A Tikhonov regularization method for Cauchy problem based on a new relaxation model

In this paper, we consider a Cauchy problem of recovering both missing value and flux on inaccessible boundary from Dirichlet and Neumann data measured on the remaining accessible boundary. Associated with two mixed boundary value problems, a regularized Kohn-Vogelius formulation is proposed. With a...

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Bibliographic Details
Published in:Nonlinear analysis: real world applications 2023-12, Vol.74, p.103935, Article 103935
Main Authors: Huang, Qin, Gong, Rongfang, Jin, Qinian, Zhang, Ye
Format: Article
Language:English
Subjects:
Cauchy problem
Error estimates
Finite element methods
Relaxation model
Tikhonov regularization
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Summary:In this paper, we consider a Cauchy problem of recovering both missing value and flux on inaccessible boundary from Dirichlet and Neumann data measured on the remaining accessible boundary. Associated with two mixed boundary value problems, a regularized Kohn-Vogelius formulation is proposed. With an introduction of a relaxation parameter, the Dirichlet boundary conditions are approximated by two Robin ones. Compared to the existing work, weaker regularity is required on the Dirichlet data. This makes the proposed model simpler and more efficient in computation. A series of theoretical results are established for the new reconstruction model. Several numerical examples are provided to show feasibility and effectiveness of the proposed method. For simplicity of the statements, we take Poisson equation as the governed equation. However, the proposed method can be applied directly to Cauchy problems governed by more general equations, even other linear or nonlinear inverse problems.
ISSN:1468-1218
1878-5719
DOI:10.1016/j.nonrwa.2023.103935