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A refined Newton’s mesh independence principle for a class of optimal shape design problems. (English) Zbl 1112.65061

The author considers an approach to solve a class of smooth optimization problems in infinite-dimensional spaces via Newton’s method applied to the discretized problems. Under additional Lipschitz continuity type assumptions he establishes various accuracy estimates for this discretized method with respect to the initial problem, including the radius of convergence.

MSC:

65K10 Numerical optimization and variational techniques
49M25 Discrete approximations in optimal control
47J05 Equations involving nonlinear operators (general)
49M15 Newton-type methods
49Q10 Optimization of shapes other than minimal surfaces
47J25 Iterative procedures involving nonlinear operators
65J15 Numerical solutions to equations with nonlinear operators
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References:

[1] E.L. Allgower, K. Böhmer, F.A. Potra and W.C. Rheinboldt: “A mesh-independence principle for operator equations and their discretizations”, SIAM J. Numer. Anal., Vol. 23, (1986).; · Zbl 0591.65043
[2] I.K. Argyros: “A mesh independence principle for equations and their discretizations using Lipschitz and center Lipschitz conditions, Pan”, Amer. Math. J., Vol. 14(1), (2004), pp. 69-82.; · Zbl 1057.65027
[3] I.K. Argyros: “A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space”, J. Math. Anal. Appl., Vol. 298(2), (2004), pp. 374-397. http://dx.doi.org/10.1016/j.jmaa.2004.04.008; · Zbl 1057.65029
[4] I.K. Argyros: Newton Methods, Nova Science Publ. Corp., New York, 2005.;
[5] L.V. Kantorovich and G.P. Akilov: Functional Analysis in Normed Spaces, Pergamon Press, Oxford, 1982.; · Zbl 0127.06102
[6] M. Laumen: “Newton’s mesh independence principle for a class of optimal design problems”, SIAM J. Control Optim., Vol. 37(4), (1999), pp. 1070-1088. http://dx.doi.org/10.1137/S0363012996303529; · Zbl 0931.65068
[7] W.C. Rheinboldt: “An adaptive continuation process for solving systems of nonlinear equations”, In: Mathematical models and Numerical Methods, Banach Center Publ., Vol. 3, PWN, Warsaw, 1978, 129-142.; · Zbl 0378.65029
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