In this paper we consider the computation of a finite eigenvalue and corresponding right eigenvector of a large sparse generalised eigenproblem Ax = λMx using inexact inverse iteration. Our convergence theory is quite general and requires few assumptions on A and M. In particular, there is no need for M to be symmetric positive definite or even nonsingular. The theory includes both fixed and va...

The modified Hermitian and skew-Hermitian splitting (MHSS) iteration method and preconditioned MHSS (PMHSS) iteration method were introduced respectively. In the paper, on the basis of the MHSS iteration method, we present a PMHSS iteration method for solving large sparse continuous Sylvester equations with non-Hermitian and complex symmetric positive definite/semi-definite matrices. Under suit...

We introduce a new Krylov subspace iteration for large scale eigen-value problems that is able to accelerate the convergence through an inexact (iterative) solution to a shift-invert equation. The method also takes full advantage of an exact solution when it is possible to apply a sparse direct method to solve the shift-invert equations. We call this new iteration the Truncated RQ iteration (TR...

The proximal point algorithm (PPA) is classical and popular in the community of Optimization. In practice, inexact PPAs which solves the involved proximal subproblems approximately subject to certain inexact criteria are truly implementable. In this paper, we first propose an inexact PPA with a new inexact criterion for solving convex minimization, and show that the iteration-complexity of this...

This paper investigates inexact Newton methods for solving systems of nonsmooth equations. We de ne two inexact Newton methods for locally Lipschitz functions and we prove local (linear and superlinear) convergence results under the assumptions of semismoothness and BD-regularity at the solution. We introduce a globally convergent inexact iteration function based method. We discuss implementati...

Based on the Hermitian and skew-Hermitian splitting (HSS) iteration technique, we establish a generalized HSS (GHSS) iteration method for solving large sparse continuous Sylvester equations with non-Hermitian and positive definite/semidefinite matrices. The GHSS method is essentially a four-parameter iteration which not only covers the standard HSS iteration but also enables us to optimize the ...

The Krasnoselskii-Mann iteration plays an important role in the approximation of fixed points of nonexpansive operators; it is is known to be weakly convergent in the infinite dimensional setting. In this present paper, we provide a new inexact Krasnoselskii-Mann iteration and prove weak convergence under certain accuracy criteria on the error resulting from the inexactness. We also show strong...

For the Hermitian inexact Rayleigh quotient iteration (RQI), we consider the local convergence of the inexact RQI with the Lanczos method for the linear systems involved. Some attractive properties are derived for the residual, whose norm is ξk, of the linear system obtained by the Lanczos method at outer iteration k + 1. Based on them, we make a refined analysis and establish new local converg...

A nonlinear optimization method is proposed for the solution of inverse scattering problems in the frequency domain, when the scattered field is governed by the Helmholtz equation. The time-harmonic inverse medium problem is formulated as a PDE-constrained optimization problem and solved by an inexact truncated Newtontype iteration. Instead of a grid-based discrete representation, the unknown w...

We consider the computation of the smallest eigenvalue and associated eigenvector of a Hermitian positive definite pencil. Rayleigh quotient iteration (RQI) is known to converge cubically, and we first analyze how this convergence is affected when the arising linear systems are solved only approximately. We introduce a special measure of the relative error made in the solution of these systems ...