Iterated Landweber method for radial basis functions interpolation with finite accuracy

Abstract

We consider the reconstruction of a function stemming from a reproducing kernel Hilbert space using data which is perturbed by a deterministic error of maximal size ε_∞. The accuracy ε_∞ ≥ 0 provides an upper bound for reconstruction error estimates. Therefore, the main emphasis of this work is an a priori coupling of the data error, the error stemming from discretization and the numerical linear algebra. The coupling should provide an optimized cost-benefit ratio, i.e., we try to spend no numerical work to solving linear systems of equations if we cannot increase the overall accuracy of the reconstruction. Following [4], we focus here on the iterated Landweber method which serves both as numerical solver of the linear system of equations and as a regularization technique accounting for the inexact data. This method introduces a regularization parameter which should be chosen small from an error estimate perspective. On the other hand this parameter stabilizes the numerical computation. We outline this balance with the example of in-exact Cholesky decompositions. Here, we also take the finite precision of number representations in the computer into account.