论文标题
牛顿算法的统一预处理，用于总变化最小化和最小表面问题
A uniform preconditioner for a Newton algorithm for total-variation minimization and minimum-surface problems
论文作者
论文摘要
Rudin-osher-Fatemi（ROF）和最小表面模型的非线性部分微分方程的解方法对于许多现代应用都是基础。已经提出了许多有效的算法。一阶方法很常见。由于它们的简单性和易于实施，它们很受欢迎。已经提出了一些牛顿型迭代方法的二阶阶，例如Chan-Golub-Mulet方法。在本文中，我们提出了一个新的牛顿 - 克里洛夫求解器，用于ROF模型的原始偶二元元素离散化。该方法是如此简单，以至于我们只需要在迭代过程中使用一些对角度调节器即可。从理论上讲，相对于网格大小，惩罚参数，正规化参数和迭代步骤，进一步证明了所提出的前提条件是鲁棒和最佳的，从本质上讲，它是参数独立的预处理。我们首先通过使用混合有限元方法来离散原始偶对系统，然后通过Newton \ TextQuoteright S方法线性化离散系统。利用配备适当规范的适当Sobolev空间上的线性化问题的适当性，我们建议使用最小残留方法求解的相应系统块对角线预处理。提出数值结果以支持理论结果。
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.
