Abstract
We study moment discretization for compact operator equations in Hilbert space with discrete noisy data. Instead of assuming that the error in the data converges strongly to 0, we only assume weak convergence of the noise as introduced by Eggermont et al. (Inverse Probl 25:115018, 2009). A specific instance would be random noise. Under the usual source conditions, we derive optimal convergence rates for Phillips-Tikhonov regularization. The analysis is based on the comparison of the discrete problem with a semi-discrete version of the problem, which is made possible by virtue of a quadrature result in a suitable reproducing kernel Hilbert space. Some numerical results using strong and weak discrepancy principles for the selection of the regularization parameter are presented.
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