论文标题
统计反问题的离散自适应正规化
Discretisation-adaptive regularisation of statistical inverse problems
论文作者
论文摘要
我们考虑在白噪声下的线性反问题。可以解决这些类型的问题，例如迭代正则化方法，主要挑战是确定适合迭代的停止索引。流行自适应方法确定停止指数的收敛结果通常会带来限制，例如关于问题的不良性类型，未知解决方案或误差分布。在最近的工作中\ cite {jahn2021optimal}对差异原理的修改（最广泛使用的自适应方法之一）提出了用于光谱截止正则化的差异，在一般设置中提供了出色的收敛性。在这里，我们通过其他基于过滤器的正则化方法研究了修改后的差异原理的性能，并在此专注于迭代的Landweber方法。我们表明该方法产生了最佳的收敛速率，并提出了一些数值实验，证实它在计算复杂性方面也很有吸引力。关键思想是以自适应方式合并和修改离散化维度。
We consider linear inverse problems under white noise. These types of problems can be tackled with, e.g., iterative regularisation methods and the main challenge is to determine a suitable stopping index for the iteration. Convergence results for popular adaptive methods to determine the stopping index often come along with restrictions, e.g. concerning the type of ill-posedness of the problem, the unknown solution or the error distribution. In the recent work \cite{jahn2021optimal} a modification of the discrepancy principle, one of the most widely used adaptive methods, applied to spectral cut-off regularisation was presented which provides excellent convergence properties in general settings. Here we investigate the performance of the modified discrepancy principle with other filter based regularisation methods and we hereby focus on the iterative Landweber method. We show that the method yields optimal convergence rates and present some numerical experiments confirming that it is also attractive in terms of computational complexity. The key idea is to incorporate and modify the discretisation dimension in an adaptive manner.
