In this paper, we analyze the convergence rates for finite-dimensional variational regularization in Banach spaces by taking into account the noisy data and operator approximations. In particular, we determine the convergence rates by incorporating the smoothness concepts of Hölder stability estimates and the variational inequalities. Additionally, we discuss two ill-posed inverse problems to complement the abstract theory presented in our main results.

中文翻译：

---

关于变分正则化：有限维和Hölder稳定性

在本文中，我们通过考虑噪声数据和算子逼近来分析Banach空间中有限维变分正则化的收敛速度。特别是，我们通过结合Hölder稳定性估计的光滑度概念和变分不等式来确定收敛速度。此外，我们讨论了两个不适定的逆问题，以补充我们主要结果中提出的抽象理论。