  We study inertial versions of primal-dual proximal splitting, also known as
the Chambolle--Pock method. Our starting point is the preconditioned proximal
point formulation of this method. By adding correctors corresponding to the
anti-symmetric part of the relevant monotone operator, using a FISTA-style gap
unrolling argument, we are able to derive gap estimates instead of merely
ergodic gap estimates. Moreover, based on adding a diagonal component to this
corrector, we are able to combine strong convexity based acceleration with
inertial acceleration. We test our proposed method on image processing and
inverse problems problems, obtaining convergence improvements for sparse
Fourier inversion and Positron Emission Tomography.
