Numerical homogenization concerns the (coarse) finite dimensional approximation of the solution space of, for example, divergence form elliptic equation with L^\infty coefficients which allows for nonseparable scales. 

Based on a Bayesian reformulation of numerical homogenization, we can propose a class of numerical homogenization methods which allow for exponential decaying bases and localisation, as well as optimal convergence rates. This can be used to construct efficient and robust fine scale fast solvers such as multi-resolution decomposition (the so-called “gamblet” decomposition) or multigrid solver with bounded condition number on each subband, and enables the resolution of boundary value problems in near-linear complexity and rigorous a-priori error bounds. The method can be generalized to time dependent problems such as wave propagation in heterogeneous media, and multi-scale eigenvalue problems. This is a joint work with Houman Owhadi (Caltech) and Hehu Xie (CAS).