An adaptive RKHS regularization for Fredholm integral equations

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.