Numerical simulation of multiscale problems with random defects

In this talk, we consider diffusion problems with multiscale coefficients subject to a certain random model describing defects e.g. in production processes. We are interested in numerically simulating the solutions for any given sample. Due to the multiscale character of the problem, computational multiscale methods based upon problem-adapted spaces should be used. To reduce the computational complexity of calculating a multiscale basis for each sample, we propose an offline-online strategy. We determine local stiffness matrices for a cleverly chosen set of coefficients offline and can then assemble the system matrix for any sample quickly in the online phase. We discuss the error analysis of the approach as well as some recent further developments and connections.