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.
K. Cao and D. Lesnic,
Reconstruction of the space-dependent perfusion coefficient from final time or time-average temperature measurements,
J. Comput. Appl. Math. 337 (2018), 150–165.
Z.-C. Deng, L. Yang and J.-N. Yu,
Identifying the radiative coefficient of heat conduction equations from discrete measurement data,
Appl. Math. Lett. 22 (2009), no. 4, 495–500.
Q. Jin and M. Zhong,
On the iteratively regularized Gauss–Newton method in Banach spaces with applications to parameter identification problems,
Numer. Math. 124 (2013), no. 4, 647–683.
B. Kaltenbacher, F. Schöpfer and T. Schuster,
Iterative methods for nonlinear ill-posed problems in Banach spaces: Convergence and applications to parameter identification problems,
Inverse Problems 25 (2009), no. 6, Article ID 065003.
V. L. Kamynin and A. B. Kostin,
Two inverse problems of the determination of a coefficient in a parabolic equation,
Differ. Uravn. 46 (2010), no. 3, 372–383.
D. A. Murio,
On the stable numerical evaluation of Caputo fractional derivatives,
Comput. Math. Appl. 51 (2006), no. 9–10, 1539–1550.
D. Trucu, D. B. Ingham and D. Lesnic,
Space-dependent perfusion coefficient identification in the transient bio-heat equation,
J. Engrg. Math. 67 (2010), no. 4, 307–315.
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.