希尔伯特空间中不适定优化问题的迭代正则化梯度法的误差估计
Error Estimates for the Iteratively Regularized Gradient Method for Ill-Posed Optimization Problems in a Hilbert Space
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中文总结 AI 辅助
本文针对希尔伯特空间不适定优化问题，建立迭代正则化梯度法的误差估计，提供两种不同证明，改进Tikhonov方法的误差估计并证明其全局极小元存在性。
中文摘要 AI 辅助
我们研究应用于希尔伯特空间上一般光滑泛函极小化的不适定问题的迭代正则化梯度法。在解满足源条件下，当目标泛函精确已知或仅近似已知时，我们建立该方法的误差估计。针对这些估计，我们提供基于本质不同思路的两种证明，并比较对应定理的假设与结论。此外，我们给出Tikhonov方法应用于泛函精确已知的不适定优化问题的误差估计的改进证明，还证明了无需假设原泛函的弱下半连续性时，Tikhonov泛函存在全局极小元的定理。
英文摘要
We study the iteratively regularized gradient method applied to the ill-posed problem of minimizing a general smooth functional on a Hilbert space. Under a source condition on the solution, we establish error estimates for this method when the objective functional is known exactly or only approximately. For these estimates we provide two proofs based on substantially different ideas and compare the assumptions and conclusions of the corresponding theorems. In addition, we present an improved proof of an error estimate for the Tikhonov method applied to ill-posed optimization problems with an exactly known functional. We also prove a theorem on the existence of a global minimizer of the Tikhonov functional without assuming weak lower semicontinuity of the original functional.
