The multiscale regularizing method for solving Ill-posed problems

Abstract: 

Using compactly supported radial basis functions (CSRBFs) of varying radii, Sloan, Wendland and LeGia have shown how a multiscale analysis can be applied to the approximation of Sobolev functions on a bounded domain and on the unit sphere. Here, we examine the application of this analysis to the solution of linear moderately ill-posed problems using Support Vector Approach regularization. Motivated by existing CSRBF-based multiscale regression methods, the multiscale reconstruction for an ill-posed problem is constructed by a sequence of residual corrections, where different support radii are employed to accommodate different scales. Convergence proof for the case of noise-free data and noisy data are derived from an appropriate choice of the Vapnik’s cut-off parameter and the regularization parameter. Numerical examples are constructed to verify the efficiency of the proposed approach and the effectiveness of the parameter choices.