An adaptive RKHS regularization for Fredholm integral equations
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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.
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