The computation of sparse solutions of large-scale linear discrete ill-posed problems remains a computationally demanding task. A powerful framework in this context is the use of iteratively reweighted schemes, which are based on constructing a sequence of quadratic tangent majorants of the $\ell_2$-$\ell_1$ regularization functional (with additional smoothing to ensure differentiability at the origin), and solving them successively. Recently, flexible Krylov-Tikhonov methods have been used to partially solve each problem in the sequence efficiently. However, in order to guarantee convergence, the complexity of the algorithm at each iteration increases with respect to more traditional methods. We propose a randomized flexible Krylov method to alleviate the increase of complexity, which leverages the adaptability of the flexible Krylov subspaces with the efficiency of `sketch-and-solve' methods. A possible caveat of the mentioned methods is their memory requirements. In this case, one needs to rely instead on inner-outer schemes. In these scenarios, we propose a `sketch-to-precondition' method to speed up the convergence of each of the subproblems in the sequence. The performance of these algorithms is shown through a variety of numerical examples.

翻译：大规模线性离散不适定问题稀疏解的计算仍然是一项计算密集的任务。在此背景下，一个强大的框架是使用迭代重加权方案，该方案基于构建$\ell_2$-$\ell_1$正则化泛函（在原点处通过额外平滑处理以确保可微性）的二次切线主函数序列，并依次求解它们。最近，灵活的Krylov-Tikhonov方法已被用于高效地部分求解序列中的每个问题。然而，为了保证收敛，算法在每次迭代中的复杂度相对于更传统的方法有所增加。我们提出了一种随机化灵活Krylov方法来缓解复杂度的增加，该方法将灵活Krylov子空间的适应性与“草图求解”方法的效率相结合。上述方法的一个潜在缺陷是其内存需求。在这种情况下，需要转而依赖内外迭代方案。在这些场景中，我们提出了一种“草图预条件”方法来加速序列中每个子问题的收敛。通过各种数值示例展示了这些算法的性能。