An energy-like error functional is introduced in the context of the ill-posed problem of boundary data recovering, which is well known as a Cauchy problem. Links with existing methods for data completion are detailed. Here the problem is converted into an optimization problem; the computation of the gradients of the energy-like functional is given for both the continuous and the discrete problems. Numerical experiments highlight the efficiency of the proposed method as well as its robustness in the model context of Laplace's equation, but also for anisotropic conductivity problems.

The problem of determining a crack by overspecified boundary data is considered. When complete data are available on the external boundary, a reciprocity gap concept is introduced. This concept formalizes the comparison of the response of the safe body to the response of the cracked one of the same physical characteristics. If the crack is known (or assumed) to be planar, explicit inversion formulae are derived determining the host plane equation and the length of an emerging crack in two-dimensional (2D) situations. A reciprocity gap functional is designed and exploited to establish a complete identification result.Numerical trials of the identification methods proposed show very good accuracies and insignificant computational costs.

In this paper we are concerned with the planar crack identification problem defined by a unique complete elastostatic overdetermined boundary datum. Based on the reciprocity gap principle, we give a direct process for locating the host plane and we establish a new constuctive identifiability result for 3D planar cracks.

We consider the inverse problems of locating pointwise or small size conductivity defaults in a plane domain, from overdetermined boundary measurements of solutions to the Laplace equation. We express these issues in terms of best rational or meromorphic approximation problems on the boundary, with poles constrained to belong to the domain. This approach furnishes efficient and original resolution schemes.

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This paper deals with a semi-explicit algorithm to reconstruct two-dimensional (2D) segment cracks, or three-dimensional (3D) planar cracks, in the framework of overspecified boundary data. The algorithm is based on the reciprocity gap concept, introduced by Andrieux and Ben Abda, which provides explicitly the line (or the plane) support of the cracks. A numerical reconstruction of the cracks, which are actually the support of the solution jump across this plane, is then performed by computing the Fourier expansion of the solution jump itself. After the numerical analysis of the method, some numerical results are presented and commented on.

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