Faten Jelassi

Extended-domain-Lavrentiev's regularization for the Cauchy problem

We investigate new issues pertaining to the regularization of the ill-posed Cauchy problem for the Laplace equation. We show how the variational formulation proposed in Ben Belgacem and El Fekih (2005 Inverse Problems 21 1915–36) allows for a meaningful interpretation of the general source condition, currently used in the analysis of ill-posed problems. This assumption turns out to be closely connected to the possibility for the exact Cauchy solution to be harmonically extended to a larger domain. Then, we turn to the consolidation of the Lavrentiev method owing to the extension of the computational domain. The study we conduct and the numerical examples we present display an important improvement of the regularization capabilities and accuracy, especially for thin domains and when the data are substantially damaged by noise.

Finite Element Methods for the Temperature in Composite Media with Contact Resistance

We consider a heat diffusion problem inside a composite medium. The contact resistance at the interface of constitutive materials allows for jumps of the temperature field. The transmission conditions need to be handled carefully and efficiently. The main concerns are accuracy and feasibility. Hybrid dual formulations are recommended here as the most popular mixed finite elements well adapted to account for the discontinuity of the temperature field. We therefore write the discretization of the heat problem by mixed finite elements and perform its numerical analysis. Of course, applying Lagrangian finite elements is possible in simple composite media but it turns out to be problematic for complex geometries. Nevertheless, we study the convergence of this finite element method to highlight some particularities related to the model under consideration and point out the effect of the contact resistance on the accuracy. Illustrative numerical experiments are finally provided to assess the theoretical findings.RésuméNous considérons une équation qui modélise la diffusion de la température dans une mousse de graphite contenant des capsules de sel. Les conditions de transition de la température entre le graphite et le sel doivent être traitées correctement. Nous effectuons l’analyse de ce modèle et prouvons qu’il est bien posé. Puis nous en proposons une discrétisation par éléments finis et effectuons l’analyse a priori du problème discret. Quelques expériences numériques confirment l’intérêt de cette approche.

Local Convergence of the Lavrentiev Method for the Cauchy Problem via a Carleman Inequality

The purpose is to perform a sharp analysis of the Lavrentiev method applied to the regularization of the ill-posed Cauchy problem, set in the Steklov-Poincaré variational framework. Global approximation results have been stated earlier that demonstrate that the Lavrentiev procedure yields a convergent strategy. However, no convergence rates are available unless a source condition is assumed on the exact Cauchy solution. We pursue here bounds on the approximation (bias) and the noise propagation (variance) errors away from the incomplete boundary where instabilities are located. The investigation relies on a Carleman inequality that enables enhanced local convergence rates for both bias and variance errors without any particular smoothness assumption on the exact solution. These improved results allows a new insight on the behavior of the Lavrentiev solution, look similar to those established for the Quasi-Reversibility method in [Inverse Problems 25, 035005, 2009]. There is a case for saying that this sort of ‘super-convergence’ is rather inherent to the nature of the Cauchy problem and any reasonable regularization procedure would enjoy the same locally super-convergent behavior.

A finite element model for the data completion problem: analysis and assessment

We investigate the computational advantages of the Steklov-Poincaré variational formulation, based on the uniqueness Holmgren theorem, for the badly ill-posed data completion problem. We study the discrete solution with a finite element approximation. The uniqueness issue is dealt with and we check that it is related to a discrete Holmgren result which requires a particular assumption on the mesh. Then, the important point is to show how the finite element variational problem may be recast into a least-squares problem. Lavrentievs method turns out to be a Tikhonov regularization of an ‘underlying’ equation that will never be explicited. When employed in conjunction with the Morozov discrepancy principle to select the regularization parameter, the overall mathematical studies realized specifically for the Tikhonov method extends as well to the Lavrentiev method to conclude to a convergent regularization strategy. Similarly, the conjugate gradient method (CGM) may be considered as applied to a normal equation of that same ‘hidden underlying’ problem. We conduct a brief discussion about this method to explain why it yields a convergent strategy. We close with several numerical simulations for different geometries to assess the Lavrentiev finite element or the PCG finite element solution of the data completion problem.

Identifiability of surface sources from Cauchy data

We explore the reconstruction of surface sources from single Cauchy data on the potential. Identifiability is first proven when the location of these sources is known and their density is to be sought. We also investigate the ill-posedness degree to establish that the problem is severely ill-posed, and exhibit the necessary orthogonality conditions fulfilled by exact Cauchy data, for which the process of recovering the density function of the sources succeeds. We later turn to the more difficult reconstruction of the location, the shape and the density function of the sources. Identifiability cannot be obtained for general geometries. Nevertheless, under some acceptable assumptions on the size and the location of these sources, we are able to state identifiability results.

On Handling the Boundary Conditions at the Infinity for Some Exterior Problems by the Alternating Schwarz Method

Partial differential equations modeling physical phenomena occurring in unbounded domains have given rise to lot of work either for their mathematical analysis or for their numerical approximation. To make them accessible to scientific computing it is necessary to write them, in an appropriate way, on truncated domains. Integral equations are affordable tools for the simulation of these problems, although, very often, the associated algebraic system turns out to be heavy to solve. An alternative is to resort to the coupled approach of volumic finite elements/integral representation, which combines the advantages of truncating the domain through the integral formula while avoiding the singularities, and of having a sparse matrix to invert coming from classical finite elements (see [9], [7]). We propose in this note an iterative algorithm to solve the coupled problem in a competitive way. This algorithm may be viewed as an alternating Schwarz method which allows us, for some elliptic exterior problems, to state a geometrical convergence rate.