论文标题
希尔伯特量表中的统计逆问题
Statistical Inverse Problems in Hilbert Scales
论文作者
论文摘要
在本文中，我们研究了希尔伯特量表中的Tikhonov正则化方案，该方案具有一般噪声的非线性统计反问题。该方案中的正规规范比希尔伯特领域的规范更强。我们专注于基于条件稳定性估计值对该方案进行理论分析。我们利用距离函数的概念来建立重现Hilbert空间设置的直接和重建误差的高概率估计。此外，对于通过适当的源条件定义的规则性类别，确定了针对过度厚度的情况和常规案例的明确收敛速率。我们的结果改善并概括了在相关设置中获得的先前结果。
In this paper, we study the Tikhonov regularization scheme in Hilbert scales for the nonlinear statistical inverse problem with a general noise. The regularizing norm in this scheme is stronger than the norm in Hilbert space. We focus on developing a theoretical analysis for this scheme based on the conditional stability estimates. We utilize the concept of the distance function to establish the high probability estimates of the direct and reconstruction error in Reproducing kernel Hilbert space setting. Further, the explicit rates of convergence in terms of sample size are established for the oversmoothing case and the regular case over the regularity class defined through appropriate source condition. Our results improve and generalize previous results obtained in related settings.
