  For a regular function f on a smooth complex quasi-projective variety, J.-D.
Yu introduced in arXiv:1203.2338 a filtration (the irregular Hodge filtration)
on the de Rham complex with twisted differential d+df, extending a definition
of Deligne in the case of curves. In this article, we show the degeneration at
E_1 of the spectral sequence attached to the irregular Hodge filtration, by
using the method of arXiv:0804.4328. We also make explicit the relation with a
complex introduced by M. Kontsevich and give details on its proof of the
corresponding E_1 degeneration, by reduction to characteristic p, when the pole
divisor of the function is reduced with normal crossings. In Appendix E, M.
Saito gives a different proof of the latter statement with a possibly non
reduced normal crossing pole divisor.
