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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.
Jiang, Daijun 1 ; Feng, Hui 1 ; Zou, Jun 1
@article{M2AN_2018__52_3_1085_0,
author = {Jiang, Daijun and Feng, Hui and Zou, Jun},
title = {Quadratic convergence of {Levenberg-Marquardt} method for elliptic and parabolic inverse robin problems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1085--1107},
publisher = {EDP-Sciences},
volume = {52},
number = {3},
year = {2018},
doi = {10.1051/m2an/2018016},
mrnumber = {3865559},
zbl = {1404.35174},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018016/}
}
TY - JOUR AU - Jiang, Daijun AU - Feng, Hui AU - Zou, Jun TI - Quadratic convergence of Levenberg-Marquardt method for elliptic and parabolic inverse robin problems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 1085 EP - 1107 VL - 52 IS - 3 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018016/ DO - 10.1051/m2an/2018016 LA - en ID - M2AN_2018__52_3_1085_0 ER -
%0 Journal Article %A Jiang, Daijun %A Feng, Hui %A Zou, Jun %T Quadratic convergence of Levenberg-Marquardt method for elliptic and parabolic inverse robin problems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 1085-1107 %V 52 %N 3 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018016/ %R 10.1051/m2an/2018016 %G en %F M2AN_2018__52_3_1085_0
Jiang, Daijun; Feng, Hui; Zou, Jun. Quadratic convergence of Levenberg-Marquardt method for elliptic and parabolic inverse robin problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 3, pp. 1085-1107. doi: 10.1051/m2an/2018016
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