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.

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关于使用 $\ell^1$-penalty 最小化 Tikhonov 泛函的注释

在本文中，我们考虑最小化具有 $\ell_1$ 惩罚的 Tikhonov 函数，以解决具有稀疏约束的线性逆问题。用于解决此问题的众多方法之一使用 Nemskii 算子将 Tikhonov 函数转换为具有 $\ell_2$ 惩罚项但非线性算子的函数。然后可以使用标准方法分析和最小化转换后的问题。然而，由于这种变换的性质，产生的泛函只能连续可微一次，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变换，它导致了一个两倍可微的泛函，现在可以使用有效的二阶方法（如牛顿法）来最小化。我们提供了我们提出的方案的收敛性分析，