An Adaptive RKHS Regularization for the Fredholm Integral Equations

ABSTRACT 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 regularize by a new reproducing kernel Hilbert space (RKHS) determined by the 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 ‐regularizers yield the same convergence rate when their optimal hyperparameters 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 commonly used ‐ and ‐regularizers in producing consistently converging estimators when the noise level decays or the observation mesh refines.