Volume 52, Number 3, May–June 2018
|Page(s)||1085 - 1107|
|Published online||13 September 2018|
Quadratic convergence of Levenberg-Marquardt method for elliptic and parabolic inverse robin problems
School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University,
430079, P.R. China
2 School of Mathematics and Statistics, Wuhan University, Wuhan 430072, P.R. China
3 Department of Mathematics, The Chinese University of Hong Kong, Shatin, P.R. China.
**** Corresponding author: email@example.com
Accepted: 9 February 2018
We study the Levenberg-Marquardt (L-M) method for solving the highly nonlinear and ill-posed inverse problem of identifying the Robin coefficients in elliptic and parabolic systems. The L-M method transforms the Tikhonov regularized nonlinear non-convex minimizations into convex minimizations. And the quadratic convergence of the L-M method is rigorously established for the nonlinear elliptic and parabolic inverse problems for the first time, under a simple novel adaptive strategy for selecting regularization parameters during the L-M iteration. Then the surrogate functional approach is adopted to solve the strongly ill-conditioned convex minimizations, resulting in an explicit solution of the minimisation at each L-M iteration for both the elliptic and parabolic cases. Numerical experiments are provided to demonstrate the accuracy, efficiency and quadratic convergence of the methods.
Mathematics Subject Classification: 31A25 / 65M12 / 90C25
Key words: Inverse Robin problems / Levenberg-Marquardt method / surrogate functional.
The work of this author was financially supported by National Natural Science Foundation of China (Nos. 11401241 and 11571265) and NSFC-RGC(China-Hong Kong, No. 11661161017).
The work of this author was supported by National Natural Science Foundation of China (No. 91130022, No. 10971159 and No. 11161130003) and The Doctoral Fund of Ministry of Education of China (No. 20130141110026).
© EDP Sciences, SMAI 2018
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