Recovery of non-smooth radiative coefficient from nonlocal observation by diffusion system

Mengmeng Zhang 1  and Jijun Liu 2
  • 1 School of Mathematics, Southeast University, S. T. Yau Center of Southeast University, Nanjing, P. R. China
  • 2 School of Mathematics, Southeast University, S. T. Yau Center of Southeast University, Nanjing, P. R. China
Mengmeng Zhang
  • School of Mathematics, S. T. Yau Center of Southeast University, Southeast University, Nanjing, 210096, P. R. China
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and Jijun Liu
  • Corresponding author
  • School of Mathematics, S. T. Yau Center of Southeast University, Southeast University, Nanjing, 210096, P. R. China
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Abstract

The heat conduction process in composite medium can be modeled by a parabolic equation with discontinuous radiative coefficient. To detect the composite medium characterized by such a non-smooth coefficient from measurable information about the heat distribution, we consider a nonlinear inverse problem for parabolic equation, with the average measurement of temperature field in some time interval as the inversion input. We firstly establish the uniqueness for this nonlinear inverse problem, based on the property of the direct problem and the known uniqueness result for linear inverse source problem. To solve the inverse problem from a nonlinear operator equation, the differentiability and the tangential condition of this nonlinear map is analyzed. An iterative process called two-point gradient method is proposed by minimizing data-fit term and the penalty term alternatively, with rigorous convergence analysis in terms of the tangential condition. Numerical simulations are presented to illustrate the effectiveness of the proposed method.

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This journal presents original articles on the theory, numerics and applications of inverse and ill-posed problems. These inverse and ill-posed problems arise in mathematical physics and mathematical analysis, geophysics, acoustics, electrodynamics, tomography, medicine, ecology, financial mathematics etc. Articles on the construction and justification of new numerical algorithms of inverse problem solutions are also published.

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