Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and allow a robust approximation of ill-posed (pseudo-) inverses. In the last two decades interest has shifted from linear to nonlinear regularization methods, even for linear inverse problems. The aim of this paper is to provide a reasonably comprehensive overview of this shift towards modern nonlinear regularization methods, including their analysis, applications and issues for future research.In particular we will discuss variational methods and techniques derived from them, since they have attracted much recent interest and link to other fields, such as image processing and compressed sensing. We further point to developments related to statistical inverse problems, multiscale decompositions and learning theory.

中文翻译：

---

逆问题的现代正则化方法

正则化方法是解决逆问题的关键工具。它们用于引入先验知识并允许对不适定（伪）逆进行稳健的近似。在过去的二十年里，人们的兴趣已经从线性正则化方法转向非线性正则化方法，即使对于线性逆问题也是如此。本文的目的是对这种向现代非线性正则化方法的转变提供一个相当全面的概述，包括它们的分析、应用和未来研究的问题。特别是我们将讨论从它们衍生出来的变分方法和技术，因为它们已经吸引了很多人最近的兴趣和与其他领域的联系，例如图像处理和压缩感知。我们进一步指出与统计逆问题、多尺度分解和学习理论相关的发展。