TV‐like regularization for backward parabolic problems. (23rd June 2016)

• Record Type: Journal Article
• Authors: Pan, Bin; Xu, Dinghua; Xu, Yinghong; Yu, Yue
• Is Part Of: Mathematical methods in the applied sciences. Volume 40:Number 4(2017)
• Page Start: 957
• Page End: 969
• Publication Date: 2016-06-23
• Subjects: inverse problems -- backward parabolic problems -- TV‐regularization -- variational adjoint method; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.4027
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : 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.