Inexact Quasi-Newton methods for sparse systems of nonlinear equations

Abstract

In this paper we present the results obtained in solving consistent sparse systems of n nonlinear equations F(x) = 0; by a Quasi-Newton method combined with a p block iterative row-projection linear solver of Cimmino-type, 1 p n: Under weak regularity conditions for F; it is proved that this Inexact Quasi-Newton method has a local, linear convergence in the energy norm induced by the preconditioned matrix HA; where A is an initial guess of the Jacobian matrix, and it may converge superlinearly too. is the Moore-Penrose pseudo inverse of the m i n block, A i is the preconditioner. A simple partitioning of the Jacobian matrix was used for solving a set of nonlinear test problems with sizes ranging from 1024 to 131072 on the CRAY T3E under the MPI environment.

Optimal direct determination of sparse Jacobian matrices

“…The sparsity structure of the Jacobian matrix yields a natural row partitioning whereby the number of matrix-vector products is reduced from n to log 2 n . A second example where the Jacobian matrix is row partitioned arises in solving a linear system (Newton equations) [2] with an iterative method where the coefficient matrix A is partitioned into q row blocks A = (A T 1 A T 2 . .…”

Inexact descent methods for elastic parameter optimization

“…The inexact idea is not new and it has been widely explored by the numerical optimization community. For most of the numerical methods, researchers have developed their inexact counterparts, including inexact Newton methods [Dembo et al 1982;Eisenstat and Walker 1994], inexact Quasi-Newton methods [Bergamaschi et al 2001;Birgin et al 2003], inexact Newton-Dogleg methods [Pawlowski et al 2008], inexact SQP methods [Byrd et al 2008], and inexact proximal point methods [Burachik and Dutta 2010;Solodov and Svaiter 2001]. It should be noted that the descent methods discussed in this work look similar to many splitting methods, such as the alternating direction method of multipliers (ADMM) and proximal point methods, but they are fundamentally different due to the strong dependency of descent methods on the exactness of the forward simulation step.…”

Inexact quasi-Newton global convergent method for solving constrained nonsmooth equations

“…Some versions of the inexact quasi-Newton method for solving smooth equations was proposed e.g. by Bergamaschi, Moret, Zilli in [2] (inexact Newton-Cimmino method for sparse systems) and Birgin, Krejić, Martínez in [3] (inexact quasi-Newton algorithm with backtracking). Another study on the inexact quasi-Newton method with preconditioners can be found in Bergamaschi, Bru, Martínez, Putti [1].…”