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TV‐like regularization for backward parabolic problems
Author(s) -
Pan Bin,
Xu Dinghua,
Xu Yinghong,
Yu Yue
Publication year - 2016
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.4027
Subject(s) - mathematics , regularization (linguistics) , uniqueness , conjugate gradient method , inverse problem , total variation denoising , parabolic partial differential equation , nonlinear system , numerical analysis , mathematical analysis , nonlinear conjugate gradient method , inverse , mathematical optimization , partial differential equation , gradient descent , computer science , geometry , artificial intelligence , physics , quantum mechanics , machine learning , artificial neural network , image (mathematics)
This paper investigates an inverse problem for parabolic equations backward in time, which is solved by total‐variation‐like (TV‐like, in abbreviation) regularization method with cost function ∥ u x ∥ 2 . The existence, uniqueness and stability estimate for the regularization problem are deduced in the linear case. For numerical illustration, the variational adjoint method, which presents a simple method to derive the gradient of the optimization functional, is introduced to reconstruct the unknown initial condition for both linear and nonlinear parabolic equations. The conjugate gradient method is used to iteratively search for the optimal approximation. Numerical results validate the feasibility and effectiveness of the proposed algorithm. Copyright © 2016 John Wiley & Sons, Ltd.