  We show that the expected solution operator of prototypical linear elliptic
partial differential operators with random coefficients is well approximated by
a computable sparse matrix. This result is based on a random localized
orthogonal multiresolution decomposition of the solution space that allows both
the sparse approximate inversion of the random operator represented in this
basis as well as its stochastic averaging. The approximate expected solution
operator can be interpreted in terms of classical Haar wavelets. When combined
with a suitable sampling approach for the expectation, this construction leads
to an efficient method for computing a sparse representation of the expected
solution operator.
