Regularization is a long-standing challenge for ill-posed linear inverse problems, and a prototype is the Fredholm integral equation of the first kind. We introduce a practical RKHS regularization algorithm adaptive to the discrete noisy 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. We prove that the RKHS-regularized estimator has a mean-square error converging linearly as the noise scale decreases, with a multiplicative factor smaller than the commonly-used $L^2$-regularized estimator. Furthermore, numerical results demonstrate that the RKHS-regularizer significantly outperforms $L^2$-regularizer when either the noise level decays or when the observation mesh refines.

翻译：正则化是解决不适定线性逆问题中长期存在的挑战，其原型是第一类Fredholm积分方程。本文提出一种实用的再生核希尔伯特空间（RKHS）正则化算法，该算法可自适应于离散含噪观测数据及底层线性算子。该RKHS自然地产生于变分框架中，其闭包是能够辨识真实解的函数空间。我们证明，RKHS正则化估计量的均方误差随噪声尺度减小呈线性收敛，其乘性因子小于常用的$L^2$正则化估计量。此外，数值结果表明，当噪声水平降低或观测网格细化时，RKHS正则化方法显著优于$L^2$正则化方法。