A parameter choice strategy for a multi-level augmentation method solving ill-posed operator equations
The problem of solving the first kind operator equation (1) \(K_x= y\), where \(K\) is a linear compact operator from a Hilbert space \(X\) to another Hilbert space \(X\) is considered. In practice, one has to solve the perturbed operator equation (2) \(K^\delta_x= y^\delta\) with \(\| y- y^\delta\|\leq\delta\), where \(\delta> 0\) is an error level. The above problem is ill-posed when the range of \(K\) is not closed and therefore regularization methods are needed to obtain a stable approximate solution. The purpose of this paper is to develop a new a posteriori regularization parameter choice strategy for the multi-level augmentation method based on a multiscale decomposition of the range space of the operator and the solution space of the equation for solving ill-posed operator equations of the first kind. An optimal order of convergence for the approximate solution obtained from the multilevel augmentation method using the a posteriori regularization parameter is established. Some numerical examples to illustrate the effectiveness of the algorithm proposed by the authors are presented.
scientific article; zbMATH DE number 5509280