Generalized Arnoldi--Tikhonov Method for Sparse Reconstruction

This paper introduces two new algorithms, belonging to the class of Arnoldi--Tikhonov regularization methods, which are particularly appropriate for sparse reconstruction. The main idea is to consider suitable adaptively defined regularization matrices that allow the usual 2-norm regularization term to approximate a more general regularization term expressed in the $p$-norm, $p\geq 1$. The regularization matrix can be updated both at each step and after some iterations have been performed, leading to two different approaches: the first one is based on the idea of the iteratively reweighted least squares method and can be obtained considering flexible Krylov subspaces; the second one is based on restarting the Arnoldi algorithm. Numerical examples are given in order to show the effectiveness of these new methods, and comparisons with some other already existing algorithms are made.