Abstract
Newton type methods are one kind of the efficient methods to solve nonlinear ill-posed problems, which have attracted extensive attention. However, computational cost of Newton type methods is high because practical problems are complicated. We propose a mixed Newton-Tikhonov method, i.e., one step Newton-Tikhonov method with several other steps of simplified Newton-Tikhonov method. Convergence and stability of this method are proved under some conditions. Numerical experiments show that the proposed method has obvious advantages over the classical Newton method in terms of computational costs.
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References
Hanke, M. A regularization Levenberg-Marquardt scheme, with application to inverse groundwater filtration problems. Inverse Problems 13(1), 79–95 (1997)
Engl, H. W., Hanke, M., and Neubauer, A. Regularization of Inverse Problem, Kluwer Academic, Dordrecht (1996)
Bakushinskii, A. B. and Kokurin, M. Y. Iterative Methods for Approximate Solution of Inverse Problems, Springer, Dordrecht (2004)
Kaltenbacher, B., Neubauer, A., and Scherzer, O. Iterative Regularization Methods for Nonlinear Ill-Posed Problems, Walter de Gruyter (2008)
Jin, Q. N. On the iteratively regularized Gauss-Newton method for solving nonlinear ill-posed problems. Math. Comp. 69(232), 1603–1623 (2000)
Jin, Q. N. A convergence analysis of the iteratively regularized Gauss-Newton method under the lipschitz condition. Inverse Problems 24(4), 1–16 (2008)
Hanke, M., Neubauer, A., and Scherzer, O. A convergence analysis of the Landweber iteration for nonlinear ill-posed problems. Numerical Mathematics 72(11), 21–37 (1995)
Deuflhard, P., Engl, H. W., and Scherzer, O. A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under affinely invariant conditions. Inverse Problems 14(5), 1081–1106 (1998)
Deuflhard, P. Newton Method for Nonlinear Problems: Affine Invariance and Adaptive Algorithms, Springer Series in Computational Mathematics, Vol. 35, Springer-Verlag, Berlin/Heidelberg (2004)
He, G. Q. and Meng, Z. H. A Newton type iterative method for heat-conduction inverse problems. Appl. Math. Mech. -Engl. Ed. 28(4), 531–539 (2007) DOI: 10.1007/s10483-007-0414-y
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(Communicated by Xing-ming GUO)
Project supported by the Key Disciplines of Shanghai Municipality (Operations Research & Cybernetics, No. S30104) and Shanghai Leading Academic Discipline Project (No. J50101)
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Kang, Cg., He, Gq. A mixed Newton-Tikhonov method for nonlinear ill-posed problems. Appl. Math. Mech.-Engl. Ed. 30, 741–752 (2009). https://doi.org/10.1007/s10483-009-0608-2
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DOI: https://doi.org/10.1007/s10483-009-0608-2
Key words
- nonlinear ill-posed problem
- inverse heat conduction problem
- mixed Newton-Tikhonov method
- convergence
- stability