Department of Mathematics

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Author: search.filters.author.Bechouat TaharStart date: 2020

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Now showing 1 - 6 of 6
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    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.
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    On the numerical solution of weakly singular integral equations of the first kind using a regularized projection method
    (International Journal of Computer Mathematics, 2024-05-28) Bechouat Tahar; Boussetila Nadjib
    This study investigates a numerical method based on the Jacobi–Gauss quadrature for solving Fredholm integral equations of the first kind with a weakly singular kernel by combining the Tikhonov regularization and projection methods. This numerical method reduces the solution of the weakly singular integral equations of the first kind to the solution of a linear system of algebraic equations. The theoretical analysis of the proposed technique is provided. Finally, several tests are presented to show the validity and efficiency of this approach.
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    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.
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    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.
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    TWO MODIFIED CONJUGATE GRADIENT METHODS FOR SOLVING UNCONSTRAINED OPTIMIZATION AND APPLICATION
    (RAIRO Operations Research, 2023) Abd Elhamid Mehamdia; Yacine Chaib; Bechouat Tahar
    Conjugate gradient methods are a popular class of iterative methods for solving linear systems of equations and nonlinear optimization problems as they do not require the storage of any matrices. In order to obtain a theoretically effective and numerically efficient method, two modified conjugate gradient methods (called the MCB1 and MCB2 methods) are proposed. In which the coefficient 𝛽𝑘 in the two proposed methods is inspired by the structure of the conjugate gradient parameters in some existing conjugate gradient methods. Under the strong Wolfe line search, the sufficient descent property and global convergence of the MCB1 method are proved. Moreover, the MCB2 method generates a descent direction independently of any line search and produces good convergence properties when the strong Wolfe line search is employed. Preliminary numerical results show that the MCB1 and MCB2 methods are effective and robust in minimizing some unconstrained optimization problems and each of these modifications outperforms the four famous conjugate gradient methods. Furthermore, the proposed algorithms were extended to solve the problem of mode function.
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    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.