  The two-loop self-energy correction to the ground state Lamb shift is
calculated for hydrogen-like ions with the nuclear charge Z=10-30 without any
expansion in the binding field of the nucleus. A calculational technique is
reported for treatment of Feynman diagrams in the mixed coordinate-momentum
representation, which yields significant improvement in numerical accuracy as
compared to previous results. An extrapolation of the all-order numerical data
towards Z=1 yields a result for the higher-order remainder function for
hydrogen. The previously reported disagreement between the all-order and the
perturbative approaches is reduced to the marginal agreement.
