  In this paper we study elliptic partial differential equations with rapidly
varying diffusion coefficient that can be represented as a perturbation of a
reference coefficient. We develop a numerical method for efficiently solving
multiple perturbed problems by reusing local computations performed with the
reference coefficient. The proposed method is based on the Petrov--Galerkin
Localized Orthogonal Decomposition (PG-LOD) which allows for straightforward
parallelization with low communcation overhead and memory consumption. We focus
on two types of perturbations: local defects which we treat by recomputation of
multiscale shape functions and global mappings of a reference coefficient for
which we apply the domain mapping method. We analyze the proposed method for
these problem classes and present several numerical examples.
