论文标题
使用多个重复测量数据的双重梯度方法用于不良问题
Dual gradient method for ill-posed problems using multiple repeated measurement data
论文作者
论文摘要
我们考虑确定线性不足问题的$ \ r $ miniminimine解决方案$ a x = y $，其中$ a：{\ mathscr x} \ to {\ mathscr y} $是来自banach space $ {\ mathscr x} $ to Hilbert space $ {\ Maths $ { {\ Mathscr X} \ to [0，\ infty] $是适当的强烈凸起惩罚函数。假设可以使用多个重复重复的独立分布的无偏数据，我们考虑了一种双梯度方法来重建$ {\ Mathcal R} $ - 使用这些数据的平均值最小化解决方案。通过{\ it先验}停止规则或差异原理的统计变体终止该方法，我们将提供收敛分析并得出收敛速率时，当寻求的解决方案满足某些变异源条件。报告了各种数值结果以测试该方法的性能。
We consider determining $\R$-minimizing solutions of linear ill-posed problems $A x = y$, where $A: {\mathscr X} \to {\mathscr Y}$ is a bounded linear operator from a Banach space ${\mathscr X}$ to a Hilbert space ${\mathscr Y}$ and ${\mathcal R}: {\mathscr X} \to [0, \infty]$ is a proper strongly convex penalty function. Assuming that multiple repeated independent identically distributed unbiased data of $y$ are available, we consider a dual gradient method to reconstruct the ${\mathcal R}$-minimizing solution using the average of these data. By terminating the method by either an {\it a priori} stopping rule or a statistical variant of the discrepancy principle, we provide the convergence analysis and derive convergence rates when the sought solution satisfies certain variational source conditions. Various numerical results are reported to test the performance of the method.
