The issue is in fact already present when working with only caplets. Especially in the SABR LMM, under particular assumptions, this parametrization doesn't work. An alternative method is described in this upcoming book (tests done in particular challenging market conditions are showed). The method proposed however can slow down the simulation although a simple workaround for this problem is available.I cannot comment on the reasons why Piterbarg and Andersen suggest to use a grid of LIBOR vols (I don't have their book with me at the moment). From my personal experience this solution simplifies a lot the simulation however it doesn't allow you to recover the forward term structure of vols. I don't think there is a simple answer - it depends a lot by the market and the product you are trying to price. Probably you might want to try both approaches and understand how your price is affected, maybe also considering in the equation simulation issues and instabilities. EDIT: typos

hi Japanstar,I m not focus on capturing the total smile, as I believe the shift will be ok to get a good slope. I am interested in the right compromised between the calibration accuracy and the time homogeneity. I am not targeting exotic products.An application would be how to price a mid curve swaption or a bermudan (fwd vols dependent).