Direct and iterative methods for the solution of linear operator equations in Hilbert space

Abstract

Introduction. During the last two decades the problem of justifying existing methods and finding new ones for the solution of the equation Lu = f, where f is a given vector in Hilbert space and L is a given operator, has been investigated by many authors(2). This investigation led to the development of a group of direct methods of Ritz, Galerkin, least squares, and moments; and the iterative methods of steepest descent or gradient method, the method with minimal residuals, and their variants. These methods have a long history and have been studied and extensively applied by earlier authors to integral and differential equations. Only recently, however, the Hilbert space operator theory has been used for their study. Thus, using the variational principle, Mikhlin [16] proved the convergence of Ritz method for a self-adjoint positive definite operator L and the convergence of the method of least squares for an invertible operator L. In case L is of the form L = A + B, where A is self-adjoint and positive definite and B satisfies some additional conditions, the method of Galerkin was investigated by Mikhlin [16], Polsky [19], and others [9; 13]. For similar differential operators the method of moments was studied by Kravchuk [11], Polsky [19], and Zdanov [22]. The convergence and the estimate of error of the gradient method for selfadjoint and positive definite operators L have been studied by Kantorovich [7] and Hayes [6]. The latter considered also a slightly more general problem. In case L is a finite, symmetric, and positive definite matrix the method with minimal residuals was developed by Krasnoselsky and Krein [10]. A powerful method related to direct methods was also developed by Murray [18]. The purpose of this paper is to extend the study and the applicability of these methods to a larger class of linear operator equations(3) than those considered by the above authors and to present these seemingly distinct methods in a more unified manner. In our investigation we do not use the variational principle