One thing I have noticed is that the DI engine seems to be kinda sensitive to shitty or low octane gas. On decent gas the car purrs, on crap gas the car hesitates, shakes, sputters, etc.
I stopped filling up with the discount gas at my local supermarket because of this. I suspect the octane was actually lower than 87. I have been filling up at the local BP and the car runs well.