Abstract
We shall derive and propose several efficient domain decomposition methods for solving the nonlinear inverse problem of identifying the Robin coefficients in elliptic and parabolic systems. The highly ill-posed inverse problems are transformed into output least-squares nonlinear and non-convex minimizations with classical Tikhonov regularization. The Levenberg–Marquardt method is applied to transform the non-convex minimizations into convex minimizations, which will be solved by several efficient domain decomposition methods. The methods are completely local and the local minimizers have explicit expressions within the subdomains. Several numerical experiments are presented to show the accuracy and efficiency of the methods; in particular, the convergence seems nearly optimal in the sense that the iteration number of the methods is nearly independent of mesh sizes.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11401241
Award Identifier / Grant number: 11571265
Award Identifier / Grant number: 11661161017
Award Identifier / Grant number: 91130022
Award Identifier / Grant number: 10971159
Award Identifier / Grant number: 11161130003
Funding source: China Postdoctoral Science Foundation
Award Identifier / Grant number: 20130141110026
Funding statement: The first author was financially supported by National Natural Science Foundation of China (nos. 11401241 and 11571265) and NSFC-RGC (China-Hong Kong, no. 11661161017). The second author was supported by National Natural Science Foundation of China (nos. 91130022, 10971159 and 11161130003), and the Doctoral Fund of Ministry of Education of China (no. 20130141110026).
Acknowledgements
The authors would like to thank the anonymous referees for their many insightful and constructive comments and suggestions, which have helped us improve the quality of the paper.
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