Solving a nonlinear inverse Robin problem through a linear Cauchy problem

ABSTRACT

Considered in this paper is an inverse Robin problem governed by a steady-state diffusion equation. By the Robin inverse problem, one wants to recover the unknown Robin coefficient on an inaccessible boundary from Cauchy data measured on the accessible boundary. In this paper, instead of reconstructing the Robin coefficient directly, we compute first the Cauchy data on the inaccessible boundary which is a linear inverse problem, and then compute the Robin coefficient through Newton's law. For the Cauchy problem, a parameter-dependent coupled complex boundary method (CCBM) is applied. The CCBM has its own merits, and this is particularly true when it is applied to the Cauchy problem. With the introduction of a positive parameter, we can prove the regularized solution is uniformly bounded with respect to the regularization parameter which is a very good property because the solution can now be reconstructed for a rather small value of the regularization parameter. For the problem of computing the Robin coefficient from the recovered Cauchy data, a least square output Tikhonov regularization method is applied to Newton's law to obtain a stable approximate Robin coefficient. Numerical results are given to show the feasibility and effectiveness of the proposed method.

COMMUNICATED BY:

• J. Zou

KEYWORDS:

• Inverse Robin problem
• Cauchy problem
• coupled complex boundary method
• Tikhonov regularization

AMS CLASSIFICATIONS:

• 65N21
• 65F22

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work of the first and fourth authors are supported by the Natural Science Foundation of China [grant numbers 11401304, 11571311]. The work of the third author is supported by the Future Fellowship of the Australian Research Council [grant number FT170100231].