  NMR is a tool of choice for the measure of diffusion coefficients of species
in solution. The DOSY experiment, a 2D implementation of this measure, has
proven to be particularly useful for the study of complex mixtures, molecular
interactions, polymers, etc. However, DOSY data analysis requires to resort to
inverse Laplace transform, in particular for polydisperse samples. This is a
known difficult numerical task, for which we present here a novel approach. A
new algorithm based on a splitting scheme and on the use of proximity operators
is introduced. Used in conjunction with a Maximum Entropy and $\ell_1$ hybrid
regularisation, this algorithm converges rapidly and produces results robust
against experimental noise. This method has been called PALMA. It is able to
reproduce faithfully monodisperse as well as polydisperse systems, and numerous
simulated and experimental examples are presented. It has been implemented on
the server http://palma.labo.igbmc.fr where users can have their datasets
processed automatically.
