标题： 迭代Golub-Kahan-Tikhonov方法 显示英文标题

标题： The iterated Golub-Kahan-Tikhonov method

摘要： 戈卢布-卡恩-蒂科诺夫方法是求解大型线性离散不适定问题的一种流行解法。 该方法首先应用部分戈卢布-卡恩双对角化来减小给定问题的规模，然后使用蒂科诺夫正则化来计算约简后问题的有意义近似解。 众所周知，该方法的迭代变体通常比标准的非迭代方法产生质量更高的近似解。 此外，当定义线性离散不适定问题的矩阵远离对称时，它产生的计算解比阿诺尔迪方法更准确。 本文从无限维希尔伯特空间中的不适定算子方程开始，对方程进行离散化，然后将迭代的戈卢布-卡恩-蒂科诺夫方法应用于后者的求解。 提供了涵盖所有离散化和近似误差的误差分析。 此外，还描述了一种选择正则化参数的新方法。 该求解方案比标准（非迭代）戈卢布-卡恩-蒂科诺夫方法和迭代的阿诺尔迪-蒂科诺夫方法产生更精确的近似解。 显示英文摘要

摘要： The Golub-Kahan-Tikhonov method is a popular solution technique for large linear discrete ill-posed problems. This method first applies partial Golub-Kahan bidiagonalization to reduce the size of the given problem and then uses Tikhonov regularization to compute a meaningful approximate solution of the reduced problem. It is well known that iterated variants of this method often yield approximate solutions of higher quality than the standard non-iterated method. Moreover, it produces more accurate computed solutions than the Arnoldi method when the matrix that defines the linear discrete ill-posed problem is far from symmetric. This paper starts with an ill-posed operator equation in infinite-dimensional Hilbert space, discretizes the equation, and then applies the iterated Golub-Kahan-Tikhonov method to the solution of the latter problem. An error analysis that addresses all discretization and approximation errors is provided. Additionally, a new approach for choosing the regularization parameter is described. This solution scheme produces more accurate approximate solutions than the standard (non-iterated) Golub-Kahan-Tikhonov method and the iterated Arnoldi-Tikhonov method.