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. Furthermore, we prove and numerically demonstrate that the RKHS-regularized estimator has a mean-square error converging linearly as the noise scale decays. In contrast, the commonly-used $L^2$-regularized estimator has a flat mean-square error.

翻译：正则化是病态线性逆问题中一项长期存在的挑战，第一类弗雷德霍姆积分方程即为典型代表。我们提出了一种自适应于离散含噪测量数据及底层线性算子的实用再生核希尔伯特空间正则化算法。该再生核希尔伯特空间自然产生于一种变分方法中，其闭包正是我们能够识别真实解的函数空间。此外，我们证明并通过数值实验表明，随着噪声尺度衰减，该再生核希尔伯特空间正则化估计量的均方误差呈线性收敛。相比之下，常用的$L^2$正则化估计量的均方误差则表现出平缓趋势。