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

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