Abstract Ill-posed problems arise in many areas of science and engineering. Their solutions, if they exist, are very sensitive to perturbations in the data. To reduce this sensitivity, the original problem may be replaced by a minimization problem with a fidelity term and a regularization term. We consider minimization problems of this kind, in which the fidelity term is the square of the l 2 -norm of a discrepancy and the regularization term is the qth power of the l q -norm of the size of the computed solution measured in some manner. We are interested in the situation when 0 q ≤ 1 , because such a choice of q promotes sparsity of the computed solution. The regularization term is determined by a regularization matrix. Novati and Russo let q = 2 and proposed in (2014) [13] a regularization matrix that is a finite difference approximation of a differential operator applied to the computed approximate solution after reordering. This gives a Tikhonov regularization problem in general form. We show that this choice of regularization matrix also is well suited for minimization problems with 0 q ≤ 1 . Applications to image restoration are presented.

中文翻译：

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一种用于图像恢复的ℓ2-ℓ最小化方法的正则化矩阵的选择

摘要 不适定问题出现在许多科学和工程领域。他们的解决方案（如果存在）对数据中的扰动非常敏感。为了降低这种敏感性，原始问题可以用带有保真项和正则化项的最小化问题代替。我们考虑这种最小化问题，其中保真项是差异的 l 2 -范数的平方，正则化项是以某种方式测量的计算解的大小的 lq -范数的 q 次方。我们对 0 q ≤ 1 的情况感兴趣，因为这样选择 q 会提高计算解的稀疏性。正则化项由正则化矩阵确定。Novati 和 Russo 让 q = 2 并在 (2014) [13] 中提出了一个正则化矩阵，该矩阵是在重新排序后应用于计算的近似解的微分算子的有限差分近似。这给出了一般形式的 Tikhonov 正则化问题。我们表明，这种正则化矩阵的选择也非常适合于 0 q ≤ 1 的最小化问题。介绍了图像恢复的应用。