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

Regularity and sparse approximation of the recursive first moment equations for the lognormal Darcy problem