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

arXiv: https://arxiv.org/abs/1309.5032 · HTML: https://mlsanity.com/paper/1309.5032/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Sep 2013.

## Built on by (most cited first, 5 of 5)

- [Solving Inverse Problems by Joint Posterior Maximization with Autoencoding Prior](https://mlsanity.com/paper/2103.01648/) (2021)
- [Accelerated primal-dual methods with enlarged step sizes and operator learning for nonsmooth optimal control problems](https://mlsanity.com/paper/2307.00296/) (2023)
- [Lifting-based variational multiclass segmentation algorithm: design, convergence analysis, and implementation with applications in medical imaging](https://mlsanity.com/paper/2202.04680/) (2022)
- [Variational multichannel multiclass segmentation using unsupervised lifting with CNNs](https://mlsanity.com/paper/2302.02214/) (2023)
- [Online optimisation for dynamic electrical impedance tomography](https://mlsanity.com/paper/2412.12944/) (2024)

## Abstract (arXiv)

We study the solution of minimax problems minx maxy G(x) + langle K(x),yrangle - 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 minx |f-T(x)|2/2 + α 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 α 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.