对病态问题的投影方法再探讨
Projection methods for ill-posed problems revisited
AI总结：
本文研究了希尔伯特空间中线性病态算子方程的最小二乘问题离散化条件，探讨了全局和局部收敛性，提出新的等价条件并利用斜投影工具分析子空间角度，提供实用条件并举例说明离散解序列可能不收敛。
摘要：
本文考虑了希尔伯特空间中线性病态算子方程的最小二乘问题的离散化。本文主要研究了离散化最小范数最小二乘解对精确解的收敛条件，使用精确可得数据。研究了全局收敛（对所有精确解收敛）或局部收敛（对特定精确解收敛）两种情况。回顾了现有结果并证明了新的等价条件，当离散解始终收敛到精确解时。重要工具是将离散解算子识别为斜投影。因此，全局收敛可以由某些子空间具有统一有界角度来表征。我们进一步推导了在该情况下成立的实用条件，并将其置于已知结果的背景下。对于局部收敛，我们推广了关于弱收敛或强收敛的表征结果，并陈述了一些新的充分条件。我们还提供了一个有界离散解序列的例子，该序列不收敛，甚至不弱收敛。
英文摘要：
The discretization of least-squares problems for linear ill-posed operator equations in Hilbert spaces is considered. The main subject of this article concerns conditions for convergence of the associated discretized minimum-norm least-squares solution to the exact solution using exact attainable data. The two cases of global convergence (convergence for all exact solution) or local convergence (convergence for a specific exact solution) are investigated. We review the existing results and prove new equivalent condition when the discretized solution always converges to the exact solution. An important tool is to recognize the discrete solution operator as oblique projection. Hence, global convergence can be characterized by certain subspaces having uniformly bounded angles. We furthermore derive practically useful conditions when this holds and put them into the context of known results. For local convergence we generalize results on the characterization of weak or strong convergence and state some new sufficient conditions. We furthermore provide an example of a bounded sequence of discretized solutions which does not converge at all, not even weakly.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
