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A globally convergent Newton-GMRES method for large sparse systems of nonlinear equations. (English) Zbl 1123.65040

A class of globally convergent inexact Newton methods, the Newton-GMRES with quasi-conjugate-gradient backtracking (NGQCGB) methods, for solving large sparse systems of nonlinear equations are presented. These methods can be considered as a suitable combination of the Newton-GMRES iteration and some efficient backtracking strategies. In some cases, known Newton-GMRES backtracking (NGB) methods stagnate for some iterations or even fail. To avoid this disadvantage of NGB methods the authors propose a new alternative strategy, called quasi-conjugate-gradient with backtracking (QCGB), using the known information such as the projection of the gradient of the merit function on a proper subspace and last nonlinear step. Numerical computations show that the NGQCGB method is more robust and efficient than both the NGB method and the Newton-GMRES with eqality curve backtracking (NGECB) method.

MSC:

65H10 Numerical computation of solutions to systems of equations

Software:

NITSOL
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[1] Bai, Z.-Z.; Duff, I.S.; Wathen, A.J., A class of incomplete orthogonal factorization methods. I: methods and theories, BIT numer. math., 41, 1, 53-70, (2001) · Zbl 0990.65038
[2] Bai, Z.-Z.; Golub, G.H.; Lu, L.-Z.; Yin, J.-F., Block triangular and skew-Hermitian splitting methods for positive-definite linear systems, SIAM J. sci. comput., 26, 3, 844-863, (2005) · Zbl 1079.65028
[3] Bai, Z.-Z.; Golub, G.H.; Ng, M.K., Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. matrix anal. appl., 24, 603-626, (2003) · Zbl 1036.65032
[4] Bai, Z.-Z.; Sun, J.-C.; Wang, D.-R., A unified framework for the construction of various matrix multisplitting iterative methods for large sparse system of linear equations, Comput. math. appl., 32, 12, 51-76, (1996) · Zbl 0870.65025
[5] Bai, Z.-Z.; Tong, P.-L., Multi-step update algorithms based on the factorization of Jacobian, J. univ. electr. sci. tech. China, 22, 3, 311-316, (1993), (in Chinese)
[6] Bai, Z.-Z.; Tong, P.-L., On the affine invariant convergence theorems of inexact Newton method and Broyden’s method, J. univ. electr. sci. tech. China, 23, 5, 535-540, (1994), (in Chinese)
[7] Bellavia, S.; Morini, B., A globally convergent Newton-GMRES subspace method for systems of nonlinear equations, SIAM J. sci. comput., 23, 940-960, (2001) · Zbl 0998.65053
[8] Bellavia, S.; Macconi, M.; Morini, B., A hybrid Newton-GMRES method for solving nonlinear equations, (), 68-75 · Zbl 0978.65041
[9] Brown, P.N.; Saad, Y., Hybrid Krylov methods for nonlinear systems of equations, SIAM J. sci. statist. comput., 11, 450-481, (1990) · Zbl 0708.65049
[10] Brown, P.N.; Saad, Y., Convergence theory of nonlinear newton – krylov algorithms, SIAM J. optim., 4, 297-330, (1994) · Zbl 0814.65048
[11] Dembo, R.S.; Eisenstat, S.C.; Steihaug, T., Inexact Newton methods, SIAM J. numer. anal., 19, 400-408, (1982) · Zbl 0478.65030
[12] Dennis, J.E.; Schnabel, R.B., Numerical methods for unconstrained optimization and nonlinear equations, (1983), Prentice-Hall Englewood Cliffs, NJ · Zbl 0579.65058
[13] Dolan, E.D.; MorĂ©, J.J., Benchmarking optimization software with performance profiles, Math. prog. ser. A, 19, 201-213, (2002) · Zbl 1049.90004
[14] Eisenstat, S.C.; Walker, H.F., Globally convergent inexact Newton methods, SIAM J. optim., 4, 393-422, (1994) · Zbl 0814.65049
[15] Eisenstat, S.C.; Walker, H.F., Choosing the forcing term in an inexact Newton method, SIAM J. sci. comput., 17, 16-32, (1996) · Zbl 0845.65021
[16] Fokkema, D.R.; Sleijpen, G.L.G.; van der Vorst, H.A., Accelerated inexact Newton schemes for large systems of nonlinear equations, SIAM J. sci. comput., 19, 657-674, (1998) · Zbl 0916.65050
[17] Friedlander, A.; Gomes-Ruggiero, M.A.; Kozakevich, D.N.; Martinez, J.M.; Santos, S.A., Solving nonlinear systems of equations by means of quasi-Newton methods with a nonmonotone strategy, Optim. methods software, 8, 25-51, (1997) · Zbl 0893.65032
[18] Kaporin, I.E.; Axelsson, O., On a class of nonlinear equation solvers based on the residual norm reduction over a sequence of affine subspaces, SIAM J. sci. comput., 16, 228-249, (1995) · Zbl 0826.65048
[19] Knoll, D.A.; Keys, D.A., Jacobian-free newton – krylov method: A survey of approaches and applications, J. comput. phys., 193, 357-397, (2004) · Zbl 1036.65045
[20] Luksan, L., Inexact trust region method for large sparse systems of nonlinear equations, J. optim. theory appl., 81, 569-591, (1994) · Zbl 0803.65071
[21] Ortega, J.M.; Rheinboldt, W.C., Iterative solution of nonlinear equations in several variables, (2000), SIAM Philadelphia, PA · Zbl 0949.65053
[22] Pernice, M.; Walker, H.F., NITSOL: A Newton iterative solver for nonlinear systems, SIAM J. sci. comput., 19, 302-318, (1998) · Zbl 0916.65049
[23] Saad, Y., Iterative methods for sparse linear systems, (1996), PWS Publishing Company Boston · Zbl 1002.65042
[24] Saad, Y.; Schultz, M.H., GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. sci. comput., 7, 856-869, (1986) · Zbl 0599.65018
[25] Tuminaro, R.S.; Walker, H.F.; Shadid, J.N., On backtracking failure in Newton-GMRES methods with a demonstration for the navier – stokes equations, J. comput. phys., 180, 549-558, (2002) · Zbl 1143.76489
[26] Yuan, Y.-X.; Stoer, J., A subspace study on conjugate gradient algorithms, Zamm, 75, 11, 69-77, (1995) · Zbl 0823.65061
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