The Jacobian-free Newton-Krylov method is widely used in solving nonlinear equations arising in many applications. However, an effective preconditioner is required each iteration and determining such may be hard or expensive. In this paper, we propose an efficient two-sided bicoloring method to determine the lower triangular half of the sparse Jacobian matrix via automatic differentiation. Then...

This article considerss a Newton-like method already used by several authors but which has not been thouroughly studied yet. We call it the robust-variance scoring (RVS) algorithm because the main version of the algorithm that we consider replaces minus the Hessian of the loglikelihood used in the 1 Newton-Raphson algorithm by a matrix G which is an estimate of the variance of the score under t...

We are interested in the determination of the shape of an elastic obstacle from its effects on known incident acoustic waves. We propose a solution methodology based on a regularized Newton-type method to solve this inverse obstacle problem. At each Newton iteration, we solve the forward problems using a finite element solver based on discontinuous Galerkin approximations, and equipped with hig...

We present a generalized Newton method and a quasiNewton method for solving H(x) := F(nc(x))+x-nc(x) = 0, when C is a polyhedral set. For both the Newton and quasi-Newton methods considered here, the subproblem to be solved is a linear system of equations per iteration. The other characteristics of the quasi-Newton method include: (i) a g-superlinear convergence theorem is established without a...

A new method for the solution of minimization problems with simple bounds is presented. Global convergence of a general scheme requiring the solution of a single linear system at each iteration is proved and a superlinear convergence rate is established without requiring the strict complementarity assumption. The theory presented covers Newton and Quasi-Newton methods, allows rapid changes in t...

We propose a generic method for iteratively approximating various second-order gradient steps - Newton, Gauss-Newton, Levenberg-Marquardt, and natural gradient - in linear time per iteration, using special curvature matrix-vector products that can be computed in O(n). Two recent acceleration techniques for on-line learning, matrix momentum and stochastic meta-descent (SMD), implement this appro...

This thesis deals with the solution of a nonlinear inverse problem arising in digital image registration. In image registration one seeks to compute a transformation between two images such that they become more similar in some sense. In the first part, we define the problem as the minimization of a regularized nonlinear least-squares functional, which measures the image difference and smoothne...

The preconditioned generalized shift-splitting (PGSS) iteration method is unconditionally convergent for solving saddle point problems with nonsymmetric coefficient matrices. By making use of the PGSS as inner solver Newton method, we establish a class Newton-PGSS large sparse nonlinear system Jacobian matrices about problems. For new presented give local convergence analysis and semilocal unde...

In the study of transverse vibrations of a hinged beam there arises a boundary value problem for fourth order ordinary differential equation, where a significant difficulty lies in a nonlinear term under integral sign. In recent years several authors considered finite approximation of the problem and proposed an iterative method for solving the system of nonlinear equations obtained. The essenc...