On Tikhonov’s method and optimal error bound for inverse source problem for a time-fractional diffusion equation

We investigate the linear but ill-posed inverse problem of determining a multi-dimensional space-dependent heat source in a time-fractional diffusion equation. We show that the problem is ill-posed in the Hilbert scale Hr(Rn) and establish global order optimal lower bound for the worst case error. N...

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Published in:Computers & mathematics with applications (1987) 2020-07, Vol.80 (1), p.61-81
Main Authors: Dien, Nguyen Minh, Hai, Dinh Nguyen Duy, Viet, Tran Quoc, Trong, Dang Duc
Format: Article
Language:English
Subjects:
Ill posed problems
Ill-posed problem
Inverse problems
Lower bounds
Numerical methods
Optimal error bound
Regularization
Regularization methods
Tikhonov regularization method
Time-fractional diffusion problem
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container_title Computers & mathematics with applications (1987)
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description We investigate the linear but ill-posed inverse problem of determining a multi-dimensional space-dependent heat source in a time-fractional diffusion equation. We show that the problem is ill-posed in the Hilbert scale Hr(Rn) and establish global order optimal lower bound for the worst case error. Next, we use the Tikhonov regularization method to deal with this problem in the Hilbert scale Hr(Rn). Locally optimal choices of parameters for the family of regularization operator in the Hilbert scales Hr(Rn) are analyzed by a-priori and a-posteriori methods. Numerical implementations are presented to illustrate our theoretical findings.
doi_str_mv 10.1016/j.camwa.2020.02.024
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subjects Ill posed problems
Ill-posed problem
Inverse problems
Lower bounds
Numerical methods
Optimal error bound
Regularization
Regularization methods
Tikhonov regularization method
Time-fractional diffusion problem
title On Tikhonov’s method and optimal error bound for inverse source problem for a time-fractional diffusion equation
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