论文标题
关于用$ \ ell^1 $ -penalty功能最小化的tikhonov的注释
A note on the minimization of a Tikhonov functional with $\ell^1$-penalty
论文作者
论文摘要
在本文中，我们考虑了tikhonov功能的最小化，并以$ \ ell_1 $罚款，以求解具有稀疏性约束的线性反问题。用于解决此问题的众多方法之一使用NEMSKII操作员将Tikhonov功能转换为具有$ \ ell_2 $罚款项但非线性操作员的功能。然后，可以使用标准方法对转化的问题进行分析和最小化。但是，根据这种转换的性质，所得的功能仅是连续差异的一次，这禁止使用二阶方法。因此，在本文中，我们提出了一种不同的转换，这将导致两倍的可区分功能，现在可以使用牛顿方法（例如牛顿方法）进行有效的二阶方法将其最小化。我们提供了对我们提出的方案的收敛分析，以及许多数值结果，显示了我们提出的方法的有用性。
In this paper, we consider the minimization of a Tikhonov functional with an $\ell_1$ penalty for solving linear inverse problems with sparsity constraints. One of the many approaches used to solve this problem uses the Nemskii operator to transform the Tikhonov functional into one with an $\ell_2$ penalty term but a nonlinear operator. The transformed problem can then be analyzed and minimized using standard methods. However, by the nature of this transform, the resulting functional is only once continuously differentiable, which prohibits the use of second order methods. Hence, in this paper, we propose a different transformation, which leads to a twice differentiable functional that can now be minimized using efficient second order methods like Newton's method. We provide a convergence analysis of our proposed scheme, as well as a number of numerical results showing the usefulness of our proposed approach.
