Simple-iteration method with alternating step size for solving operator equations in Hilbert space

摘要

We introduce a new explicit iterative method with alternating step size for solving ill-posed operator equations of the first kind: Ax=y. We investigate the basic properties of the method for a positive bounded self-conjugate operator A:H→H in Hilbert space H under the assumption that the error for the right part of the equation is available. We discuss the convergence of the method, for a given number of iterations, in the original Hilbert space norm, estimate its precision and formulate recommendations for choosing the stopping criterion. Furthermore, we prove the convergence of the method with respect to the stopping criterion and estimate the remaining error. In case the equation has multiple solutions, we prove that the method converges to the minimum norm solution.