We consider approximate and truncated Newton methods for solving nonlinear Poisson-type systems of equations with emphasis on vector computers. We have selected two model differential equations for our analysis. The first, after finite difference discretization, yields a nonlinear system with a symmetric positive definite Jacobian matrix, while the second yields a nonsymmetric Jacobian matrix which is not, in general, positive definite.
In the case of a symmetric positive definite Jacobian matrix, the linearized Newton systems are solved by (preconditioned) conjugate gradient methods. In particular, we consider incomplete Cholesky factorization in conjunction with the red-black ordering scheme and symmetric diagonal scaling as preconditioners. In the case of a nonsymmetric Jacobian matrix, we construct symmetric positive definite approximations and analyze the effect of these approximations.
We develop, and present theory for, a truncated approximate Newton method. This method uses an approximate Jacobian matrix in addition to truncation in the solution of the Newton system. Convergence properties and work reduction of this new method have been found to be very competitive.
[1]
Henk A. van der Vorst,et al.
ICCG and related methods for 3D problems on vector computers
,
1989
.
[2]
R. Dembo,et al.
INEXACT NEWTON METHODS
,
1982
.
[3]
J. Ortega.
Introduction to Parallel and Vector Solution of Linear Systems
,
1988,
Frontiers of Computer Science.
[4]
R. Grimes,et al.
On vectorizing incomplete factorization and SSOR preconditioners
,
1988
.
[5]
C.-C. Jay Kuo,et al.
Parallel elliptic preconditioners: Fourier analysis and performance on the connection machine
,
1989
.
[6]
James M. Ortega,et al.
Iterative solution of nonlinear equations in several variables
,
2014,
Computer science and applied mathematics.
[7]
I. Duff,et al.
The effect of ordering on preconditioned conjugate gradients
,
1989
.
[8]
Y. Saad,et al.
GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
,
1986
.