  The light speed anisotropy, i.e., the variation of the light speed with
respect to the direction in an "absolute" reference frame, is a profound issue
in physics. The one-way experiment, performed at the GRAAL facility of the
European Synchrotron Radiation Facility (ESRF) in Grenoble, reported results on
the light speed anisotropy by Compton scattering of laser photons on
high-energy electrons. So far, most articles concerned with the GRAAL data have
established only the upper bounds on the anisotropy parameters based on
available theories. We use a new theory of the Lorentz invariance violation to
analyse the available GRAAL data and obtain the stringent upper limit of the
order $2.4\times10^{-14}$ on the Lorentz violation parameters. In the meantime,
we also can reproduce the allowed light speed anisotropy appearing in the
azimuthal distribution of the GRAAL experimental data, and find that the
best-fit parameters are compatible with the competitive upper bounds.
