Department of Mathematics

Permanent URI for this communityhttps://dspace.univ-soukahras.dz/handle/123456789/40

Browse

Now showing 1 - 20 of 71

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.
A fast iterative regularizationmethod for ill-posed problems
(Springer, 2024-11-27) Bechouat Tahar
Ill-posed problems manifest in a wide range of scientific and engineering disciplines. The solutions to these problems exhibit a high degree of sensitivity to data perturbations. Regularization methods strive to alleviate the sensitivity exhibited by these solutions. This paper presents a fast iterative scheme for addressing linear ill-posed problems, similar to nonstationary iterated Tikhonov regularization. Both the a-priori and a-posteriori choice rules for regularization parameters are provided, and both rules yield error estimates that are order optimal. In comparison to the nonstationary iterated Tikhonov method, the newly introduced method significantly reduces the required number of iterations to achieve convergence based on an appropriate stopping criterion. The numerical computations provide compelling evidence regarding the efficacy of our new iterative regularization method. Furthermore, the versatility of this method extends to image restorations.
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.
A new hybrid CG method as convex combination
(Mathematical Foundations of Computing, 2023) Amina Hallal; Mohammed Belloufi; Badreddine Sellami
Conjugate gradient methods are among the most efficient methods for solving optimization models. In this paper, a newly proposed conjugate gradient method is proposed for solving optimization problems as a convex combination of the Harger-Zhan and Dai-Yaun nonlinear conjugate gradient methods, which is capable of producing the sufficient descent condition with global convergence properties under the strong Wolfe conditions. The numerical results demonstrate the efficiency of the proposed method with some benchmark problems.
A new hybrid conjugate gradient algorithm based on the Newton direction to solve unconstrained optimization problems
(Journal of Applied Mathematics and Computing, 2023-03-25) Naima Hamel, Noureddine Benrabia, Mourad Ghiat, Hamza Guebbai
In this paper, we propose a new hybrid conjugate gradient method to solve unconstrained optimization problems. This new method is defined as a convex combination of DY and DL conjugate gradient methods. The special feature is that our search direction respects Newton’s direction, but without the need to store or calculate the second derivative (the Hessian matrix), due to the use of the secant equation that allows us to remove the troublesome part required by the Newton method. Our search direction not only satisfies the descent property, but also the sufficient descent condition through the use of the strong Wolfe line search, the global convergence is proved. The numerical comparison shows the efficiency of the new algorithm, as it outperforms both the DY and DL algorithms.
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.
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.
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
A novel conformable fractional approach to the Brusselator system with numerical simulation
(2024-03-12) Mohamed Lamine Merikhi · Hamza Guebbai · Noureddine Benrabia · Mohamed Moumen Bekkouche
In this study, we delve into a comprehensive analysis of the Brusselator system, combining both analytical and numerical approaches. In summary, our initial approach involves revisiting the classic Brusselator system using a conformable fractional derivative-based approach. Starting from this innovative reformulation, we obtain a nonlinear Volterra-type equation. This transformation allows us to simultaneously demonstrate the existence and uniqueness of the solution, while providing us with the necessary tools to develop an efficient numerical approximation method to solve the problem. Subsequently, we present a numerical simulation based on the Nyström method.
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.
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.
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.
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.
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.
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.
Calcul du volume et surface des sous variétés cylindriques par la méthode de rotation
(2024) Dorsaf Zeraoulia
In thisWorkswediscussed how to calculate the area and volume of bodies revolutionwhichappear in the spiral form, using the rotation technique around a relative axis in an orthonormalcoordinate system. wecut.apiece of the curve of a continuous, defined and boundedfunction, thenwerotateitaround the axis of ,abscissae, generating for us a body of rotation in space, has a lateral surface and twolower and upper bases whichrepresent discs. The dissertation iscomposed of threechapters The first chapteris a reminder and fundamental concepts on rotation in the plane and in space and itselements. The second chapteris the heart of ourwork, whichconsists of designing an expression or a law for the area and volume of arotating body. The third chapterisapresentation of somewell-known spiral-shaped bodies and an explanation of how to calculate their volume and surface area using the method of revolution. ----------------------------------------------------------------------------------- في عملنا هذا تطرقنا الى كيفية حساب مساحة وحجم الاجسام الدورانية التي تظهر على شكل لولبي وذلك باستعمال تقنية الدوران حول محور منسوب في معلم متعامد ومتجانس وتتم على النحو التالي نأخذ جزء من المنحنى لدالةمستمرة ومعرفةومحدودة ثم نقوم بتدويرها حول محور الفواصل فتولد لنا جسما دورانيا في الفضاء له مساحة جانبية وقاعدتين سفلية وعلوية تمثلان قوسين. المذكرة تتكون من ثلاثة فصول: • الفصل الاول وهو تذكير ومفاهيم عامة حول الدوران في مستوي والفضاء وعناصره • الفصل الثاني وهو لب المذكرة وهو استنباط عبارة او قانون لمساحة وحجم الجسم الدوراني • الفصل الثالث وهو عرض بعض الاجسام اللولبية المعروفة والمشهورة وتبيين كيفية حساب حجمها ومساحتها باستعمال طريقه الدوران ----------------------------------------------------------------------------------- Dans ce travail, nous avons discuté de la manière de calculer l'aire et le volume des corps en révolution qui apparaissent sous la forme spiroïdal, en utilisant la technique de rotation autour d'un axe relatif dans un repère orthonormée. nouscoupons. Un morceau de la courbe d'une fonction continue, définie et bornée, puis nous la faisons tourner autour de l'axe des abscisses , générant pour nous un corps de rotation dans l'espace, possédé une surface latérale et deux bases inférieure et supérieure qui représentent des disques. Le mémoire est composée de trois chapitres Le premier chapitre est un rappel et concepts fondamentaux sur la rotation dans le plan et dans l'espace et ses éléments. Le deuxième chapitre est le cœur de notre travail, qui consiste à concevoir une expression ou une loi pour l'aire et le volume d'un corps en rotation. Le troisième chapitre est une présentation de quelques corps en forme spirale bien connus et une explication de la façon de calculer leur volume et leur surface à l'aide de la méthode de révolution.
Comportement asymptotique desolutions pour les systèmes de réaction-diffusion non linéaires à matrice de diffusion de Toeplitz
(2024) Abir Soualmia
The objective of this memory is the study of a nonlinear reaction-diffusion system with toeplitz diffusion matrix. Note that reaction-diffusion systems are system of partial differential equations of parabolic type. In this work, we have show the asymptotic behavior of a positive solution of a nonlinear reaction- diffusion system using the invariant region method and a priori estimates which made it possible to show the asymptotic behavior of the solution. ----------------------------------------------------------------------------------- الهدف من هذه المذكرة هو دراسة نظام التفاعل-الانتاار يرر طي ااستطدام مصفوفة انتاار توالرتز. ان أنظمة التفاعل-الانتاار ه أنظمة معادلات تفاضلرة جزئرة من النوع المكافئ . ف هذا العمل نارهن السلوك التقارا لحل موجب لنظام معادلات التفاعل-الانتاار يرر الطيرة ااستطدام يررقة المنيقة الثااتة وتقدررات معرنة الت تمكننا من اظهار السلوك التقارا للحل . ---------------------------------------------------------------------------------- L'objectif de ce mémoire est l'étude d'un système de réaction-diffusion non linéaire à matrice de diffusion de toeplitz. Notons que les systèmes de réaction-diffusion sont des systèmes d'équations aux dérivées par- tielles de type parabolique. Dans ce travail, on a montré le comportement asymptotique d'une solution positive d'un système d'équations de réaction-diffusion non linéaires en utilisant la méthode de région invariante et des estimations a priori qui ont permis de montrer le comportement asymptotique de la solution.
Convergence des méthodes du gradient conjugué pour minimisation sans contrainte
(2024-03-17) Ben hanachi Sabrina
Abstract The aim of this thesis is to present a new and fundamentally different conjugate gradient method which, when applied to solve unconstrained optimization problems, gives good convergence results and ensures the sufficient descent condition. To achieve this, we will use a well-known technique, the hybrid method based on a convex combination of three classical conjugate gradient methods. Furthermore, numerical experiments were performed to test the effectiveness of the proposed method, which confirmed its promising potential. ---------- ملخص الهدف من هذه الأطروحة هو تقديم طريقة تدرج مترافق جديدة ومختلفة اختلافا جذريا، والتي عند تطبيقها لحل مشكلات التحسين غير المقيدة، تعطي نتائج تقارب جيدة وتضمن حالة الانحدار الكافية. لتحقيق ذلك، سوف نستخدم تقنية معروفة جيدا، وهي الطريقة الهجينة التي تعتمد على مزيج محدب من ثلاث طرق التدرج المترافق. كما يتم إجراء تجارب عددية لاختبار فاعلية الطريقة المقترحة مما يؤكد إمكاناتها الواعدة ---------------- Résumé L’objectif de cette thèse est de présenter une nouvelle méthode de gradient conjugué fondamentalement différente qui, lorsqu’elle est appliquée à la résolution de problèmes d’optimisation sans contrainte, donne de bons résultats de convergence et assure la condition de descente suffisante. Pour ce faire, nous utiliserons une technique bien connue, la méthode hybride basée sur la combinaison convexe de trois méthodes standard de gradient conjugué. Des expériences numériques sont également réalisées pour tester l’efficacité de la méthode proposée, ce qui confirme son potentiel prometteur. -------------------- Mots clés : Optimisation sans contraintes, Gradient conjuguée, Algorithme, Convergence globale, Recherche linéaire.
DEVELOPING A NEW CONJUGATE GRADIENT ALGORITHM WITH THE BENEFIT OF SOME DESIRABLE PROPERTIES OF THE NEWTON ALGORITHM FOR UNCONSTRAINED OPTIMIZATION
(Journal of Applied Analysis & Computation, 2024-02-15) Naima Hamel, Noureddine Benrabia, Mourad Ghiat, Hamza Guebbai
The conjugate gradient method and the Newton method are both numerical optimization techniques. In this paper, we aim to combine some desirable characteristics of these two methods while avoiding their drawbacks, more specifically, we aim to develop a new optimization algorithm that preserves some essential features of the conjugate gradient algorithm, including the simplicity, the low memory requirements, the ability to solve large scale problems and the convergence to the solution regardless of the starting vector (global convergence). At the same time, this new algorithm approches the quadratic convergence behavior of the Newton method in the numerical sense while avoiding the computational cost of evaluating the Hessian matrix directly and the sensitivity of the selected starting vector. To do this, we propose a new hybrid conjugate gradient method by linking (CD) and (WYL) methods in a convex blend, the hybridization paramater is computed so that the new search direction accords with the Newton direction, but avoids the computational cost of evaluating the Hessian matrix directly by using the secant equation. This makes the proposed algorithm useful for solving large scale optimization problems. The sufficient descent condition is verified, also the global convergence is proved under a strong Wolfe Powel line search. The numerical tests show that, the proposed algorithm provides the quadratic convergence behavior and confirm its efficiency as it outperformed both the (WYL) and (CD) algorithms.
Estimation paramétrique d’une distribution de probabilité à l’aide de quelque méthodes d’optimisation
(2024) Hanane Bensoltane
Data analysis is algebraic mathematical logic (sets, groups, inclusion, exclusion), it has a theoretical component as well as an applied component, the theoretical component is based on probability theory and forms with the latter, the analysis of random phenomena. Applied statistics is used in all areas of human activity: engineering, management, economics, biology, computer science, physics (fundamentals of quantum physics, for example). The Maximum Likelihood estimator is our objective in our study for this we used optimization methods the gradient method and the conjugate gradient method where the latter we used Fletcher-Reeves, Polak-Ribière-Polyak , Dai-Yuan , to have our estimators of the parameters of the Pareto probability distributions for some distribution (Béta, weibull , lindly....). --------------------------------------------------------------------------------- تحليل البيانات هو منطق رياضي جبري (مجموعات، جزء من مجموعة، تضمين، استبعاد)، له مكون نظري بالإضافة إلى مكون تطبيقي، المكون النظري يعتمد على نظرية الاحتمالات والأشكال مع الأخير، تحليل الظواهر العشوائية. يتم استخدام الإحصاء التطبيقي في جميع مجالات النشاط البشري: الهندسة، والإدارة، والاقتصاد، والأحياء، وعلوم الكمبيوتر، والفيزياء (أساسيات فيزياء الكم، على سبيل المثال). مقدر الاحتمالية القصوى هو هدفنا في دراستنا لذلك استخدمنا طرق التحسين وهي طريقة التدرج وطريقة التدرج المترافق حيث استخدمنا الأخيرة فليتشرز-ريفرز، بولاك-ريبير-بولياك، داي-يوان ، للحصول على مقدراتنا لمعلمات توزيعات باريتو الاحتمالية لبعض التوزيعات (بيتا، وليبل، لندلي.....). --------------------------------------------------------------------------------- L'analyse de données est la logique mathématiques algébrique (ensembles, groupes, inclusion, exclusion), elle possède une composante théorique ainsi qu'une composante appliquée, la composante théorique s'appuie sur la théorie des probabilités et forme avec cette dernière, l'analyse de phénomènes aléatoires. La statistique appliquée est utilisée dans tous les domaines des activités humaine : ingénierie, management, économie, biologie, informatique, la physique (fondamentaux de la physique quantique, par exemple). L'estimateur du Maximum de vraisemblance est notre objectif dans notre étude, pour cela on a fait appelle a des méthodes d'optimisation, la méthode de gradient et la méthode de gradient conjugué ou dans cette dernière, on a utilisée Fletcher-Reeves, Polak-Ribière-Polyak , Dai-Yuan, pour avoir notre estimateur des paramètres des distribution de probabilités de paréto pour quelque distribution (Béta,weibull, lindly....).