Choosing an appropriate regularization term is necessary to obtain a meaningful solution to an ill-posed linear inverse problem contaminated with measurement errors or noise. The $\ell_p$ norm covers a wide range of choices for the regularization term since its behavior critically depends on the choice of $p$ and since it can easily be combined with a suitable regularization matrix. We develop an efficient algorithm that simultaneously determines the regularization parameter and corresponding $\ell_p$ regularized solution such that the discrepancy principle is satisfied. We project the problem on a low-dimensional Generalized Krylov subspace and compute the Newton direction for this much smaller problem. We illustrate some interesting properties of the algorithm and compare its performance with other state-of-the-art approaches using a number of numerical experiments, with a special focus of the sparsity inducing $\ell_1$ norm and edge-preserving total variation regularization.

中文翻译：

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用于噪声约束 $\ell_p$ 正则化的投影牛顿法

选择合适的正则化项对于获得对受测量误差或噪声污染的不适定线性逆问题的有意义的解是必要的。$\ell_p$ 范数涵盖了正则化项的广泛选择，因为它的行为严重依赖于 $p$ 的选择，并且可以很容易地与合适的正则化矩阵相结合。我们开发了一种有效的算法，可以同时确定正则化参数和相应的 $\ell_p$ 正则化解，以满足差异原则。我们将问题投影到低维广义 Krylov 子空间上，并为这个小得多的问题计算牛顿方向。