Introduction to numerical homogenization of PDEs with arbitrary rough coefficients

This talk is concerned with homogenization problems from the perspective of numerical analysis and approximation theory. By means of a linear elliptic model diffusion problem, we will introduce a numerical homogenization method that is based in the computation of operator-dependent subspaces with a quasi-local basis and uniform approximation properties for arbitrary rough diffusion coefficients. The key result in the corresponding error analysis is the exponential decay of the Green's function associated with numerical corrector problems. A non-standard application of the numerical homogenization method and its analysis is the sparse approximability of the expected solution operator in prototypical random diffusion problems.