The author describes a method for proving global uniqueness theorems for one broad class of multidimensional coefficient inverse problems. This method is based on Carleman estimates, and it does not depend essentially on the order or type of differential operator.

https://iopscience.iop.org/article/10.1088/0266-5611/8/4/009/pdf

Inverse problems and Carleman estimates

We consider the problem of detecting corrosion damage on an inaccessible part of a metallic specimen. Electrostatic data are collected on an accessible part of the boundary. The adoption of a simplified model of corrosion appearance reduces our problem to recovering a functional coefficient in a Robin boundary condition for Laplace's equation. We review theoretical results and numerical methods based on the thin-plate approximation and the Galerkin method. Moreover, we introduce a numerical algorithm based on the quasi-reversibility method.

An inverse Robin problem for Laplace's equation: theoretical results and numerical methods

In this paper we combine the method of fundamental solutions with various regularization techniques to solve Cauchy problems of elliptic differential operators. The main idea is to approximate the unknown solution by a linear combination of fundamental solutions whose singularities are located outside the solution domain. To solve effectively the discrete ill-posed resultant matrix, we use three regularization strategies under three different choices for the regularization parameter. Several examples on problems with smooth and non-smooth geometries in 2D and 3D spaces using under-, equally, and over-specified Cauchy data on an accessible boundary are given. Numerical results indicate that the generalized cross-validation and L -curve choice rulers for Tikhonov regularization and damped singular value decomposition strategy are most effective when using the same numbers of collocation and source points. It has also been observed that the use of more Cauchy data will greatly improve the accuracy of the approximate solution.

Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators

We consider the problem of determining quantitative information about corrosion occurring on an inaccesible part of a specimen. The data for the problem consist of prescribed current flux and voltage measurements on an accessible part of the specimen boundary. The problem is modelled by Laplace's equation with an unknown term in the boundary conditions. Our goal is recovering from the data. We prove uniqueness under certain regularity assumptions and construct a regularized numerical method for obtaining approximate solutions to the problem. The numerical method, which is based on the assumption that the specimen is a thin plate, is tested in numerical experiments using synthetic data.

https://iopscience.iop.org/article/10.1088/0266-5611/13/4/006/pdf

An inverse problem in corrosion detection

This is a review paper of the role of Carleman estimates in the theory of Multidimensional Coefficient Inverse Problems since the first inception of this idea in 1981.

Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems

This work concerns the use of the method of quasi-reversibility for solving Cauchy problems for Laplace’s equation. The paper begins with a derivation of an error estimate for this method using Car...

Fadil Santosa

Papers

A computational quasi-reversiblility method for Cauchy problems for Laplace's equation