  We study and characterize the optimal rates of convergence in periodic
homogenization of linear elliptic equations in non-divergence form. We obtain
that the optimal rate of convergence is either $O(\varepsilon)$ or
$O(\varepsilon^2)$ depending on the diffusion matrix $A$, source term $f$, and
boundary data $g$. Moreover, we show that the set of diffusion matrices $A$
that give optimal rate $O(\varepsilon)$ is open and dense in the set of
$C^{2,\alpha}$ periodic, symmetric, and positive definite matrices, which means
that generically, the optimal rate is $O(\varepsilon)$.
