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We study the solution of minimax problems $\min_x \max_y G(x) + \langleK(x),y\rangle - F^*(y)$ in finite-dimensional Hilbert spaces. The functionals$G$ and $F^*$ we assume to be convex, but the operator $K$ we allow to benon-linear. We formulate a natural extension of the modified primal-dual hybridgradient method (PDHGM), originally for linear $K$, due to Chambolle and Pock.We prove the local convergence of the method, provided various technicalconditions are satisfied. These include in particular the Aubin property of theinverse a monotone operator at the solution. Of particular interest to us isthe case arising from reformulation of regularisation problems $\min_x\|f-T(x)\|^2/2 + \alpha R(x)$ with the operator $T$ non-linear. For suchproblems, we show that our general local convergence result holds when thenoise level of the data $f$ is low, and the regularisation parameter $\alpha$is correspondingly small. We verify the numerical performance of the method byapplying it to problems from magnetic resonance imaging (MRI) in chemicalengineering and medicine. The specific applications are in diffusion tensorimaging (DTI) and MR velocity imaging. These numerical studies show verypromising performance.

A primal-dual hybrid gradient method for non-linear operators with applications to MRI