In this study, we address the ill-posed stationary Stokes data assimilation (DA) problem using the fading regularization method (FRM). It involves reconstructing the fluid velocity field throughout the study domain in R , where is the dimension of the space, as well as the boundary conditions, using knowledge of some observations of the fluid velocity field measured within a limited domain included in . Using the FRM, the main ill-posed problem is transformed into a sequence of well-posed constraint optimization problems and simplifies the resolution of DA problem. Additionally, we prove the convergence of both the continuous and the discrete formulations. This method is implemented numerically using the method of fundamental solutions (MFS) and several numerical simulations are shown to illustrate the performance of the algorithm in terms of efficiency, accuracy, convergence, stability, and robustness to noisy data, as well as its ability to deblur the data in .

In the present work, we focus on the resolution of the Cauchy Laplace problem using an energetic variational minimization approach in the framework of a finite element method. A new strategy of regularization, called a filtering procedure regularization, is developed. The advantage of using this new regularization is that it does not require a regularization parameter and is easy to implement. An optimal a priori error estimate is proven, for the first time up to our knowledge, in the context of the finite element method. Some numerical results are presented to illustrate the performance of our approach.

In this paper, the Cauchy problem for the Helmholtz equation is investigated. The objective is to recover the missing data on some part of the boundary of a bounded domain from overspecified data on the remaining part of the boundary. We propose a preconditioned Krylov algorithm to solve this ill-posed problem, based on the representation of the solution with surface integral equations and the Steklov–Poincaré operator. We give a theoretical and numerical validation of the proposed method conducted in the 3D setting. We show the fast convergence of the proposed algorithm tested on various synthetic examples. The numerical results show a high precision of the reconstruction obtained for different levels of noisy data.

In this paper, we investigate the application of the fading regularization method with the method of fundamental solutions (MFS) to the ill-posed Cauchy-Stokes problem. For both smooth and piecewise smooth two-dimensional geometries, we present a numerical reconstruction of the missing velocity and normal stress tensor on an inaccessible part of the boundary based on knowledge of over-prescribed noisy data acquired on the remaining accessible boundary part. Three numerical examples demonstrate the proposed numerical algorithm’s accuracy, convergence, stability, and efficiency, as well as its ability to deblur the data.

In this chapter, we consider the optimization problem of a heat distribution on a bounded domain Ω containing a heat source at an unknown location ω⊂Ω. More precisely, we are interested in the best location of ω allowing a suitable thermal environment. For this propose, we consider the minimization of the maximum temperature and its L2 mean oscillations. We extend the notion of topological derivative to the case of local coated perturbation and we perform the asymptotic expansion of the considered shape functionals. In order to reconstruct the location of ω, we propose a one-shot algorithm based on the topological derivative. Finally, we present some numerical experiments in two dimensional case, showing the efficiency of the proposed method.

This work is concerned with the boundary data completion problem related to the heat equation in the special case of an annular domain. We first reformulate this inverse problem into an interfacial equation involving Steklov‐Poincaré operator based on fictitious domain decomposition techniques. We present some theoretical results. For solving the problem under consideration, we suggest a new numerical point of view which helps to reduce the computational cost using the Schur complement algorithm. We perform then the convergence analysis of this new approach in the annular domain. Several numerical experiments are shown to illustrate the efficiency of the proposed method.

In this paper, we define explicit posteriori error estimates for data completion method for the Cauchy problem for Laplace equation, which is known as a highly ill-posed problem. The estimator is obtained for data without noise and noisy data problems. This estimator is used as a stopping criterion and gives information of the stability of the problem. Numerical applications are given for two methods of data completion: the conjugate gradient method and the method based on Legendre moments.

Abstract This work is devoted to the study of the void identification problem from partially overdetermined boundary data in the 2D-elastostatic case. In a first part, a shape identifiability result from a Cauchy data is presented, i.e. with traction field and boundary displacement as measurements. Then this geometric inverse problem is tackled by the minimization of two cost functionals, an energy gap functional and an L2{L^{2}}-gap functional, which enable the reconstruction of voids under Navier’s boundary conditions. The shape derivatives of these cost functionals are computed for the purpose of sensitivity analysis.

We consider the two-dimensional steady-state heat conduction on a bounded domain Omega containing a heat source at an unknown location omega subset of Omega. We are interested in determining the locations omega allowing a suitable thermal environment. The resulting shape optimization problem consists of a geometric control of either the maximum of the temperature or its L-2 mean oscillations in Omega. We derive the topological asymptotic expansion of the considered shape functionals with respect to the insertion of small circular coated inclusion characterized by a discontinuous thermal conductivity and their radius. We propose a reconstruction algorithm based on this topological gradient to identify the locations. Some numerical simulations are presented to show the efficiency of the algorithm.

In contrast to the conventional Cauchy–Stokes problems, in which the velocity and the stress force data are given on the accessible boundary, in the present paper, we reduce the accessible boundary data information and we consider a problem which deals only with shear stress data. We refer to this problem as a sub-Cauchy–Stokes problem. This problem is ill-posed because of severe instability and even uniqueness is unknown. We first address the uniqueness issues associated with this problem. Resorting to the domain decomposition techniques together with the duplication process of Vogelius (Kohn and Vogelius, 1985), we propose new Lagrange multiplier methods to solve the sub-Cauchy–Stokes problem. These methods consist in recasting the problem in terms of interfacial equations, by equalizing two solutions of the sub-Cauchy–Stokes problem using matching conditions defined on the inaccessible boundary. The matching is based on second order conditions and the types of the interfacial equations depend on the equations used to match the values of the unknowns on the inaccessible boundary. The interfacial problems are then solved by iterative procedures in which coefficients can be optimized to improve convergence rates. A complete analysis of the methods is presented, and intensive numerical results illustrate the effectiveness and the performance of the proposed approaches.

We consider the Cauchy-Stokes problem. We use a new method based on Nash game theory to recover the missing velocity and normal stress on some inaccessible part of the boundary. This method is used with two different approaches. The first one is compared to a control type one. The numerical study attests that both approaches give accurate results. We compare these results with those of the energy-like minimization method.

This work is devoted to some geometric inverse problems in li near elasticity. The problem considered is the cavities identification in mechanical structures from the knowledge of partially overdetermined boundary data, namely the disp lacement field and the normal component of the normal stress. We state a uniq ueness result from a single pair of data under some geometric al assumptions. We propose an iterative method based on the cou pling of the data completion process through the Steklov-Po incaré operator to reconstruct the shear stress and of the shape gra dient method combined with the level set method to identify c avities. Numerical simulations highlight the algorithm efficiency.

The aim of thiswork is an analysis of some geometrical inverse problems related to the identification of cavities in linear elasticity framework. We rephrase the inverse problem into a shape optimization one using an energetic least-squares functional. The shape derivative of this cost functional is combined with the level set method in a steepest descent algorithm to solve the shape optimization problem. The efficiency of this approach is illustrated by several numerical results.

In this initial study, we propose a numerical method for identifying multiple leak zones in a saturated unsteady flow. Using the conventional saturated groundwater flow equation, the leak identification problem is modeled as a Cauchy problem for the heat equation and the aim is to find the regions on the boundary of the solution domain where the solution vanishes because the leak zones correspond to null pressure values. This problem is ill-posed and to reconstruct the solution in a stable way, we modify it and employ a previously proposed iterative regularizing method. In this method, mixed well-posed problems obtained by changing the boundary conditions are solved for the heat operator as well as for its adjoint to obtain a sequence of approximations to the original Cauchy problem. The mixed problems are solved using a finite element method and the numerical results indicate that the leak zones can be identified with the proposed method. (C) 2016 Elsevier Inc. All rights reserved.

We are interested in this paper with the ill-posed Cauchy-Stokes problem. We consider a data completion problem in which we aim recovering lacking data on some part of a domain boundary , from the knowledge of partially overspecified data on the other part. The inverse problem is formulated as an optimization one using an energy-like misfit functional. We give the first order opti-mality condition in terms of an interfacial operator. Displayed numerical results highlight its accuracy. Nous nous intéressons à un problème de Cauchy mal posé, celui de la complétion de données frontières pour les équations de Stokes. Nous voulons reconstituer les données manquantes sur une partie non accessible de la frontière du domaine à partir de données peu surdéterminées sur la partie accessible. Nous formulons ce problème inverse sous forme de minimisation d'une fonctionnelle de type énergie. Les conditions d'optimalité du premier ordre sont écrites en termes d'équation d'interface utilisant les opérateurs de Stecklov-Poincaré. Nous donnons des résultats numériques attestant l'efficacité de la méthode.

In this work, we propose an approximate optimal control formulation of the Cauchy problem for the Stokes system. Here the problem is converted into an optimization one. In order to handle the instability of the solution of this ill-posed problem, a regularization technique is developed. We add a term in the least square function which happens to vanish while the algorithm converges. The efficiency of the proposed method is illustrated by numerical experiments.

The Bernoulli problem is rephrased into a shape optimization problem. In particular, the cost function, which turns out to be a constitutive law gap functional, is borrowed from inverse problem formulations. The shape derivative of the cost functional is explicitly determined. The gradient information is combined with the level set method in a steepest descent algorithm to solve the shape optimization problem. The efficiency of this approach is illustrated by numerical results for both interior and exterior Bernoulli problems.