arXiv subjects

Abdul Halim

Publications and source records attributed to Abdul Halim.

10 recordsLinked to original sources

arXiv2024

A $\ell_2-\ell_p$ regulariser based model for Poisson noise removal using augmented Lagrangian method

In this article, we propose a variational PDE model using $\ell_2-\ell_p$ regulariser for removing Poisson noise in presence of blur. The proposed minimization problem is solved using augmented Lagrangian method. The convergence of the sequence of minimizers have been carried out. Numerical simulations on some standard test images have been shown. The numerical results are compared with that of a few models existed in literature in terms of image quality metric such as SSIM, PSNR and SNR.

math.NA↗

arXiv2023

A data-driven method for parametric PDE Eigenvalue Problems using Gaussian Process with different covariance functions

We use a Gaussian Process Regression (GPR) strategy that was recently developed [3,16,17] to analyze different types of curves that are commonly encountered in parametric eigenvalue problems. We employ an offline-online decomposition method. In the offline phase, we generate the basis of the reduced space by applying the proper orthogonal decomposition (POD) method on a collection of pre-computed, full-order snapshots at a chosen set of parameters. Then, we generate our GPR model using four different Mat\'{e}rn covariance functions. In the online phase, we use this model to predict both eigenvalues and eigenvectors at new parameters. We then illustrate how the choice of each covariance function influences the performance of GPR. Furthermore, we discuss the connection between Gaussian Process Regression and spline methods and compare the performance of the GPR method against linear and cubic spline methods. We show that GPR outperforms other methods for functions with a certain regularity.

math.NA↗

arXiv2023

On the effect of different samplings to the solution of parametric PDE Eigenvalue Problems

In this article we apply reduced order techniques for the approximation of parametric eigenvalue problems. The effect of the choice of sampling points is investigated. Here we use the standard proper orthogonal decomposition technique to obtain the basis of the reduced space and Galerking orthogonal technique is used to get the reduced problem. We present some numerical results and observe that the use of sparse sampling is a good idea for sampling if the dimension of parameter space is high.

math.NA↗

arXiv2023

Reduced basis approximation of parametric eigenvalue problems in presence of clusters and intersections

In this paper we discuss reduced order models for the approximation of parametric eigenvalue problems. In particular, we are interested in the presence of intersections or clusters of eigenvalues. The singularities originating by these phenomena make it hard a straightforward generalization of well known strategies normally used for standards PDEs. We investigate how the known results extend (or not) to higher order frequencies.

math.NA↗

arXiv2023

Data-driven reduced order modeling for parametric PDE eigenvalue problems using Gaussian process regression

In this article, we propose a data-driven reduced basis (RB) method for the approximation of parametric eigenvalue problems. The method is based on the offline and online paradigms. In the offline stage, we generate snapshots and construct the basis of the reduced space, using a POD approach. Gaussian process regressions (GPR) are used for approximating the eigenvalues and projection coefficients of the eigenvectors in the reduced space. All the GPR corresponding to the eigenvalues and projection coefficients are trained in the offline stage, using the data generated in the offline stage. The output corresponding to new parameters can be obtained in the online stage using the trained GPR. The proposed algorithm is used to solve affine and non-affine parameter-dependent eigenvalue problems. The numerical results demonstrate the robustness of the proposed non-intrusive method.

math.NA↗

arXiv2022

A hybrid Haar wavelet collocation method for nonlocal hyperbolic partial differential equations

In this paper, we propose a hybrid collocation method based on finite difference and Haar wavelets to solve nonlocal hyperbolic partial differential equations. Developing an efficient and accurate numerical method to solve such problem is a difficult task due to the presence of nonlocal boundary condition. The speciality of the proposed method is to handle integral boundary condition efficiently using the given data. Due to various attractive properties of Haar wavelets such as closed form expression, compact support and orthonormality, Haar wavelets are efficiently used for spatial discretization and second order finite difference is used for temporal discretization. Stability and error estimates have been investigated in order to ensure the convergence of the method. Finally, numerical results are compared with few existing results and it is shown that numerical results obtained by the proposed method is better than few existing results.

math.NA↗

arXiv2022

A Fractional Image Inpainting Model Using a Variant of Mumford-Shah Model

In this paper, we propose a fourth order PDE model for image inpainting based on a variant of the famous Mumford-Shah (MS) image segmentation model. Convexity splitting is used to discrtised the time and we replace the Laplacian by its fractional counterpart in the time discretised scheme. Fourier spectral method is used for space discretization. Consistency, stability and convergence of the time discretised model has been carried out. The model is tested on some standard test images and compared them with the result of some models existed in the literature.

math.NA↗

arXiv2022

On the matching of eigensolutions to parametric partial differential equations

In this paper a novel numerical approximation of parametric eigenvalue problems is presented. We motivate our study with the analysis of a POD reduced order model for a simple one dimensional example. In particular, we introduce a new algorithm capable to track the matching of eigenvalues when the parameters vary.

math.NA↗

arXiv2022

A reduced order model for the finite element approximation of eigenvalue problems

In this paper we consider a reduced order method for the approximation of the eigensolutions of the Laplace problem with Dirichlet boundary condition. We use a time continuation technique that consists in the introduction of a fictitious time parameter. We use a POD approach and we present some theoretical results showing how to choose the optimal dimension of the POD basis. The results of our computations, related to the first eigenvalue, confirm the optimal behavior of our approximate solution.

math.NA↗

arXiv2021

The Rohingyas of Rakhine State: Social Evolution and History in the Light of Ethnic Nationalism

Recent event of ousting Rohingyas from Rakhine State by the Tatmadaw provoked worldwide public-and-academic interest in history and social evolution of the Rohingyas, and this is to what the article is devoted. As the existing literature presents a debate over Who are the Rohingyas?, and How legitimate is their claim over Rakhine State?, the paper reinvestigates the issues using a qualitative research method. Compiling a detailed history, the paper finds that Rohingya community developed through historically complicated processes marked by invasions and counter-invasions. The paper argues many people entered Bengal from Arakan before British brought people into Rakhine state. The Rohingyas believe Rakhine State is their ancestral homeland and they developed a sense of Ethnic Nationalism. Their right over Rakhine State is as significant as other groups. The paper concludes that the UN must pursue solution to the crisis and the government should accept the Rohingyas as it did the land or territory.

econ.GN↗