arXiv subjects

Anas El Hachimi

Publications and source records attributed to Anas El Hachimi.

6 recordsLinked to original sources

arXiv2026

Stable full-field simulation of a multiscale elliptic equation by means of Quantized Tensor Trains

In this article, we design an original solver based on Quantized Tensor Trains (QTT) for linear elliptic equations with heterogeneous coefficient field, that allows for extremely fine meshes. It can achieve full-field simulations in dimensions $d=2$ and $d=3$ with a number of Degrees of Freedom (DoFs) up to $20$ orders of magnitude beyond the classical solvers, recovering accurately the solution as well as its gradient in the $\LL^2$ norm. For treating such an enormous amount of data, the solver crucially relies on the exponential compression properties of QTTs. This significantly improves upon the existing literature. The main ingredient of the proposed solver consists in the introduction of a penalization term involving the Helmholtz--Leray projector in the equation governing the gradient unknown. For practical reasons related to the expression of the Helmholtz--Leray projector, the penalized equation is solved in Fourier space. The primal solution is then obtained from the gradient via the Green operator. A core property of the solver is that it is unconditionally stable with respect to the mesh size. Based on numerical evidence supported by mathematical analysis, we show that reliable gradients and solutions can be obtained, and guaranteed by the proposed a posteriori error estimator. As an illustration, we successfully solve an elliptic equation in a microstructured material with up to $10^{37}$ virtual degrees of freedom in dimension $d=3$.

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arXiv2025

Low-Rank Regularized Convex-Non-Convex Problems for Image Segmentation or Completion

This work proposes a novel convex-non-convex formulation of the image segmentation and the image completion problems. The proposed approach is based on the minimization of a functional involving two distinct regularization terms: one promotes low-rank structure in the solution, while the other one enforces smoothness. To solve the resulting optimization problem, we employ the alternating direction method of multipliers (ADMM). A detailed convergence analysis of the algorithm is provided, and the performance of the methods is demonstrated through a series of numerical experiments.

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arXiv2024

Non-negative Einstein tensor factorization for unmixing hyperspectral images

In this manuscript, we introduce a tensor-based approach to Non-Negative Tensor Factorization (NTF). The method entails tensor dimension reduction through the utilization of the Einstein product. To maintain the regularity and sparsity of the data, certain constraints are imposed. Additionally, we present an optimization algorithm in the form of a tensor multiplicative updates method, which relies on the Einstein product. To guarantee a minimum number of iterations for the convergence of the proposed algorithm, we employ the Reduced Rank Extrapolation (RRE) and the Topological Extrapolation Transformation Algorithm (TEA). The efficacy of the proposed model is demonstrated through tests conducted on Hyperspectral Images (HI) for denoising, as well as for Hyperspectral Image Linear Unmixing. Numerical experiments are provided to substantiate the effectiveness of the proposed model for both synthetic and real data.

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arXiv2023

Tensor Golub Kahan based on Einstein product

The Singular Value Decomposition (SVD) of matrices is a widely used tool in scientific computing. In many applications of machine learning, data analysis, signal and image processing, the large datasets are structured into tensors, for which generalizations of SVD have already been introduced, for various types of tensor-tensor products. In this article, we present innovative methods for approximating this generalization of SVD to tensors in the framework of the Einstein tensor product. These singular elements are called singular values and singular tensors, respectively. The proposed method uses the tensor Lanczos bidiagonalization applied to the Einstein product. In most applications, as in the matrix case, the extremal singular values are of special interest. To enhance the approximation of the largest or the smallest singular triplets (singular values and left and right singular tensors), a restarted method based on Ritz augmentation is proposed. Numerical results are proposed to illustrate the effectiveness of the presented method. In addition, applications to video compression and facial recognition are presented.

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arXiv2023

Spectral computation with third-order tensors using the t-product

The tensor t-product, introduced by Kilmer and Martin [26], is a powerful tool for the analysis of and computation with third-order tensors. This paper introduces eigentubes and eigenslices of third-order tensors under the t-product. The eigentubes and eigenslices are analogues of eigenvalues and eigenvectors for matrices. Properties of eigentubes and eigenslices are investigated and numerical methods for their computation are described. The methods include the tensor power method, tensor subspace iteration, and the tensor QR algorithm. Computed examples illustrate the performance of these methods.

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arXiv2023

A tensor bidiagonalization method for higher-order singular value decomposition with applications

The need to know a few singular triplets associated with the largest singular values of third-order tensors arises in data compression and extraction. This paper describes a new method for their computation using the t-product. Methods for determining a couple of singular triplets associated with the smallest singular values also are presented. The proposed methods generalize available restarted Lanczos bidiagonalization methods for computing a few of the largest or smallest singular triplets of a matrix. The methods of this paper use Ritz and harmonic Ritz lateral slices to determine accurate approximations of the largest and smallest singular triplets, respectively. Computed examples show applications to data compression and face recognition.

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