  We present a novel algorithm based on the ensemble Kalman filter to solve
inverse problems involving multiscale elliptic partial differential equations.
Our method is based on numerical homogenization and finite element
discretization and allows to recover a highly oscillatory tensor from
measurements of the multiscale solution in a computationally inexpensive
manner. The properties of the approximate solution are analysed with respect to
the multiscale and discretization parameters, and a convergence result is shown
to hold. A reinterpretation of the solution from a Bayesian perspective is
provided, and convergence of the approximate conditional posterior distribution
is proved with respect to the Wasserstein distance. A numerical experiment
validates our methodology, with a particular emphasis on modelling error and
computational cost.
