Ogus, Vologodsky and Schepler established a correspondence between the categories of nilpotent logarithmic flat bundles and of nilpotent logarithmic Higgs bundles. Under this correspondence, they generalized the decomposition theorem of Deligne-Illusie in a large extent. We give an intersection version of this generalized decomposition theorem with some applications. This is a joint work with Mao Sheng.