Geodesic
Matrix-free iterative processes with least-squares error damping for nonlinear systems of equations
Journal of the Belarusian State University. Mathematics and Informatics, Tome 3 (2017), pp. 73-84.

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New iterative processes for numerical solution of big nonlinear systems of equations are considered. The processes do not require factorization and storing of Jacobi matrix and employ a special technique of convergence acceleration which is called least-squares error damping and requires solution of auxiliary linear least-squares problems of low dimension. In linear case the resulting method is similar to the general minimal residual method (GMRES) with preconditioning. In nonlinear case, in contrast to popular Newton – Krylov method, the computational scheme do not involve operation of difference approximation of derivative operator. Numerical experiments include three nonlinear problems originating from two-dimensional elliptic partial differential equations and exhibit advantage of the proposed method compared to Newton – Krylov method.
Keywords: nonlinear systems of equations; matrix-free methods; acceleration of convergence; least-squares; Newton – Krylov method; difference schemes. 
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I. V. Bondar; B. V. Faleichik. Matrix-free iterative processes with least-squares error damping for nonlinear systems of equations. Journal of the Belarusian State University. Mathematics and Informatics, Tome 3 (2017), pp. 73-84. http://geodesic.mathdoc.fr/item/BGUMI_2017_3_a7/

[1] B. Faleichik, I. Bondar, V. Byl, “Generalized Picard iterations: A class of iterated Runge – Kutta methods for stiff problems”, J. Comput. Appl. Math, 262 (2014), 37–50 | DOI | MR | Zbl

[2] Y. Saad, “Iterative methods for Sparse Linear Systems”, Philadelphia: Siam, 2003 | MR | Zbl

[3] D. A. Knoll, D. E. Keyes, “Jacobian-free Newton – Krylov methods: a survey of approaches and applications”, J. Comput. Phys, 193 (2004), 357–397 | DOI | MR | Zbl

[4] D. K. Faddeev, V. N. Faddeeva, “Vychislitelnye metody lineinoi algebry”, Fizmatgiz, 1960 | MR

[5] D. Ortega, V. Reinboldt, “Iteratsionnye metody resheniya nelineinykh uravnenii so mnogimi neizvestnymi”, Mir, 1975 | MR | Zbl

[6] V. P. Shapeev, E. V. Vorozhtsov, V. I. Isaev, “Metod kollokatsii i naimenshikh nevyazok dlya trekhmernykh uravnenii Nave – Stoksa”, Vychisl. metody i programmirovanie, 14 (2013), 306–322

[7] A. G. Kurosh, “Kurs vysshei algebry”, Glavnaya redaktsiya fiziko-matematicheskoi literatury, 1968 | MR

[8] L. N. Trefethen, D. Bau, “Numerical Linear Algebra”, Philadelphia: Siam, 1997 | MR | Zbl

[9] A. A. Samarskii, “Teoriya raznostnykh skhem”, Nauka, 1977

[10] A. H. Baker, E. R. Jessup, T. Manteuffel, “A Technique for Accelerating the Convergence of Restarted GMRES”, SIAM J. Matrix Anal. Appl, 26 (2005), 962–984 | DOI | MR | Zbl

[11] H. Walker, N. Peng, “Anderson acceleration for fixed-point iterations”, SIAM J. Numer. Anal, 49 (2011), 1715–1735 | DOI | MR | Zbl

[12] A. Toth, C. T. Kelley, “Convergence Analysis for Anderson Acceleration”, SIAM J. Numer. Anal, 5 (2015), 805–819 | DOI | MR