  We study the Darcy boundary value problem with log-normal permeability field.
We adopt a perturbation approach, expanding the solution in Taylor series
around the nominal value of the coefficient, and approximating the expected
value of the stochastic solution of the PDE by the expected value of its Taylor
polynomial. The recursive deterministic equation satisfied by the expected
value of the Taylor polynomial (first moment equation) is formally derived.
Well-posedness and regularity results for the recursion are proved to hold in
Sobolev space-valued H\"older spaces with mixed regularity. The recursive first
moment equation is then discretized by means of a sparse approximation
technique, and the convergence rates are derived.
