  Online optimisation revolves around new data being introduced into a problem
while it is still being solved; think of deep learning as more training samples
become available. We adapt the idea to dynamic inverse problems such as video
processing with optical flow. We introduce a corresponding predictive online
primal-dual proximal splitting method. The video frames now exactly correspond
to the algorithm iterations. A user-prescribed predictor describes the
evolution of the primal variable. To prove convergence we need a predictor for
the dual variable based on (proximal) gradient flow. This affects the model
that the method asymptotically minimises. We show that for inverse problems the
effect is, essentially, to construct a new dynamic regulariser based on infimal
convolution of the static regularisers with the temporal coupling. We finish by
demonstrating excellent real-time performance of our method in computational
image stabilisation and convergence in terms of regularisation theory.
