A uniform preconditioner for a Newton algorithm for total-variation minimization and minimum-surface problems

TL;DR

This paper introduces a simple, robust, and parameter-independent preconditioner for a Newton-Krylov solver addressing total-variation minimization and minimum-surface problems, demonstrating optimal performance across various parameters.

Contribution

The paper proposes a new diagonal preconditioner for a Newton-Krylov solver that is theoretically proven to be robust and optimal, independent of mesh size and regularization parameters.

Findings

Preconditioner is robust and optimal across mesh sizes and parameters.

Numerical results confirm theoretical robustness and efficiency.

The method simplifies implementation with diagonal preconditioners.

Abstract

Solution methods for the nonlinear partial differential equation of the Rudin-Osher-Fatemi (ROF) and minimum-surface models are fundamental for many modern applications. Many efficient algorithms have been proposed. First order methods are common. They are popular due to their simplicity and easy implementation. Some second order Newton-type iterative methods have been proposed like Chan-Golub-Mulet method. In this paper, we propose a new Newton-Krylov solver for primal-dual finite element discretization of the ROF model. The method is so simple that we just need to use some diagonal preconditioners during the iterations. Theoretically, the proposed preconditioners are further proved to be robust and optimal with respect to the mesh size, the penalization parameter, the regularization parameter, and the iterative step, essentially it is a parameter independent preconditioner. We first discretize the primal-dual system by using mixed finite element methods, and then linearize the discrete system by Newton\textquoteright s method. Exploiting the well-posedness of the linearized problem on appropriate Sobolev spaces equipped with proper norms, we propose block diagonal preconditioners for the corresponding system solved with the minimum residual method. Numerical results are presented to support the theoretical results.