General scheme of one-step variational-gradient methods for linear equations

Approximate methods developed during the last decades essentially accelerate the convergence of gradient methods. They enjoy many applications and are more stable to perturbations than variational methods. These methods are known in the literature as variational-gradient methods [1-3]. Like gradient methods, they do not require any prior information about the iterative operators, other than some conditions of a general form. In this article we propose a general scheme for the construction of one-step variational-gradient methods to solve the linear equation Au = f, where A is a bounded operator that acts in a Hilbert space H. A theoretical justification of the scheme is given.