Semi-discrete Tikhonov regularization in RKHS with large randomly distributed noise

Abstract

Abstract The concern of this paper is the numerical solution of the moderately ill-posed first kind integral equations in reproducing kernel Hilbert spaces. Different with previous works in (Krebs et al 2009 J. Inverse Ill-Posed Problems 17 845–69; Zhong et al 2012 Inverse Problems 28 065019), the amount of observation data is large, and corrupted by randomly distributed noise with large variance. In this paper, a purely data driven semi-discrete Tikhonov regularization method is proposed, effectively reconstructing the sought solution. The choice rules of parameters are provided and the rigorous upper bound estimation of confidence interval of the error in L 2 norm is established. Some numerical examples are provided to illustrate the appropriateness of the parameter choice and show the computational performances of the method.