In this paper, we investigate the regularization and numerical solution of geometric inverse problems related to linear elasticity with minimal assumptions on the geometry of the solution. In particular, we consider the probably severely ill-posed reconstruction problem of a two-dimensional inclusion from a single boundary measurement.In order to avoid parametrizations, which would introduce a priori assumptions on the geometric structure of the solution, we employ the level set method for the numerical solution of the reconstruction problem. With this approach, we construct an evolution of shapes with a normal velocity chosen depending on the shape derivative of the corresponding least-squares functional in order to guarantee its descent. Moreover, we analyse penalization by perimeter as a regularization method, based on recent results on the convergence of Neumann problems and a generalization of Golab's theorem. The behaviour of the level set method and of the regularization procedure in the presence of noise are tested in several numerical examples. It turns out that reconstructions of good quality can be obtained only for simple shapes or for unreasonably low noise levels. However, it seems reasonable that the quality of reconstructions improves by using more than a single boundary measurement, which is an interesting topic for future research.(Some figures in this article are in colour only in the electronic version) 4 This work was started during the visit to the JK university (Linz) as a research fellow 2002. 5 On leave from Johannes Kepler Universität Linz.

When estimating hydraulic transmissivity the question of parameterization is of great importance. The transmissivity is assumed to be a piecewise constant space dependent function and the unknowns are both the transmissivity values and the zonation, the partition of the domain whose parts correspond to the zones where the transmissivity is constant. Refinement and coarsening indicators, which are easy to compute from the gradient of the least squares misfit function, are introduced to construct iteratively the zonation and to prevent overparameterization.

The purpose of this work is to identify two-dimensional (2D) cracks by means of elastic boundary measurements. A uniqueness result is first proved in the general case, as well as the local Lipschitzian stability in the case of line segment emergent cracks. In this last case, the search for the unique zero of the reciprocity gap functional related to the singular solution of the elasticity problem provides a fast algorithm to determine the unknown crack tip.

The estimation of distributed parameters in a partial differential equation (PDE) from measures of the solution of the PDE may lead to underdetermination problems. The choice of a parameterization is a frequently used way of adding a priori information by reducing the number of unknowns according to the physics of the problem. The refinement indicators algorithm provides a fruitful adaptive parameterization technique that parsimoniously opens the degrees of freedom in an iterative way. We present a new general form of the refinement indicators algorithm that is applicable to the estimation of distributed multidimensional parameters in any PDE. In the linear case, we state the relationship between the refinement indicator and the decrease of the usual least-squares data misfit objective function. We give numerical results in the simple case of the identity model, and this application reveals the refinement indicators algorithm as an image segmentation technique.

- This paper is concerned with the ill-posed problem of identifying a hydraulic transmissivity in an isotropic and confined aquifer in two space dimensions. To define a regularization by adaptive discretization of the parameter, we use refinement and coarsening indicators, which give the first order effect on the optimal data misfit of adding or removing degrees of freedom to a current set of parameters. The direct problem is discretized by a mixed hybrid finite element method. A combination of the direct problem discretization with the adaptive discretrization for the transmissivity allows us to prove convergence of the algorithm as the mesh size goes to zero, stability of the finite-dimensional approximation, and convergence for noisy data with an appropriate stopping rule.