A primal-dual hybrid gradient method for nonlinear operators with applications to MRI.

*(English)*Zbl 1310.47081This is a very extensive and detailed paper concerning the solution of minimax problems
\[
\min_x \max_y \{G(x)+\langle K(x),y\rangle-F^*(y)\}
\]
in the context of iterative methods which can be applied, for example, in chemical engineering and medicine to magnetic resonance imaging (MRI) when the stable approximate solution of such problems is required. Here, \(G\) and \(F^*\) are convex, proper and lower-semicontinuous functionals defined on finite dimensional Hilbert spaces, and \(K\) is a nonlinear operator possessing \(C^2\)-smoothness. In this sense, the paper under review tries to extend the seminal paper by A. Chambolle and T. Pock [J. Math. Imaging Vis. 40, No. 1, 120–145 (2011; Zbl 1255.68217)], which has suggested and analyzed a primal-dual hybrid gradient method, but with focus on linear operators \(K\).

In the well-written Introduction, the author explains the advantage of handling the minimax problem instead of generalized versions of the Tikhonov regularization with one regularization parameter, e.g., in the case of TV-penalty. As outlined in Section 5 in more detail, for the recovery of complex-valued images in MRI, nonlinear operators \(K\) are needed, and it seems to be better to regularize phase and amplitude differently. After presenting the basics in Section 2, a comprehensive analysis of the method can be found in Section 3. Section 4 is devoted to Lipschitz estimates, where the focus of two specific subsections is on regularization functionals with \(L^1\)-type norms and on squared \(L^2\) cost functionals with \(L^1\)-type regularization, respectively. The final section concerning MRI applications and diffusion tensor imaging provides the reader also with corresponding computational experiments, including illustrations.

In the well-written Introduction, the author explains the advantage of handling the minimax problem instead of generalized versions of the Tikhonov regularization with one regularization parameter, e.g., in the case of TV-penalty. As outlined in Section 5 in more detail, for the recovery of complex-valued images in MRI, nonlinear operators \(K\) are needed, and it seems to be better to regularize phase and amplitude differently. After presenting the basics in Section 2, a comprehensive analysis of the method can be found in Section 3. Section 4 is devoted to Lipschitz estimates, where the focus of two specific subsections is on regularization functionals with \(L^1\)-type norms and on squared \(L^2\) cost functionals with \(L^1\)-type regularization, respectively. The final section concerning MRI applications and diffusion tensor imaging provides the reader also with corresponding computational experiments, including illustrations.

Reviewer: Bernd Hofmann (Chemnitz)

##### MSC:

47J06 | Nonlinear ill-posed problems |

49J35 | Existence of solutions for minimax problems |

90C52 | Methods of reduced gradient type |

92C55 | Biomedical imaging and signal processing |