Department of Mathematics

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Applying the Powell's Symmetrical Technique to Conjugate Gradient Methods with the Generalized Conjugacy Condition
(Numerical Functional Analysis and Optimization, 2016-07-02) Noureddine Benrabia, Yamina Laskri, Hamza Guebbai, Mehiddin Al-Baali
This article proposes new conjugate gradient method for unconstrained optimization by applying the Powell symmetrical technique in a defined sense. Using the Wolfe line search conditions, the global convergence property of the method is also obtained based on the spectral analysis of the conjugate gradient iteration matrix and the Zoutendijk condition for steepest descent methods. Preliminary numerical results for a set of 86 unconstrained optimization test problems verify the performance of the algorithm and show that the Generalized Descent Symmetrical Hestenes-Stiefel algorithm is competitive with the Fletcher-Reeves (FR) and Polak-Ribiére-Polyak (PRP+) algorithms.
On the regularization method for Fredholm integral equations with odd weakly singular kernel
(Computational and Applied Mathematics, 2018-03-23) Noureddine Benrabia & Hamza Guebbai
In this paper, we propose a numerical method to approach the solution of a Fredholm integral equation with a weakly singular kernel by applying the convolution product as a regularization operator and the Fourier series as a projection. Preliminary numerical results show that the order of convergence of the method is better than the one of conventional projection methods.
A Variant of Projection-Regularization Method for ill-posed linear operator equations
(International Journal of Computational Methods, 2020-09-20) Bechouat Tahar; Boussetila Nadjib; Rebbani Faouzia
In the present paper, we report on a strategy for computing the numerical approximate solution for a class of ill-posed operator equations in Hilbert spaces: Kf = g. This approach is a combination of Tikhonov regularization method and the finite rank approximation of K. Finally, numerical results are given to show the effectiveness of this method.
A new optimization method based on Perry's idea through the use of the matrix power
(Journal of Applied Mathematics and Computational Mechanics, 2021-04-03) Imane Hafaidia, Noureddine Benrabia, Mourad Ghiat, Hamza Guebbai
The purpose of this paper is to present a new conjugate gradient method for solving unconstrained nonlinear optimization problems, based on Perry’s idea. An accelerated adaptive algorithm is proposed, where our search direction satisfies the sufficient descent condition. The global convergence is analyzed using the spectral analysis. The numerical results are described for a set of standard test problems, and it is shown that the performance of the proposed method is better than that of the CG-DESCENT, the mBFGS and the SPDOC
accélération méthodes itératives appliquées a l'optimisation non inéaire
(2021-06-27) DELLADJI Sarra
ملخص في هذه الأطروحة التي تهدف إلى تسريع تقارب طرق التدرج المترافق الكلاسيكية، اقترحنا ثلاث خوارزميات تعمل على هذا المبدأ، إذ اعتمدنا بشكل خاص على تقنية تسريع شهيرة و هي تهجين خوارزميتين من خلال مزج محدب لمعاملاتهما المحددان لمختلف طرق التدرج المترافق. بعد إثبات تقارب الخوارزميات المقترحة وباستخدام دوال تجريبية، أثبتنا من خلال التجارب العددية أن هذه الخوارزميات أكثر كفاءة و نجاعة مقارنة بالخوارزميات المدمجة. ------------------------------ Abstract In this thesis aims to accelerate the convergence of classical conjugate gradient methods, we have proposed three algorithms based on this concept, where we specifically relied on a famous acceleration technique which is the hybridization of two algorithms, by convex combination of their coefficients that determine the different standard conjugate gradient methods. After having proven the convergence of the proposed algorithms, using experimental functions, we have shown through numerical experiments that these algorithms are more efficient and perform than the combined algorithms. ----------------- Résumé Dans cette thèse, qui vise à accélérer la convergence des méthodes classiques de gradient conjugué, nous avons proposé trois algorithmes qui fonctionnent sur ce principe, où nous nous sommes spécifiquement appuyés sur une célèbre technique d'accélération qui est l'hybridation de deux algorithmes, par combinaison convexe de leurs coefficients qui déterminent les différentes méthodes de gradient conjugué standard. Après avoir prouvé la convergence des algorithmes proposés, en utilisant des fonctions expérimentales, nous avons montré à travers des expériences numériques que ces algorithmes sont plus efficaces et performants que les algorithmes combinés.
Sur Quelques Développements Récents de la méthode du gradient conjugué
(2021-07-07) HAMDI Amira
Abstract A new hybrid conjugate gradient algorithm is proposed for solving unconstrained optimization problems, the conjugate gradient parameter β_kis computed as a convex combination of Hager& Zhang et Dai and Yuan , producing a descent direction at each iteration and globally converges provided the linear search meets the requirements of Wolfe. Numerical experiments are performed to test the effectiveness of the new method, which confirms the potential of this method. -------------- Résumé Un nouvel algorithme de hybride du gradient conjuguée est proposé pour résoudre des problèmes d’optimisation sans contrainte, le paramètre de gradient conjugué β_k est calculé comme une combinaison convexe de Hager& Zhang et Dai and Yuan ce qui produit une direction de descente à chaque itération et converge globalement à condition que la recherche linéaire satisfait aux conditions de Wolfe. Les expériences numériques sont effectuées pour tester l’efficacité de la nouvelle méthode, ce qui confirme les potentiels de cette méthode. ---------------------- ملخص تم اقتراح خوارزمية التدرج المترافق الجديدة لحل مشاكل التحسين غير المقيدة، ويتم حساب معامل التدرج المترافق〖 β〗_k كمجموعة محدبة لهجار وهونغ وداي يونغ والتي تحافظ على خصائص النزول في كل مرحلة كما انها تعطي التقارب المطلق باستعمال شروط البحث الخطي لوولف تظهر النتائج العددية ان هذه الطريقة فعالة بالنسبة للمسائل المختبرة مما تثبت افضليتها مقارنة مع الطرق الاخرى.
A new class of nonlinear conjugate gradient coefficients for unconstrained optimization
(Asian-European Journal of Mathematics, 2022) Amina Boumediene; Tahar Bechouat; Rachid Benzine; Ghania Hadji
The nonlinear Conjugate gradient method (CGM) is a very effective way in solving largescale optimization problems. Zhang et al. proposed a new CG coefficient which is defined by BNPRPk . They proved the sufficient descent condition and the global convergence for nonconvex minimization in strong Wolfe line search. In this paper, we prove that this CG coefficient possesses sufficient descent conditions and global convergence properties under the exact line search.
Benrabia distribution: properties and applications
(2022) Mohammed Benrabia; Loai M. A. AlZoubi
In this paper, we propose a new two parameter continuous distribution. It is called a Benrabia distribution. Some statistical properties are derived such as: the moment generating function, the moments and related measures, the reliability analysis and related functions. Also, the distribution of order statistics and the quantile function are presented and the R´ enyi entropy is derived. The method of maximum likelihood estimation is used to estimate the distribution parameters. A simulation is performed to investigate the performance of MLE, real data applications show that the proposed distribution can provide a better fit than several well-known distributions.
A new hybrid conjugate gradient method of unconstrained optimization methods
(Asian-European Journal of Mathematics, 2022) Chenna Nasreddine; Badreddine Sellami; Belloufi Mohammed
In this paper, we present a new hybrid method to solve a nonlinear unconstrained optimization problem by using conjugate gradient, which is a convex combination of Liu–Storey (LS) conjugate gradient method and Hager–Zhang (HZ) conjugate gradient method. This method possesses the sufficient descent property with Strong Wolfe line search and the global convergence with the strong Wolfe line search. In the end of this paper, we illustrate our method by giving some numerical examples.
Alzoubi Distribution: Properties and Applications
(2022) Mohammed Benrabia; Loai M. A. Alzoubi
In this article, a new two parameters distribution named Alzoubi distribution (AzD) is suggested. Its moments have been obtained. Reliability analysis including hazard rate, cumulative hazard rate and reversed hazard rate functions and the entropy have been discussed, the deviation about mean and median is derived, and the distribution of order statistics is obtained. A simulation study is performed to estimate the model parameters using the maximum likelihood and the ordinary and weighted least squares methods. The goodness of fit to real data set shows the superiority of the new distribution. Keywords: Mixing distribution, Alzoubi distribution, moments, reliability analysis, R´ enyi entropy, maximum likelihood estimation, moment generating function.
A new hybrid conjugate gradient method of unconstrained optimization methods
(2022) Chenna Nasreddine; Sellami Badreddine; Belloufi Mohammed
In this paper, we present a new hybrid method to solve a nonlinear unconstrained optimization problem by using conjugate gradient, which is a convex combination of Liu–Storey (LS) conjugate gradient method and Hager–Zhang (HZ) conjugate gradient method. This method possesses the sufficient descent property with Strong Wolfe line search and the global convergence with the strong Wolfe line search. In the end of this paper, we illustrate our method by giving some numerical examples.
Une variante de la méthode de Projection régularisée pour une classe d’équations intégrales de Fredholm de première espèce
(HAL, 2022) Nadjib Boussetila; Tahar Bechouat
Dans ce travail, on propose une procédure de régularisation et d’approximation numérique pour une classe d’équations intégrales de Fredholm de première espèce, basée sur la méthode LSQ projetée, où le sous-espace de projection est formé par les fonctions d’ondes sphéroïdales. Pour le paramètre de régularisation on adopte un choix a posteriori basé sur la méthode de Morozov cubique amortie. L’étude est justifiée par une analyse théorique détaillée (convergence, estimations d’erreur) sous certaines hypothèses de régularité imposées sur le noyau et les données du problème en question.
Loai Distribution: Properties, Parameters Estimation and Application to Covid-19 Real Data
(2022) Loai M. A. Alzoubi; Mohammad M. Gharaibeh; Ahmad M. Alkhazaalh; Mohammed Benrabia
In this paper, we propose a new two parameter continuous distribution. It is called Loai distribution. Some statistical properties of this distribution are derived such as: the moment generating function, the moments and related measures, the reliability analysis, and associated functions. Also, the distribution of order statistics and the quantile function are presented. Shannon, Re’nyi and Tsallis entropies are derived. The method of maximum likelihood and some other methods of estimation are used to estimate the distribution parameters. A simulation study is performed to investigate the performance of different methods of estimation. Covid-19 real data applications show that the proposed distribution can provide a better t than several well-known distributions.
New iterative conjugate gradient method for nonlinear unconstrained optimization
(RAIRO-Operations Research, 2022) Sabrina Ben Hanachi; Badreddine Sellami; Mohammed Belloufi
Conjugate gradient methods (CG) are an important class of methods for solving unconstrained optimization problems, especially for large-scale problems. Recently, they have been much studied. In this paper, we propose a new conjugate gradient method for unconstrained optimization. This method is a convex combination of Fletcher and Reeves (abbreviated FR), Polak–Ribiere–Polyak (abbreviated PRP) and Dai and Yuan (abbreviated DY) methods. The new conjugate gradient methods with the Wolfe line search is shown to ensure the descent property of each search direction. Some general convergence results are also established for this method. The numerical experiments are done to test the efficiency of the proposed method, which confirms its promising potentials.
Exponential decay and numerical solution of nonlinear Bresse-Timoshenko system with second sound
(Journal of Thermal Stresses, 2022) Salim Adjemi; Ahmed Berkane; Salah Zitouni; Tahar Bechouat
This paper aims to study the one-dimensional nonlinear Bresse-Timoshenko system with second sound where the heat conduction given by Cattaneo’s law is effective in the second equation. We prove that the system is exponentially stable by using the energy method that requires constructing a suitable Lyapunov functional through exploiting the multipliers method. Furthermore, the result does not depend on any condition on the coefficients of the system. Finally, we validate our theoretical result by performing some numerical approximations based on the standard finite elements method, by using the backward Euler scheme for the temporal and spatial discretization.
An Implicit Iteration Method for Solving Linear Ill-Posed Operator Equations
(Numerical Analysis and Applications, 2023) Bechouat Tahar
In this work, we study a new implicit method to computing the solutions of ill-posed linear operator equations of the first kind under the setting of compact operators. The regularization theory can be used to demonstrate the stability and convergence of this scheme. Furthermore, we obtain convergence results and effective stopping criteria according to Morozov’s discrepancy principle. The numerical performances are conducted to show the validity of our implicit method and demonstrate its applicability to deblurring problems.
A Collocation Method for Fredholm Integral Equations of the First Kind via Iterative Regularization Scheme
(Mathematical Modelling and Analysis, 2023) Bechouat Tahar
To solve the ill-posed integral equations, we use the regularized collocation method. This numerical method is a combination of the Legendre polynomials with non-stationary iterated Tikhonov regularization with fixed parameter. A theoretical justification of the proposed method under the required assumptions is detailed. Finally, numerical experiments demonstrate the efficiency of this method.
Généralisation du résultat sur l’existence des solutions faibles positives des systèmes de réaction diffusion avec une matrice de diffusion 4×4
(2023) Ferdaous Bouhama
In this memory, we study the global existence in time of positive weak solutions for 4 - 4 parabolic reaction diffusion systems with a matrix of diffusion on abounded domain. For which two main properties hold: Positivity solutions and the total mass of the components are Conserved overtime.More overweassume that the nonlinearity of the function of reaction has a critical growth with respect to the gradient. The technique use disbased on semigroupme thods and certain estimations. Our objective in this memory, is to show, with appropriat eassumptions, that the proposed model has a global solution with multiple choices of the terme of reaction non linear. ------------------------------------------------------------------------------------- في هذه المذكرة، ندرس الوجود الكلي بالنسبة للزمن المناسب للحلول الضعيفة الموجبة لأنظمة التفاعل الانتشارللقطع المكافئ مع مصفوفة الانتشار 4×4 في مجال محدود، حيث توجد خاصيتان رئيسيتان: الحلول الموجبة والكتلة الإجمالية يتم حفظ المكونات بمرور الوقت. علاوة على ذلك، نفترض أن اللاخطية لها نمو حرج فيما يتعلق بالتدرج. تعتمد التقنية المستخدمة على طرق نصف زمر وتقديرات معينة. هدفنا أن نظهر، مع افتراضات مناسبة، أن النموذج المقترح له حل كلي مع خيارات متعددة من اللاخطية. --------------------------------------------------------------------------- Dans ce mémoire, nous étudions l’existence globale en temps des solutions faibles positives pour des systèmes de réaction-diffusion paraboliques 4- 4 avec une matrice de diffusion en un domaine borné. Pour les quels deux propriétés principales sont vérifiées : la positivité des solutions et la masse totale des composants se conservent dans le temps .De plus nous supposons que la non linéarité de les fonctions de réaction au ne croissance critique par rapport au gradient. La technique utilisée est basée sur des méthodes de semi-groupes et sur certaines estimations. Notre objectif dans ce mémoire est démontré, avec des hypothèses appropriées, que le modèle proposé possède une solution globale avec de multiple choix de terme de réaction qui non- linéaire.
UNE NOUVELLE MÈTHODE D’HYBRIDATION DU GRADIENT CONJUGUÈ
(2023) Chahrazed Haddad
In this memory, we propose a new hybrid conjugate gradient method for unconstrained optimization methods to solve large problems. This method is a combination of two existing nonlinear conjugate gradient methods (the LS method and the HZ method), which produces a direction of descent at each iteration and converges globally with the strong Wolfe conditions. Numerical experiments are performed to test the efficiency of the new method, which confirms the power of this method. --------------------------------------------------------------------------- طرق التدرج السلمي مهمة جدا من اجل حل المسائل امثلة غير خطية، خاصة المسائل ذات الابعاد الكبيرة. في هذه المذكرة نقترح عائلة جديدة هجينة بين طريقتين و هما HZوLS . اذن انها تحافظ على نتائج البحث الخطي لوولف تحافظ على خصائص النزول و بعض نتائج التقارب المطلق لهذه الطريقة مستقرة و مضمونة. تظهر النتائج العددية ان هذه الطريقة فعالة بالنسبة للمسائل المختبرة. بالاضافة لذلك الطرق الهجينة المرتبطة بهذه العائلة تمت مناقشتها على نطاق واسع. --------------------------------------------------------------------------- On propose dans ce mémoire une nouvelle méthode hybride du gradient conjugué pour les méthodes d'optimisation sans contraintess pour résoudre les problèmes de grande taille. Cette méthode est une combinaison de deux méthodes du gradient conjugué non linéaires déja existantes (la méthode de LS et la méthode HZ), ce qui produit une direction de descente à chaque itération et converge globalement avec les conditions de Wolfe forte. Les expériences numériques sont effectués pour tester l'efficacité de la nouvelle méthode, ce qui confirme la puissance de cette méthode.
Sur l'existence de solutions positives pour une classe de systèmes de réaction diffusion parabolique avec différentes matrices de diffusion
(2023) Chaima AMAMRA
In this work, we prove the existence of continuous positive solutions over time to a nonlinear parabolic system, which couples a non-homogeneous reaction-diffusion equation withdifferent diffusion matrices and initial conditions, using techniques from functional analysis andpotential analysis. ------------------------------------------------------------------------------- في هذا العمل، يمكننا إيجاد حلول إيجابية بالنسبة للزمن لنظام قطع مكافئ، الذي يجمع بين معادلة التفاعل و الانتشار غير المتجانس و مصفوفة انتشار مختلفة مع شروط ابتدائية باستخدام تقنيات التحليل الوظيفي و التحليل المعتمد. ------------------------------------------------------------------------------- Dans ce travail, nous prouvons l’existence de solutions positives continués dans le temps àun système parabolique non linéaire, qui couple une équation de type réaction-diffusionnon homogène avec une matrice de diffusion différente et avec des conditions initiales en utilisant des techniques d’analyse fonctionnelle et d’analyse de potentiel.