Frédéric Pons

Solution of the Cauchy problem using iterated Tikhonov regularization

We are interested in this paper in recovering lacking data on some part of a domain boundary, from the knowledge of Cauchy data on the other part. It is first proved that the desired solution is the unique fixed point of some appropriate operator, which naturally gives rise to an iterative process that is proved to be convergent. Discretization provides an additional regularization: the algorithm reads as a least square fitting of the given data, with a regularization term the effect of which fades as iterations go on. Displayed numerical results highlight its accuracy, as well as its robustness.

An inversion method for harmonic functions reconstruction

Abstract We are interested in this paper in recovering an harmonic function from the knowledge of Cauchy data on some part of the boundary. The inverse scheme reduces the Cauchy problem resolution to a fixed point process. It is proved, when the data are compatible, this fixed point process converges to the Cauchy problem solution. The algorithm is implemented in both situations, firstly in the framework of boundary elements and secondly in a finite element one. Displayed numerical results highlight their accuracy, as well as their robustness to noisy data. Finally we present the contribution of the method to solve an engineering problem which consists in determining temperature and heat flux on the inner cylinder wall from the knowledge of temperatures and heat flux on the whole outer wall.

Recovery of Cracks from Incomplete Boundary Data

We are interested in this article in recovering line segment (or planar) cracks from incomplete boundary measurements. Most recovery algorithms, and especially fast ones based on the reciprocity gap, however need a full set of boundary data, which compels us to extending the available data prior to achieving recovery. Both the extension and recovery problems being severely ill posed, their stacking might drive the whole process to a blowing up, unless the instabilities are properly handled. Numerical results actually prove the so-built composite algorithm to hold good accuracy and robustness features.