Two new hybrid algorithms are proposed for large-scale linear discrete ill-posed problems in general-form regularization. They are both based on Krylov subspace inner-outer iterative algorithms. At each iteration, they need to solve a linear least squares problem. It is proved that the inner linear least squares problems, which are solved by LSQR, become better conditioned as k increases, so LSQR converges faster. We also prove how to choose the stopping tolerance for LSQR in order to guarantee that the computed solutions have the same accuracy with the exact best regularized solutions. Numerical experiments are given to show the effectiveness and efficiency of our new hybrid algorithms, and comparisons are made with the existing algorithm.

翻译：针对大规模线性离散不适定问题的一般形式正则化，提出了两种新的混合算法。这两种算法均基于Krylov子空间内外迭代框架，每次迭代需求解一个线性最小二乘问题。理论证明，由LSQR求解的内部线性最小二乘问题随着迭代步数k的增加将呈现更好的条件数，从而使LSQR加速收敛。同时，我们给出了LSQR停止容限的选择准则，以确保计算解与精确最优正则化解具有相同的精度。数值实验验证了新混合算法的有效性与高效性，并与现有算法进行了对比分析。