Error estimates for Arnoldi–Tikhonov regularization for ill-posed operator equations
Error estimates for Arnoldi–Tikhonov regularization for ill-posed operator equations
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DOI：
10.1088/1361-6420/ab0663
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发表时间：
2018-12
期刊：
影响因子：
2.1
通讯作者：
R. Ramlau;L. Reichel
中科院分区：
文献类型：
--
作者：
R. Ramlau;L. Reichel
大多数关于线性不适定算子方程的解或其离散化的文献仅关注无限维设置或仅关注通过离散化获得的代数线性方程组的解。本文讨论了离散误差对计算解的影响。我们考虑当使用的离散化产生具有大矩阵的代数线性方程组时的情况。该系统的近似解是通过首先执行 Arnoldi 过程的几个步骤来确定尺寸相当小的简化系统来计算的。将吉洪诺夫正则化应用于简化问题，并根据差异原理确定正则化参数。讨论了求解过程的每个步骤中产生的错误。计算示例说明了导出的误差范围。
Most of the literature on the solution of linear ill-posed operator equations, or their discretization, focuses only on the infinite-dimensional setting or only on the solution of the algebraic linear system of equations obtained by discretization. This paper discusses the influence of the discretization error on the computed solution. We consider the situation when the discretization used yields an algebraic linear system of equations with a large matrix. An approximate solution of this system is computed by first determining a reduced system of fairly small size by carrying out a few steps of the Arnoldi process. Tikhonov regularization is applied to the reduced problem and the regularization parameter is determined by the discrepancy principle. Errors incurred in each step of the solution process are discussed. Computed examples illustrate the error bounds derived.