Low-rank update of preconditioners for the inexact Newton method with SPD Jacobian

Abstract

In this note preconditioners for the Conjugate Gradient method are studied to solve the Newton system with a symmetric positive definite Jacobian. In particular, we define a sequence of preconditioners built by means of BFGS rank-two updates. Reasonable conditions are derived which guarantee that the preconditioned matrices are not far from the identity in a matrix norm. Some notes on the implementation of the corresponding inexact Newton method are given and some numerical results on a number of model problems illustrate the efficiency of the proposed preconditioners

Low-rank updates of balanced incomplete factorization preconditioners

“…Let Z and V s be the matrices whose columns are the vectors z j and v j , and D s a diagonal matrix containing the scalars r j as defined in (3). Then, the expression…”

“…1. An analysis of the expressions (3) shows that there is a computational influence between the entries of the matrices L and L −1 (U and U −1 as well), that is, the computed factors do influence each other numerically. Furthermore, the computation of the upper triangular part of the matrices V s andṼ s can be simplified, as it is stated in Lemma 3.1 of [12] v pj = sṽ…”

“…Very often, it is coupled with the problem of solving a sequence of linear systems as, for example, the solution of nonlinear equations by a Newton-like method [26]. Quasi-Newton methods involve low-rank updates of an approximate factorization of an initial Jacobian matrix [3,4]. Other approaches are based on an estimation of the difference between a reference matrix and the updated one [16].…”

Asynchronous partial update of the Restricted Additive Schwarz preconditioner to solve nonlinear CFD problems

“…The preconditioner can be frozen for a few linear systems, or it can be subject to small rank modifications. Therefore, the quasi-Newton updates of the preconditioner have also been discussed in [5][6][7]. The update of factorized preconditioners has also been considered in [8,9].…”

Preconditioner updates for solving sequences of linear systems in matrix-free environment

“…Early work which uses the Broyden formula to update the preconditioner was introduced in [14]. Other updates based on matrices of small rank were considered in [15], [16], [17] and in limited-memory variable metric methods [18] for smooth optimization, to name just of few. An important class of algebraically-motivated strategies to accelerate the convergence of preconditioned iterative methods is based on constructing or improving the preconditioner by adaptive spectral information obtained directly from the Krylov subspace methods, see, e.g., [19,20,21,22,23,24,25,26].…”