Convergence results are provided for inexact two-sided inverse and Rayleigh quotient iteration, which extend the previously established results to the generalized eigenproblem, and inexact solves with a decreasing solve tolerance. Moreover, the simultaneous solution of the forward and adjoint problem arising in two-sided methods is considered and the successful tuning strategy for preconditione...

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...

We incorporate our recent preconditioning techniques into the classical inverse power (Rayleigh quotient) iteration for computing matrix eigenvectors. Every loop of this iteration essentially amounts to solving an ill conditioned linear system of equations. Due to our modification we solve a well conditioned linear system instead. We prove that this modification preserves local quadratic conver...

We consider a regularized Levenberg–Marquardt method for solving nonlinear ill-posed inverse problems. We use the discrepancy principle to terminate the iteration. Under certain conditions, we prove the convergence of the method and obtain the order optimal convergence rates when the exact solution satisfies suitable source-wise representations. Mathematics Subject Classification (2000) 65J15 ·...

SUMMARY Recently the GMRESR inner-outer iteration scheme for the solution of linear systems of equations has been proposed by Van der Vorst and Vuik. Similar methods have been proposed by Axelsson and Vassilevski 1] and Saad (FGMRES) 10]. The outer iteration is GCR, which minimizes the residual over a given set of direction vectors. The inner iteration is GMRES, which at each step computes a ne...

We introduce a dynamically weighted Halley (DWH) iteration for computing the polar decomposition of a matrix, and prove that the new method is globally and asymptotically cubically convergent. For matrices with condition number no greater than 1016, the DWH method needs at most 6 iterations for convergence with the tolerance 10−16. The Halley iteration can be implemented via QR decompositions w...

A new method for correcting charged-particle spectra for thick target effects is described. Starting with a trial function, inverse response functions are found by an iterative procedure. The variances corresponding to the measured spectrum are treated similiarly and in parallel. Oscillations of the solution are avoided by rebinning the data to finer bins during a correction iteration and back ...

In this paper we introduce a derivative-free, iterative method for solving nonlinear illposed problems Fx = y, where instead of y noisy data yδ with ‖y − yδ‖ ≤ δ are given and F : D(F ) ⊆ X → Y is a nonlinear operator between Hilbert spaces X and Y . This method is defined by splitting the operator F into a linear part A and a nonlinear part G, such that F = A + G. Then iterations are organized...

We present some advances, both from a theoretical and from a computational point of view, on a quadratic vector equation (QVE) arising in Markovian Binary Trees. Concerning the theoretical advances, some irreducibility assumptions are relaxed, and the minimality of the solution of the QVE is expressed in terms of properties of the Jacobian of a suitable function. From the computational point of...

This paper considers a special but broad class of convex programming (CP) problems whose feasible region is a simple compact convex set intersected with the inverse image of a closed convex cone under an affine transformation. It studies the computational complexity of quadratic penalty based methods for solving the above class of problems. An iteration of these methods, which is simply an iter...