve a class of time-dependent multiscale stochastic diffusion equations. A
dynamically bi-orthogonal (DyBO) method was developed to explore
low-dimensional structures of stochastic partial differential equations (SPDEs)
and solve them efficiently. However, when the SPDEs have multiscale features in
physical space, the original DyBO method becomes expensive. To address this
issue, we construct multiscale basis functions within each coarse grid block
for dimension reduction in the physical space. To further improve the accuracy,
we also perform online procedure to construct online adaptive basis functions.
In the stochastic space, we use the generalized polynomial chaos (gPC) basis
functions to represent the stochastic part of the solutions. Numerical results
are presented to demonstrate the efficiency of the proposed method in solving
time-dependent PDEs with multiscale and random features.
