Multiscale Support Vector Approach for Solving Ill-Posed Problems

Abstract

Based on the use of compactly supported radial basis functions, we extend in this paper the support vector approach to a multiscale support vector approach (MSVA) scheme for approximating the solution of a moderately ill-posed problem on bounded domain. The Vapnik's -intensive function is adopted to replace the standard l 2 loss function in using the regularization technique to reduce the error induced by noisy data. Convergence proof for the case of noise-free data is then derived under an appropriate choice of the Vapnik's cut-off parameter and the regularization parameter. For noisy data case, we demonstrate that a corresponding choice for the Vapnik's cut-off parameter gives the same order of error estimate as both the a posteriori strategy based on discrepancy principle and the noise-free a priori strategy. Numerical examples are constructed to verify the efficiency of the proposed MSVA approach and the effectiveness of the parameter choices.

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

“…Such idea has been extended to support vector approach [12]. The multiscale regularization based on scaled radial basis functions were considered in [29,30] and in non-point evaluations in [27,28]. An alternative method to solve the (1.4) can be found in [13,20], where the approximate inverse for (1.4) was built with appropriately chosen mollifiers and corresponding reconstruction kernels.…”