We consider the shape derivative formula for a volume cost functional studied in previous papers ... more We consider the shape derivative formula for a volume cost functional studied in previous papers where we used the Minkowski deformation and support functions in the convex setting. In this work, we extend it to some non-convex domains, namely the star-shaped ones. The formula happens to be also an extension of a well-known one in the geometric Brunn-Minkowski theory of convex bodies. At the end, we illustrate the formula by applying it to some model shape optimization problem.
This paper is devoted to the numerical resolution of an inverse Cauchy problem governed by Stokes... more This paper is devoted to the numerical resolution of an inverse Cauchy problem governed by Stokes equation modeling the airflow in the lungs. It consists in determining the air velocity and pressure on the artificial boundaries of the bronchial tree. This data completion problem is one of the highly ill-posed problems in the Hadamard sense (Hadamard in Lectures on Cauchy’s problem in linear partial differential equations. Dover, New York, 1953). This gives great importance to its numerical resolution and in particular to carry out stable numerical approaches, mostly in the case of noisy data. The main idea of this work is to extend some regularizing, stable and fast iterative algorithms for solving this problem based on the domain decomposition approach (Chakib et al. in Inverse Prob 35(1):015008, 2018). We discuss the efficiency and the feasibility of the proposed approach through some numerical tests performed using different domain decomposition algorithms. Finally, we opt for the Robin–Robin algorithm, which showed its performance, for the numerical simulation of the airflow in the bronchial tree configuration.
In this paper, we are interested to an inverse Cauchy problem governed by Stokes equation, called... more In this paper, we are interested to an inverse Cauchy problem governed by Stokes equation, called the data completion problem. It consists in determining the unspecified fluid velocity, or one of its components over a part of its boundary, by introducing given measurements on its remaining part. As it’s known, this problem is one of highly ill-posed problem in the Hadamard’s sense [14], it is then an interesting challenge to carry out a numerical procedure for approximating their solutions, mostly in the particular case of noisy data. To solve this problem, we propose here a regularizing approach based on a coupled complex boundary method, originally proposed in [8], for solving an inverse source problem. We show the existence of the regularization optimization problem and prove the convergence of subsequence of optimal solutions of Tikhonov regularization formulations to the solution of the Cauchy problem. Then we suggest the numerical approximation of this problem using the adjoin...
In this work, we consider the inverse problem of identifying a Robin coefficient in a nonlinear e... more In this work, we consider the inverse problem of identifying a Robin coefficient in a nonlinear elliptic equation with mixed boundary conditions. We firstly reformulate the inverse problem as a regularized optimal control one, where the functional cost is of type L 1 - L 2 L^{1}-L^{2} ; then we prove the existence and uniqueness of a minimizer to the resulting optimization problem in a suitable functional space. Finally, we provide a primal-dual algorithm to solve the variational problem and give some numerical results that prove the accuracy of the proposed method in the identification of the Robin coefficient.
Annals of the University of Craiova - Mathematics and Computer Science Series, 2015
This work deals with the homogenization of heat transfer nonlinear parabolic problem in a periodi... more This work deals with the homogenization of heat transfer nonlinear parabolic problem in a periodic composite medium consisting in two-component (fluid/solid). This problem presents some difficulties due to the presence of a nonlinear Neumann condition modeling a radiative heat transfer on the interface between the two parts of the medium and to the fact that the problem is strongly coupled. In order to justify rigorously the homogenization process, we use two scale convergence. For this, we show first the existence and uniqueness of the homogenization problem by topological degree of Leray-Schauder, Then we establish the two scale convergence, and identify the limit problems.
In this paper, we consider an inverse problem in hydrology governed by a highly nonlinear parabol... more In this paper, we consider an inverse problem in hydrology governed by a highly nonlinear parabolic equation called Richards equation. This inverse problem consists to determine a set of hydrological parameters describing the flow of water in porous media, from some additional observations on pressure. We propose an approximation method of this problem based on its optimal control formulation and a temporal discretization of its state problem. The obtained discrete nonlinear state problem is approached by the finite difference method and solved by Picard's method. Then, for the resolution of the discrete associated optimization problem, we opt for an evolutionary algorithm. Finally, we give some numerical results showing the efficiency of the proposed approach.
We are interested in an optimal shape design formulation for a class of free boundary problems of... more We are interested in an optimal shape design formulation for a class of free boundary problems of Bernoulli type. We show the existence of the optimal solution of this problem by proving continuity of the solution of the state problem with respect to the domain. The main tools in establishing such a continuity are a result concerning uniform continuity of the trace operator with respect to the domain and a recent result on the uniform Poincare inequality for variable domains.
ELECTROCARDIOGRAPHY (ECG) investigates the relationship between the electrical activity of the he... more ELECTROCARDIOGRAPHY (ECG) investigates the relationship between the electrical activity of the heart and its induced voltages measured on the torso surface. This relationship can be characterized mathematically as an inverse problem where the goal is to non invasively estimate cardiac electrical activity from voltage distributions measured on the body surface. In order to solve this problem we suggest a new approach based on domain decomposition technique. We approximate our approach by a Finite Element Method. Numerical experiments with 2D domains highlight the efficiency of the proposed methods as well as their robustness in the model context.
We consider the shape derivative formula for a volume cost functional studied in previous papers ... more We consider the shape derivative formula for a volume cost functional studied in previous papers where we used the Minkowski deformation and support functions in the convex setting. In this work, we extend it to some non-convex domains, namely the star-shaped ones. The formula happens to be also an extension of a well-known one in the geometric Brunn-Minkowski theory of convex bodies. At the end, we illustrate the formula by applying it to some model shape optimization problem.
This paper is devoted to the numerical resolution of an inverse Cauchy problem governed by Stokes... more This paper is devoted to the numerical resolution of an inverse Cauchy problem governed by Stokes equation modeling the airflow in the lungs. It consists in determining the air velocity and pressure on the artificial boundaries of the bronchial tree. This data completion problem is one of the highly ill-posed problems in the Hadamard sense (Hadamard in Lectures on Cauchy’s problem in linear partial differential equations. Dover, New York, 1953). This gives great importance to its numerical resolution and in particular to carry out stable numerical approaches, mostly in the case of noisy data. The main idea of this work is to extend some regularizing, stable and fast iterative algorithms for solving this problem based on the domain decomposition approach (Chakib et al. in Inverse Prob 35(1):015008, 2018). We discuss the efficiency and the feasibility of the proposed approach through some numerical tests performed using different domain decomposition algorithms. Finally, we opt for the Robin–Robin algorithm, which showed its performance, for the numerical simulation of the airflow in the bronchial tree configuration.
In this paper, we are interested to an inverse Cauchy problem governed by Stokes equation, called... more In this paper, we are interested to an inverse Cauchy problem governed by Stokes equation, called the data completion problem. It consists in determining the unspecified fluid velocity, or one of its components over a part of its boundary, by introducing given measurements on its remaining part. As it’s known, this problem is one of highly ill-posed problem in the Hadamard’s sense [14], it is then an interesting challenge to carry out a numerical procedure for approximating their solutions, mostly in the particular case of noisy data. To solve this problem, we propose here a regularizing approach based on a coupled complex boundary method, originally proposed in [8], for solving an inverse source problem. We show the existence of the regularization optimization problem and prove the convergence of subsequence of optimal solutions of Tikhonov regularization formulations to the solution of the Cauchy problem. Then we suggest the numerical approximation of this problem using the adjoin...
In this work, we consider the inverse problem of identifying a Robin coefficient in a nonlinear e... more In this work, we consider the inverse problem of identifying a Robin coefficient in a nonlinear elliptic equation with mixed boundary conditions. We firstly reformulate the inverse problem as a regularized optimal control one, where the functional cost is of type L 1 - L 2 L^{1}-L^{2} ; then we prove the existence and uniqueness of a minimizer to the resulting optimization problem in a suitable functional space. Finally, we provide a primal-dual algorithm to solve the variational problem and give some numerical results that prove the accuracy of the proposed method in the identification of the Robin coefficient.
Annals of the University of Craiova - Mathematics and Computer Science Series, 2015
This work deals with the homogenization of heat transfer nonlinear parabolic problem in a periodi... more This work deals with the homogenization of heat transfer nonlinear parabolic problem in a periodic composite medium consisting in two-component (fluid/solid). This problem presents some difficulties due to the presence of a nonlinear Neumann condition modeling a radiative heat transfer on the interface between the two parts of the medium and to the fact that the problem is strongly coupled. In order to justify rigorously the homogenization process, we use two scale convergence. For this, we show first the existence and uniqueness of the homogenization problem by topological degree of Leray-Schauder, Then we establish the two scale convergence, and identify the limit problems.
In this paper, we consider an inverse problem in hydrology governed by a highly nonlinear parabol... more In this paper, we consider an inverse problem in hydrology governed by a highly nonlinear parabolic equation called Richards equation. This inverse problem consists to determine a set of hydrological parameters describing the flow of water in porous media, from some additional observations on pressure. We propose an approximation method of this problem based on its optimal control formulation and a temporal discretization of its state problem. The obtained discrete nonlinear state problem is approached by the finite difference method and solved by Picard's method. Then, for the resolution of the discrete associated optimization problem, we opt for an evolutionary algorithm. Finally, we give some numerical results showing the efficiency of the proposed approach.
We are interested in an optimal shape design formulation for a class of free boundary problems of... more We are interested in an optimal shape design formulation for a class of free boundary problems of Bernoulli type. We show the existence of the optimal solution of this problem by proving continuity of the solution of the state problem with respect to the domain. The main tools in establishing such a continuity are a result concerning uniform continuity of the trace operator with respect to the domain and a recent result on the uniform Poincare inequality for variable domains.
ELECTROCARDIOGRAPHY (ECG) investigates the relationship between the electrical activity of the he... more ELECTROCARDIOGRAPHY (ECG) investigates the relationship between the electrical activity of the heart and its induced voltages measured on the torso surface. This relationship can be characterized mathematically as an inverse problem where the goal is to non invasively estimate cardiac electrical activity from voltage distributions measured on the body surface. In order to solve this problem we suggest a new approach based on domain decomposition technique. We approximate our approach by a Finite Element Method. Numerical experiments with 2D domains highlight the efficiency of the proposed methods as well as their robustness in the model context.
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