Abstract Based on the joint bidiagonalization (JBD) process of the matrix pair { A , L } , an iterative regularization algorithm, called JBDQR, is proposed and developed for large scale linear discrete ill-posed problems in general-form Tikhonov regularization. It is proved that the JBDQR iterates take the form of attractive filtered generalized singular value decomposition (GSVD) expansions, where the filters are given explicitly and insightful. This result and a detailed analysis on it show that JBDQR must have the desired semi-convergence property, where the iteration number k plays the role of the regularization parameter. Embedded with the L-curve criterion or the discrepancy principle that is used to estimate the optimal k ⁎ at which the semi-convergence occurs, JBDQR can compute a satisfying good regularized solution. JBDQR is theoretically solid and effective, and it is simple to implement. Numerical experiments confirm our results and the robustness of JBDQR.

中文翻译：

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一种基于联合双对角化的大规模通用型Tikhonov正则化迭代算法

摘要 基于矩阵对 { A , L } 的联合双对角化 (JBD) 过程，提出并开发了一种迭代正则化算法，称为 JBDQR，用于一般形式的 Tikhonov 正则化中的大规模线性离散不适定问题。事实证明，JBDQR 迭代采用有吸引力的滤波广义奇异值分解 (GSVD) 扩展的形式，其中明确且有洞察力地给出了过滤器。该结果及其详细分析表明 JBDQR 必须具有所需的半收敛特性，其中迭代次数 k 起着正则化参数的作用。嵌入 L 曲线准则或用于估计发生半收敛的最佳 k ⁎ 的差异原理，JBDQR 可以计算出令人满意的良好正则化解。JBDQR 理论上扎实有效，实现起来也很简单。数值实验证实了我们的结果和 JBDQR 的稳健性。