arXiv subjects

Ahmed Ratnani

Publications and source records attributed to Ahmed Ratnani.

At least 19 recordsLinked to original sources

arXiv2026

Block Schwarz methods and preconditioning strategies using Generalized locally Toeplitz tools - part I: analysis of the preconditioners and numerical validation

In the current work we present a spectral analysis of the additive and multiplicative Schwarz methods within the framework of domain decomposition techniques, by investigating the spectral properties of these classical Schwarz preconditioning matrix-sequences, with emphasis on their convergence behavior and on the effect of transmission operators. In particular, after a general presentation of various options, we focus on restricted variants of the Schwarz methods aimed at improving parallel efficiency, while preserving their convergence features. In order to rigorously describe and analyze the convergence behavior, we employ the theory of generalized locally Toeplitz (GLT) sequences, which provides a robust framework for studying the asymptotic spectral distribution of the discretized operators arising from Schwarz iterations. By associating each operator sequence with the appropriate GLT symbol, we derive explicit expressions for the GLT symbols of the convergence factors, for both additive and multiplicative Schwarz methods. The GLT-based spectral approach offers a unified and systematic understanding of how the spectrum evolves with mesh refinement and overlap size (in the algebraic case). Our analysis not only deepens the theoretical understanding of classical Schwarz methods, but also establishes a foundation for examining future restricted or hybrid Schwarz variants using GLT symbolic spectral tools. Numerical experiments are presented, while, based on the study in the current work, the analysis of preconditioned matrix-sequences and proposals of new Schwarz preconditioners are given in a twin paper, ideally part II of the present work.

math.NA↗

arXiv2026

Multilevel Isogeometric Projection Stabilization via Quasi-Interpolation for Advection-Dominated Problems

This paper presents a novel multilevel projection-based stabilization method for advection-dominated convection--diffusion problems within the framework of Isogeometric Analysis. The proposed approach extracts and penalizes fine-scale fluctuations using continuous B-spline quasi-interpolants, avoiding both the highly sensitive parameters used in residual-based stabilization methods and the discontinuous auxiliary spaces required by classical Local Projection Stabilization. Stabilization is applied hierarchically across nested levels of the discrete space via explicit mesh-dependent weights. We establish the theoretical foundation of the method by deriving a priori error estimates, supplemented by a discrete inf-sup condition established for the one-dimensional setting with constant advection under a numerically validated stability hypothesis that ensures robust streamline derivative control. Numerical experiments on stringent benchmarks demonstrate the method's ability to significantly reduce spurious oscillations across a variety of regimes, including the limiting cases of pure advection and advection--reaction. Notably, despite being a fully linear formulation, the method achieves significant reduction of undershoots near sharp layers, providing an effective stabilization mechanism without requiring complex nonlinear terms. Furthermore, by utilizing a robust global parameter scaling, the proposed approach significantly alleviates the parameter sensitivity that typically affects residual-based alternatives, reducing the strong dependence on problem-dependent tuning.

math.NA↗

arXiv2025

Higher order multi-dimension reduction methods via Einstein product

This paper explores the extension of dimension reduction (DR) techniques to the multi-dimension case by using the Einstein product. Our focus lies on graph-based methods, encompassing both linear and nonlinear approaches, within both supervised and unsupervised learning paradigms. Additionally, we investigate variants such as repulsion graphs and kernel methods for linear approaches. Furthermore, we present two generalizations for each method, based on single or multiple weights. We demonstrate the straightforward nature of these generalizations and provide theoretical insights. Numerical experiments are conducted, and results are compared with original methods, highlighting the efficiency of our proposed methods, particularly in handling high-dimensional data such as color images.

math.NA↗

arXiv2025

High-dimensional multi-view clustering methods

Multi-view clustering has been widely used in recent years in comparison to single-view clustering, for clear reasons, as it offers more insights into the data, which has brought with it some challenges, such as how to combine these views or features. Most of recent work in this field focuses mainly on tensor representation instead of treating the data as simple matrices. This permits to deal with the high-order correlation between the data which the based matrix approach struggles to capture. Accordingly, we will examine and compare these approaches, particularly in two categories, namely graph-based clustering and subspace-based clustering. We will conduct and report experiments of the main clustering methods over a benchmark datasets.

cs.LG↗

arXiv2025

Trace Ratio vs Ratio Trace Methods for Multidimensional Dimensionality Reduction

We propose a higher-order dimensionality reduction framework based on the Trace Ratio (TR) optimization problem. We establish conditions for existence and uniqueness of solutions and clarify the theoretical connection between the Trace Ratio and its surrogate, the Ratio Trace (RT) formulation. Building on these foundations, we design a Newton-type iterative algorithm that operates directly in the tensor domain via the Einstein product, avoiding data flattening and preserving multi-dimensional structure. This approach extends classical Linear Discriminant Analysis (LDA) to higher-order tensors, offering a natural generalization of trace-based dimensionality reduction from matrices to tensors. Numerical experiments on several benchmark datasets confirm the efficiency and robustness of the proposed methods, showing consistent improvements over existing matrix- and tensor-based techniques.

math.NA↗

arXiv2025

A low-rank non-convex norm method for multiview graph clustering

This study introduces a novel technique for multi-view clustering known as the "Consensus Graph-Based Multi-View Clustering Method Using Low-Rank Non-Convex Norm" (CGMVC-NC). Multi-view clustering is a challenging task in machine learning as it requires the integration of information from multiple data sources or views to cluster data points accurately. The suggested approach makes use of the structural characteristics of multi-view data tensors, introducing a non-convex tensor norm to identify correlations between these views. In contrast to conventional methods, this approach demonstrates superior clustering accuracy across several benchmark datasets. Despite the non-convex nature of the tensor norm used, the proposed method remains amenable to efficient optimization using existing algorithms. The approach provides a valuable tool for multi-view data analysis and has the potential to enhance our understanding of complex systems in various fields. Further research can explore the application of this method to other types of data and extend it to other machine-learning tasks.

cs.LG↗

arXiv2025

Quaternion tensor low rank Quaternion tensor low-rank approximation using a family of non-convex norms

In this paper, we propose a new approaches for low rank approximation of quaternion tensors \cite{chen2019low,zhang1997quaternions,hamilton1866elements}. The first method uses quasi-norms to approximate the tensor by a low-rank tensor using the QT-product \cite{miao2023quaternion}, which generalizes the known L-product to N-mode quaternions. The second method involves Non-Convex norms to approximate the Tucker and TT-rank for the completion problem. We demonstrate that the proposed methods can effectively approximate the tensor compared to the convexifying of the rank, such as the nuclear norm. We provide theoretical results and numerical experiments to show the efficiency of the proposed methods in the Inpainting and Denoising applications.

math.NA↗

arXiv2025

IGA Laplace Eigenfrequencies Distributions and Estimations: Impact of Reparametrization on Eigenfrequency Behavior

This work addresses the Galerkin isogeometric discretization of the one-dimensional Laplace eigenvalue problem subject to homogeneous Dirichlet boundary conditions on a bounded interval. We employ GLT theory to analyze the behavior of the eigenfrequencies when a reparametrization is applied to the computational domain. Under suitable assumptions on the reparametrization transformation, we prove that a structured pattern emerges in the distribution of eigenfrequencies when the problem is reframed through GLT-symbol analysis. Additionally, we establish results that refine and extend those of [3], including a uniform discrete Weyl's law. Furthermore, we derive several eigenfrequency estimates by establishing that the symbol exhibits asymptotically linear behavior near zero.

math.NA↗

arXiv2025

Directional Motional Control via Engineered Conical Intersections in Trapped Rydberg Ions

We demonstrate coherent control of motional dynamics in trapped Rydberg ions engineered to exhibit a conical intersection between adiabatic potential-energy surfaces. Using quantum optimal control, an optimally shaped electric field drives the motional wave packet between prescribed spatial configurations on microsecond timescales. Localized nonadiabatic coupling breaks the symmetry of the dynamics and produces a directed trajectory: after only a few early passages through the conical intersection region, the packet proceeds toward the target with high fidelity. In contrast, in the Born-Oppenheimer limit, where such coupling is absent, the optimized control yields symmetric, multi-cycle oscillations rather than a comparably directed displacement. While both approaches reach the target at the chosen final time, the underlying trajectories are qualitatively different. This work demonstrates that engineered conical intersections can serve as a new control resource for directional motional dynamics, complementing pulse-shaping methods in trapped-ion systems. This directional motion of the ions, enabled by the conical intersection, is expected to have important applications for quantum information processing, where controlled motional states underpin high-fidelity gate operations and scalable architectures.

quant-ph↗

arXiv2025

Parameter-Efficient Fine-Tuning of Multispectral Foundation Models for Hyperspectral Image Classification

Foundation models have achieved great success across diverse domains, including remote sensing (RS), thanks to their versatility and strong generalization abilities. However, most RS foundation models are designed for multispectral data, while hyperspectral imagery (HSI) - with its hundreds of spectral bands - remains less explored. Fine-tuning such models for downstream tasks is also challenging, often demanding considerable memory and storage. In this paper, we propose an efficient framework to fine-tune SpectralGPT, a multispectral foundation model, for hyperspectral image classification (HSIC). We explore several Parameter-Efficient Fine-Tuning (PEFT) methods, including Low-Rank Adaptation (LoRA), Kronecker-based adaptation (KronA), Low-Rank Kronecker (LoKr), and the recent LoRA+, which uses distinct learning rates for low-rank adapters scaled by a factor lambda. Inspired by LoRA+, we introduce KronA+, which applies a similar mechanism to the Kronecker matrices. We evaluate our approach on five datasets from different sensors, showing competitive performance with state-of-the-art HSI models. Our full fine-tuning (FFT) setup for SpectralGPT even outperforms a dedicated hyperspectral foundation model on some datasets while requiring only a quarter of the training epochs. Under the same number of epochs, KronA+ reaches similar performance with far fewer trainable parameters - just 0.056 percent - and adds only approximately 0.2 megabytes of storage, making it the most effective PEFT method tested.

cs.CV↗

arXiv2025

Outlier-free isogeometric discretizations for Laplace eigenvalue problems: closed-form eigenvalue and eigenvector expressions

We derive explicit closed-form expressions for the eigenvalues and eigenvectors of the matrices resulting from isogeometric Galerkin discretizations based on outlier-free spline subspaces for the Laplace operator, under different types of homogeneous boundary conditions on bounded intervals. For optimal spline subspaces and specific reduced spline spaces, represented in terms of B-spline-like bases, we show that the corresponding mass and stiffness matrices exhibit a Toeplitz-minus-Hankel or Toeplitz-plus-Hankel structure. Such matrix structure holds for any degree p and implies that the eigenvalues are an explicitly known sampling of the spectral symbol of the Toeplitz part. Moreover, by employing tensor-product arguments, we extend the closed-form property of the eigenvalues and eigenvectors to a d-dimensional box. As a side result, we have an algebraic confirmation that the considered optimal and reduced spline spaces are indeed outlier-free.

math.NA↗

arXiv2025

Computational investigations of a multi-class traffic flow model: mean-field and microscopic dynamics

We address a multi-class traffic model, for which we computationally assess the ability of mean-field games (MFGs) to yield approximate Nash equilibria for traffic flow games of intractable large finite-players. We introduce ad hoc numerical methodologies, with recourse to techniques such as High-Performance Computing (HPC) and regularization of Loose Generalized Minimal Residual (LGMRES) solvers. The developed apparatus allows us to perform simulations at significantly larger space and time discretization scales. For three generic scenarios of cars and trucks, and three cost functionals, we provide numerous numerical results related to the autonomous vehicles (AVs) traffic dynamics, which corroborate for the multi-class case the effectiveness of the approach emphasized in [22]. We additionally provide several original comparisons of macroscopic Nash mean-field speeds with their microscopic versions, allowing us to computationally validate the so-called $ε-$Nash approximation, with a rate slightly better than theoretically expected.

math.OC↗

arXiv2025

Enhancing multigrid solvers for isogeometric analysis of nonlinear problems using polynomial extrapolation techniques

When used to accelerate the convergence of fixed-point iterative methods, such as the Picard method, which is a kind of nonlinear fixed-point iteration, polynomial extrapolation techniques can be very effective. The numerical solution of nonlinear problems is further investigated in this study. Particularly, using multigrid with isogeometric analysis as a linear solver of the Picard iterative method, which is accelerated by applying vector extrapolation techniques, is how we address the nonlinear eigenvalue Bratu problem and the Monge-Ampère equation. This paper provides quadratic convergence results for polynomial extrapolation methods. Specifically, a new theoretical result on the correlation between the residual norm and the error norm, as well as a new estimation for the generalized residual norm of some extrapolation methods, are given. We perform an investigation between the Picard method, the Picard method accelerated by polynomial extrapolation techniques, and the Anderson accelerated Picard method. Several numerical experiments show that the Picard method accelerated by polynomial extrapolation techniques can solve these nonlinear problems efficiently.

math.NA↗

arXiv2024

Vector Extrapolation Methods Applied To Geometric Multigrid Solvers For Isogeometric Analysis

In the present work, we study how to develop an efficient solver for the fast resolution of large and sparse linear systems that occur while discretizing elliptic partial differential equations using isogeometric analysis. Our new approach combines vector extrapolation methods with geometric multigrid schemes. Using polynomial-type extrapolation methods to speed up the multigrid iterations is our main focus. Several numerical tests are given to demonstrate the efficiency of these polynomial extrapolation methods in improving multigrid solvers in the context of isogeometric analysis.

math.NA↗

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.

math.NA↗

arXiv2024

New preconditioner strategy for solving block four-by-four linear systems: An application to the saddle-point problem from 3D Stokes equation

We have presented a fast method for solving a specific type of block four-by-four saddlepoint problem arising from the finite element discretization of the generalized 3D Stokes problem. We analyze the eigenvalue distribution and the eigenvectors of the preconditioned matrix. Furthermore, we suggested utilizing the preconditioned global conjugate gradient method (PGCG) as a block iterative solver for handling multiple right-hand sides within the sub-system and give some new convergence results. Numerical experiments have shown that our preconditioned iterative approach is very efficient for solving the 3D Stokes problem

math.NA↗

arXiv2024

Trace Ratio Based Manifold Learning with Tensor Data

In this paper, we propose an extension of trace ratio based Manifold learning methods to deal with multidimensional data sets. Based on recent progress on the tensor-tensor product, we present a generalization of the trace ratio criterion by using the properties of the t-product. This will conduct us to introduce some new concepts such as Laplacian tensor and we will study formally the trace ratio problem by discuting the conditions for the exitence of solutions and optimality. Next, we will present a tensor Newton QR decomposition algorithm for solving the trace ratio problem. Manifold learning methods such as Laplacian eigenmaps, linear discriminant analysis and locally linear embedding will be formulated in a tensor representation and optimized by the proposed algorithm. Lastly, we will evaluate the performance of the different studied dimension reduction methods on several synthetic and real world data sets.

math.NA↗

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.

math.NA↗