In this paper, we study an inexact inverse iteration with inner-outer iterations for solving the generalized eigenvalue problem Ax = Bx; and analyze how the accuracy in the inner iterations aaects the convergence of the outer iterations. By considering a special stopping criterion depending on a threshold parameter, we show that the outer iteration converges linearly with the threshold paramete...

In our papers [Inverse Problems, 15, 309-327,1999] and [Numer. Math., 88, 347-365, 2001] we proposed algorithm REGINN being an inexact Newton iteration for the stable solution of nonlinear ill-posed problems. REGINN consists of two components: the outer iteration, which is a Newton iteration stopped by the discrepancy principle, and an inner iteration, which computes the Newton correction by so...

The computation of a right eigenvector and corresponding finite eigenvalue of a large sparse generalised eigenproblem Ax = λMx using preconditioned Rayleigh quotient iteration and the simplified JacobiDavidson method is considered. Both methods are inner-outer iterative methods and we consider GMRES and FOM as iterative algorithms for the (inexact) solution of the inner systems that arise. The ...

For a given blur, we apply a fixed point method to solve the total variation-based image restoration problem. A new algorithm for the discretized system is presented. Convergence of outer iteration is efficiently improved by adding a linear term on both sides of the system of nonlinear equations. In inner iteration, an algebraic multigrid (AMG) method is applied to solve the linearized systems ...

We analyze inexact fixed point iterations where the generating function contains an inexact solve of an equation system to answer the question of how tolerances for the inner solves influence the iteration error of the outer fixed point iteration. Important applications are the Picard iteration and partitioned fluid structure interaction. We prove that the iteration converges irrespective of ho...

In this paper, we present an effective preconditioning technique for solving nonsymmetric saddle-point problems. In particular, we consider those saddlepoint problems that arise in the numerical simulation of particulate flows—flow of solid particles in incompressible fluids, using mixed finite element discretization of the Navier–Stokes equations. These indefinite linear systems are solved usi...

By making use of duality mappings and the Bregman distance, we propose a regularizing Levenberg-Marquardt scheme to solve nonlinear inverse problems in Banach spaces, which is an extension of the one proposed in [6] in Hilbert space setting. The method consists of two components: an outer Newton iteration and an inner scheme. The inner scheme involves a family of convex minimization problems in...

Saddle point problems arise in the modelling of many important practical problems. Preconditioners for the corresponding matrices on block triangular form, based on coupled inner-outer iteration methods are analyzed and applied to a Darcy flow problem, possibly with strong heterogeneity and to non-symmetric saddle point problems. Using proper regularized forms of the given matrix and its precon...

A general inner-outer iteration for computing extreme eigenpairs of symmetric/positive-definite matrix pencils is proposed. The principle of the method is to produce a sequence of p-dimensional bases {Xk} that converge to a minimizer of a generalized Rayleigh quotient. The role of the inner iteration is to produce an “update” vector by (approximately) minimizing a quadratic model of the Rayleig...

This paper proposes a dual-iterative method, a hierarchical inner and outer iteration method (HIO), to acquire concept words from a large-scale, un-segmented Chinese corpus. It has two levels of iteration: the EM-CLS algorithm and the Viterbi-C/S algorithm constitute the inner iteration for generating concept words, and the concept word validation constitutes the outer iteration together with t...