Iterative exponential filtering for large discrete ill-posed problems
This paper presents an iterative method for the solution of discrete linear ill-posed problems with a symmetric, possible indefinite or singular matrix. The rigth-hand side vector of the linear system \(Ax= g\), represents the given data and is assumed to be contaminated by measurement errors. In order to reduce the errors in the computation of the solution an exponential filter function of the form \(\varphi_\beta(t)= 1-\exp(-\beta t)\), with the regularization parameter \(\beta\geq 0\) is applied. The iterative method is derived by expanding the filter function in terms of Chebyshev polynomials. The possibilities of the extension of the iterative method by using the Morozov discrepancy principle in order to determine both a suitable value of \(\beta\) and an approximation of the associated regularized approximate solution \(x_\beta\) are discussed. The application to image restoration illustrates the performance of the proposed method.
scientific article; zbMATH DE number 1398752