  We propose several variants of the primal-dual method due to Chambolle and
Pock. Without requiring full strong convexity of the objective functions, our
methods are accelerated on subspaces with strong convexity. This yields mixed
rates, $O(1/N^2)$ with respect to initialisation and $O(1/N)$ with respect to
the dual sequence, and the residual part of the primal sequence. We demonstrate
the efficacy of the proposed methods on image processing problems lacking
strong convexity, such as total generalised variation denoising and total
variation deblurring.
