  We develop block structure adapted primal-dual algorithms for non-convex
non-smooth optimisation problems whose objectives can be written as
compositions $G(x)+F(K(x))$ of non-smooth block-separable convex functions $G$
and $F$ with a non-linear Lipschitz-differentiable operator $K$. Our methods
are refinements of the non-linear primal-dual proximal splitting method for
such problems without the block structure, which itself is based on the
primal-dual proximal splitting method of Chambolle and Pock for convex
problems. We propose individual step length parameters and acceleration rules
for each of the primal and dual blocks of the problem. This allows them to
convergence faster by adapting to the structure of the problem. For the squared
distance of the iterates to a critical point, we show local $O(1/N)$,
$O(1/N^2)$ and linear rates under varying conditions and choices of the step
lengths parameters. Finally, we demonstrate the performance of the methods on
practical inverse problems: diffusion tensor imaging and electrical impedance
tomography.
