The iterative aggregation method for the solution of linear systems is extended in several directions: to operators on Banach spaces; to the method with inexact correction, i.e., to methods where the (inner) linear system is in turn solved iteratively; and to the problem of nding stationary distributions of Markov operators. Local convergence is shown in all cases. Convergence results apply to ...

The Cayley transform method is a Newton-like method for solving inverse eigenvalue problems. If the problem is large, one can solve the Jacobian equation by iterative methods. However, iterative methods usually oversolve the problem in the sense that they require far more (inner) iterations than is required for the convergence of the Newton (outer) iterations. In this paper, we develop an inexa...

Dynamical systems are mathematical models characterized by a set of differential or difference equations. Due to the increasing demand for more accuracy, the number of equations involved may reach the order of thousands and even millions. With so many equations, it often becomes computationally cumbersome to work with these large-scale dynamical systems. Model reduction aims to replace the orig...

Newton’s method is a classical method for solving a nonlinear equation F (z) = 0. We derive inexact Newton steps that lead to an inexact Newton method, applicable near a solution. The method is based on solving for a particular F (zk′) during p consecutive iterations k = k′, k′ + 1, . . . , k′ + p − 1. One such p-cycle requires 2 − 1 solves with the matrix F (zk′). If matrix factorization is us...

Consider an under-determined system of nonlinear equations F (x) = 0, F : IR → IR, where F is continuously differentiable and m > n. This system appears in a variety of applications, including parameter–dependent systems, dynamical systems with periodic solutions, and nonlinear eigenvalue problems. Robust, efficient numerical methods are often required for the solution of this system. Newton’s ...

In this paper we analyze the use of structured quasi-Newton formulae as preconditioners of iterative linear methods when the inexact-Newton approach is employed for solving nonlinear systems of equations. We prove that superlinear convergence and bounded work per iteration is obtained if the preconditioners satisfy a Dennis-Moré condition. We develop a theory of LeastChange Secant Update precon...

We propose and analyze an accelerated iterative dual diagonal descent algorithm for the solution of linear inverse problems with strongly convex regularization general data-fit functions. develop inertial approach which we both convergence stability properties. Using tools from inexact proximal calculus, prove early stopping results optimal rates additive data terms further consider more cases,...

In this paper we show that metric regularity and strong metric regularity of a set-valued mapping imply convergence of inexact iterative methods for solving a generalized equation associated with this mapping. To accomplish this, we first focus on the question how these properties are preserved under changes of the mapping and the reference point. As an application, we consider discrete approxi...

Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are methods specifically tailored to systems with special properties such as special forms of symmetry and...