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An inverse problem of identifying the diffusion coefficient in a coupled parabolic‐elliptic system
Author(s) -
Deng ZuiCha,
Cai Chao,
Yang Liu
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4833
Subject(s) - mathematics , uniqueness , inverse problem , parabolic partial differential equation , elliptic partial differential equation , partial differential equation , nonlinear system , mathematical analysis , optimal control , stability (learning theory) , diffusion , work (physics) , mathematical optimization , computer science , mechanical engineering , physics , quantum mechanics , machine learning , engineering , thermodynamics
This work studies a nonlinear inverse problem of reconstructing the diffusion coefficient in a parabolic‐elliptic system using the final measurement data, which has important application in a large field of applied science. Being different from other works, which are governed by single partial differential equations, the underlying mathematical model in this paper is a coupled parabolic‐elliptic system, which makes theoretical analysis rather difficult. On the basis of the optimal control framework, the identification problem is transformed into an optimization problem. Then the existence of the minimizer is proved, and the necessary condition that must be satisfied by the minimizer is also given. Since the optimal control problem is nonconvex, one may not expect a unique solution universally. However, the local uniqueness and stability of the minimizer are deduced in this paper.