Regularization is a long-standing challenge for ill-posed linear inverse problems, and a prototype is the Fredholm integral equation of the first kind with additive Gaussian measurement noise. We introduce a new RKHS regularization adaptive to measurement data and the underlying linear operator. This RKHS arises naturally in a variational approach, and its closure is the function space in which we can identify the true solution. Also, we introduce a small noise analysis to compare regularization norms by sharp convergence rates in the small noise limit. Our analysis shows that the RKHS- and $L^2$-regularizers yield the same convergence rate when their optimal hyper-parameters are selected using the true solution, and the RKHS-regularizer has a smaller multiplicative constant. However, in computational practice, the RKHS regularizer significantly outperforms the $L^2$-and $l^2$-regularizers in producing consistently converging estimators when the noise level decays or the observation mesh refines.

翻译：正则化是病态线性逆问题中长期存在的挑战，其典型代表为带有加性高斯测量噪声的第一类弗雷德霍姆积分方程。我们提出了一种自适应于测量数据和底层线性算子的新型再生核希尔伯特空间正则化方法。该再生核希尔伯特空间通过变分方法自然产生，其闭包是能够识别真实解的函数空间。此外，我们引入小噪声分析，通过小噪声极限下的尖锐收敛速率比较不同正则化范数的性能。分析表明，当使用真实解选择最优超参数时，再生核希尔伯特空间正则化与$L^2$正则化具有相同的收敛速率，且前者具有更小的乘法常数。然而在实际计算中，当噪声水平衰减或观测网格细化时，再生核希尔伯特空间正则化在生成一致收敛估计量方面显著优于$L^2$和$l^2$正则化方法。