We investigate the direct initiation of cylindrical detonations in free space by performing large-scale computations on a supercomputer. The two-dimensional (2D) compressible reactive Euler equations with a one-step chemical reaction model are solved by a well-validated upwind CE/SE scheme using up to 1.6 billion mesh points. Numerical results imply that one-dimensional (1D) approaches can only interpret the direct initiation mechanism of stable detonations. Inherent multi-dimensional instabilities have a significant influence on the direct initiation of unstable detonations. On one hand, multi-dimensional instabilities make the detonation more unstable and increase the risk of failure of the detonation. On the other hand, the collision of transverse waves generated from multi-dimensional instabilities leads to the initiation of local overdriven detonations that can enhance the overall self-sustainability of the global process. The competition between these two effects is an important mechanism to interpret the direct initiation of multi-dimensional detonations.