  The primal--dual hybrid gradient method (PDHGM, also known as the
Chambolle--Pock method) has proved very successful for convex optimization
problems involving linear operators arising in image processing and inverse
problems. In this paper, we analyze an extension to nonconvex problems that
arise if the operator is nonlinear. Based on the idea of testing, we derive new
step length parameter conditions for the convergence in infinite-dimensional
Hilbert spaces and provide acceleration rules for suitably (locally and/or
partially) monotone problems. Importantly, we prove linear convergence rates as
well as global convergence in certain cases. We demonstrate the efficacy of
these step length rules for PDE-constrained optimization problems.
