Papers by abdelkrim chakib
An improved numerical approach for solving shape optimization problems on convex domains
Numerical Algorithms, Oct 13, 2023
A new numerical approach for solving shape optimization fourth-order spectral problems among convex domains
Computers & Mathematics with Applications
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific r... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
HAL (Le Centre pour la Communication Scientifique Directe), Oct 17, 2021
We consider the shape derivative formula for a volume cost functional studied in preceding papers... more We consider the shape derivative formula for a volume cost functional studied in preceding papers and using 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.
On a numerical approach for solving some geometrical shape optimization problems in fluid mechanics
Communications in Nonlinear Science and Numerical Simulation
On a shape derivative formula for star-shaped domains using Minkowski deformation
AIMS Mathematics
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.
Approximation of a Cauchy inverse problem
International audienc
On an effective approach in shape optimization problem for Stokes equation
Optimization Letters
On numerical resolution of an inverse Cauchy problem modeling the airflow in the bronchial tree
Computational and Applied Mathematics, 2021
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.
ArXiv, 2021
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...
On numerical study of constrained coupled shape optimization problems based on a new shape derivative method
Numerical Methods for Partial Differential Equations
A primal-dual approach for the Robin inverse problem in a nonlinear elliptic equation: The case of the 𝐿1 − 𝐿2 cost functional
Journal of Inverse and Ill-posed Problems
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.
On a shape derivative formula with respect to convex domains
Journal of Convex Analysis
On Numerical Approaches for Solving an Inverse Cauchy Stokes Problem
Applied Mathematics & Optimization, 2022
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.
Applications of Mathematics
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.
New Perspectives In Solving Cardiac Electrophysiological Activity Reconstruction
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.
Computational Optimization and Applications, 2020
This paper is devoted to a numerical method for the approximation of a class of free boundary pro... more This paper is devoted to a numerical method for the approximation of a class of free boundary problems of Bernoulli's type, reformulated as optimal shape design problems with appropriate shape functionals. We show the existence of the shape derivative of the cost functional on a class of admissible domains and compute its shape derivative by using the formula proposed in [6, 7], that is, by means of support functions. On the numerical level, this allows us to avoid the tedious computations of the method based on vector fields. A gradient method combined with boundary element method are performed for the approximation of this problem, in order to overcome the re-meshing task required by the finite element method. Finally, we present some numerical results and simulations concerning practical applications, showing the effectiveness of the proposed approach.
Knowledge and Information Systems, 2020
This paper is concerned by solving supervised machine learning problem as an inverse problem. Rec... more This paper is concerned by solving supervised machine learning problem as an inverse problem. Recently, many works have focused on defining a relationship between supervised learning and the well-known inverse problems. However, this connection between the learning problem and the inverse one has been done in the particular case where the inverse problem is reformulated as a minimization problem with a quadratic cost functional (L 2 cost functional). Although, it is well known that the cost functional can be L 1 , L 2 or any positive function that measures the gap between the predicted data and the observed one. Indeed, the use of L 1 loss function for supervised learning problem gives more consistent results (see Rosasco et al. in Neural Comput 16:1063-1076, 2004). This strengthens the idea of reformulating the inverse problem, associated to machine learning problem, into a minimization problem using L 1 functional. However, the L 1 loss function is non-differentiable, which precludes the use of standard optimization tools. To overcome this difficulty, we propose in this paper a new technique of approximation based on the reformulation of the associated inverse problem into a minimizing one of a slanting cost functional Chen et al. (MIS Q Manag Inf Syst 36:1165-1188, 2012), which is solved using Tikhonov regularization and Newton's method. This approach leads to an efficient numerical algorithm allowing us to solve supervised learning problem in the most general framework. To confirm this, we present some numerical results showing the efficiency of the proposed approach. Furthermore, the numerical experiment validation is made through academic and real-life data. Thus, the comparison with existing methods and numerical stability of the algorithm is presented in order to show that our approach is better in terms of convergence speed and quality of predicted models.
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Papers by abdelkrim chakib