  We develop a domain-decomposition model reduction method for linear
steady-state convection-diffusion equations with random coefficients. Of
particular interest to this effort are the diffusion equations with random
diffusivities, and the convection-dominated transport equations with random
velocities. We investigate the equations with two types of random fields, i.e.,
colored noises and discrete white noises, both of which can lead to
high-dimensional parametric dependence. The motivation is to use domain
decomposition to exploit low-dimensional structures of local problems in the
sub-domains, such that the total number of expensive PDE solves can be greatly
reduced. Our objective is to develop an efficient model reduction method to
simultaneously handle high-dimensionality and irregular behaviors of the
stochastic PDEs under consideration. The advantages of our method lie in three
aspects: (i) online-offline decomposition, i.e., the online cost is independent
of the size of the triangle mesh; (ii) operator approximation for handling
non-affine and high-dimensional random fields; (iii) effective strategy to
capture irregular behaviors, e.g., sharp transitions of the PDE solution. Two
numerical examples will be provided to demonstrate the advantageous performance
of our method.
