  Observations of microlensing transit events can be used to measure the limb
darkening of the lensed star. We discuss the advantages and drawbacks of
several microlensing light curve inversion methods. The method of choice in
this work is inversion by means of decomposition of the stellar surface
brightness profile. We construct an ideal basis by principal component analysis
of brightness profiles obtained from model atmosphere calculations. Limb
darkening approximations using such a basis are superior to those using
standard power-law limb darkening laws. We perform a full analysis of simulated
single-lens microlensing transit events including a detailed error analysis of
the method. In realistic events with a low impact parameter the brightness
profile of the source can be recovered with a relative accuracy of 2% from the
center of the source disk to 0.9 of the disk radius. We show that in the
particular case of the observed MACHO Alert 95-30 event the intrinsic complex
variability of the lensed red giant hinders efforts to recover its surface
features.
