Explanipedia

Deep learning, transformers and graph neural networks: a linear algebra perspective Open

Abdelkader Baggag, Yousef Saad · 2025

In an age where Artificial Intelligence (AI) is being integrated into nearly every domain of science and engineering, it has become essential for experts in Numerical Linear Algebra to explore the foundational elements of deep learning and…

Acceleration methods for fixed point iterations Open

Yousef Saad · 2025

A pervasive approach in scientific computing is to express the solution to a given problem as the limit of a sequence of vectors or other mathematical objects. In many situations these sequences are generated by slowly converging iterative…

Acceleration methods for fixed-point iterations Open

Yousef Saad · 2025

A pervasive approach in scientific computing is to express the solution to a given problem as the limit of a sequence of vectors or other mathematical objects. In many situations these sequences are generated by slowly converging iterative…

Coffee waste utilization as an eco-friendly disposal for pollutants removal from wastewater Open

Mohamed Abdallah, Mohamed Mohsen, Yousef Khaled, Yousef Saad, Mona A. Naghmash · 2025

Mixed Precision Orthogonalization-Free Projection Methods for Eigenvalue and Singular Value Problems Open

Tianshi Xu, Ziqi Zhang, Jie Chen, Yousef Saad, Yuanzhe Xi · 2025

Mixed-precision arithmetic offers significant computational advantages for large-scale matrix computation tasks, yet preserving accuracy and stability in eigenvalue problems and the singular value decomposition (SVD) remains challenging. T…

Combinative model compression approach for enhancing 1D CNN efficiency for EIT-based Hand Gesture Recognition on IoT edge devices Open

Mahdi Mnif, Salwa Sahnoun, Yousef Saad, Ahmed Fakhfakh, Olfa Kanoun · 2024

Joint Approximate Partial Diagonalization of Large Matrices Open

Abd‐Krim Seghouane, Yousef Saad · 2024

Given a set of $p$ symmetric (real) matrices, the Orthogonal Joint Diagonalization (OJD) problem consists of finding an orthonormal basis in which the representation of each of these $p$ matrices is as close as possible to a diagonal matri…

Straggler-tolerant stationary methods for linear systems Open

Vassilis Kalantzis, Yuanzhe Xi, Lior Horesh, Yousef Saad · 2024

In this paper, we consider the iterative solution of linear algebraic equations under the condition that matrix-vector products with the coefficient matrix are computed only partially. At the same time, non-computed entries are set to zero…

Anderson Acceleration with Truncated Gram-Schmidt Open

Ziyuan Tang, Tianshi Xu, Huan He, Yousef Saad, Yuanzhe Xi · 2024

Anderson Acceleration (AA) is a popular algorithm designed to enhance the convergence of fixed-point iterations. In this paper, we introduce a variant of AA based on a Truncated Gram-Schmidt process (AATGS) which has a few advantages over …

Gradient-type subspace iteration methods for the symmetric eigenvalue problem Open

Foivos Alimisis, Yousef Saad, Bart Vandereycken · 2023

This paper explores variants of the subspace iteration algorithm for computing approximate invariant subspaces. The standard subspace iteration approach is revisited and new variants that exploit gradient-type techniques combined with a Gr…

NLTGCR: A class of Nonlinear Acceleration Procedures based on Conjugate Residuals Open

Huan He, Ziyuan Tang, Shifan Zhao, Yousef Saad, Yuanzhe Xi · 2023

This paper develops a new class of nonlinear acceleration algorithms based on extending conjugate residual-type procedures from linear to nonlinear equations. The main algorithm has strong similarities with Anderson acceleration as well as…

On the tubular eigenvalues of third-order tensors Open

Fatemeh P. A. Beik, Yousef Saad · 2023

This paper introduces the notion of tubular eigenvalues of third-order tensors with respect to T-products of tensors and analyzes their properties. A focus of the paper is to discuss relations between tubular eigenvalues and two alternativ…

An Efficient Nonlinear Acceleration method that Exploits Symmetry of the Hessian Open

Huan He, Shifan Zhao, Ziyuan Tang, Joyce C. Ho, Yousef Saad , et al. · 2022

Nonlinear acceleration methods are powerful techniques to speed up fixed-point iterations. However, many acceleration methods require storing a large number of previous iterates and this can become impractical if computational resources ar…

parGeMSLR: A parallel multilevel Schur complement low-rank preconditioning and solution package for general sparse matrices Open

Tianshi Xu, Vassilis Kalantzis, Ruipeng Li, Yuanzhe Xi, Geoffrey Dillon , et al. · 2022

Table of Contents Open

Manish Parashar, Sven Hammarling, Nicholas J. Higham, Hatem Ltaief, Marc G. Genton , et al. · 2022

parGeMSLR: A Parallel Multilevel Schur Complement Low-Rank Preconditioning and Solution Package for General Sparse Matrices Open

Tianshi Xu, Vassilis Kalantzis, Ruipeng Li, Yuanzhe Xi, Geoffrey Dillon , et al. · 2022

This paper discusses parGeMSLR, a C++/MPI software library for the solution of sparse systems of linear algebraic equations via preconditioned Krylov subspace methods in distributed-memory computing environments. The preconditioner impleme…

A Non-perturbative Approach to Computing Seismic Normal Modes in Rotating Planets Open

Jia Shi, Ruipeng Li, Yuanzhe Xi, Yousef Saad, Maarten V. de Hoop · 2022

Graph coarsening: from scientific computing to machine learning Open

Jie Chen, Yousef Saad, Zechen Zhang · 2022

The general method of graph coarsening or graph reduction has been a remarkably useful and ubiquitous tool in scientific computing and it is now just starting to have a similar impact in machine learning. The goal of this paper is to take …

GDA-AM: On the effectiveness of solving minimax optimization via Anderson Acceleration Open

Huan He, Shifan Zhao, Yuanzhe Xi, Joyce C. Ho, Yousef Saad · 2021

Many modern machine learning algorithms such as generative adversarial networks (GANs) and adversarial training can be formulated as minimax optimization. Gradient descent ascent (GDA) is the most commonly used algorithm due to its simplic…

Solve Minimax Optimization by Anderson Acceleration Open

Huan He, Shifan Zhao, Yuanzhe Xi, Joyce C. Ho, Yousef Saad · 2021

Many modern machine learning algorithms such as generative adversarial networks (GANs) and adversarial training can be formulated as minimax optimization. Gradient descent ascent (GDA) is the most commonly used algorithm due to its simplic…

Table of Contents Open

William H. Frishman, Joseph S. Alpert, Jean Terrier, Ed Eric Gaspar, Pierre Fontana , et al. · 2021

Shanks and Anderson-type acceleration techniques for systems of nonlinear equations Open

Claude Brezinski, Stefano Cipolla, Michela Redivo‐Zaglia, Yousef Saad · 2021

This paper examines a number of extrapolation and acceleration methods and introduces a few modifications of the standard Shanks transformation that deal with general sequences. One of the goals of the paper is to lay out a general framewo…

Graph coarsening: From scientific computing to machine learning Open

Jie Chen, Yousef Saad, Zechen Zhang · 2021

The general method of graph coarsening or graph reduction has been a remarkably useful and ubiquitous tool in scientific computing and it is now just starting to have a similar impact in machine learning. The goal of this paper is to take …

Planetary Normal Mode Computation: Parallel Algorithms, Performance, and Reproducibility Open

Jia Shi, Ruipeng Li, Yuanzhe Xi, Yousef Saad, Maarten V. de Hoop · 2021

This report is an extension of work entitled “Computing planetary interior normal modes with a highly parallel polynomial filtering eigensolver.” by Shi et al., [1] originally presented at the SC18 conference. A highly parallel polynomial …

Spectrum-Adapted Polynomial Approximation for Matrix Functions with Applications in Graph Signal Processing Open

Tiffany Fan, David I Shuman, Shashanka Ubaru, Yousef Saad · 2020

We propose and investigate two new methods to approximate f(A)b for large, sparse, Hermitian matrices A. Computations of this form play an important role in numerous signal processing and machine learning tasks. The main idea behind both m…

Shanks and Anderson-type acceleration techniques for systems of\n nonlinear equations Open

Claude Brezinski, Stefano Cipolla, Michela Redivo‐Zaglia, Yousef Saad · 2020

This paper examines a number of extrapolation and acceleration methods, and\nintroduces a few modifications of the standard Shanks transformation that deal\nwith general sequences. One of the goals of the paper is to lay out a general\nfra…

Shanks and Anderson-type acceleration techniques for systems of nonlinear equations Open

Claude Brezinski, Stefano Cipolla, Michela Redivo‐Zaglia, Yousef Saad · 2020

This paper examines a number of extrapolation and acceleration methods, and introduces a few modifications of the standard Shanks transformation that deal with general sequences. One of the goals of the paper is to lay out a general framew…

A power Schur complement Low-Rank correction preconditioner for general sparse linear systems Open

Qingqing Zheng, Yuanzhe Xi, Yousef Saad · 2020

An effective power based parallel preconditioner is proposed for general large sparse linear systems. The preconditioner combines a power series expansion method with some low-rank correction techniques, where the Sherman-Morrison-Woodbury…

Iterative methods for linear systems of equations: A brief historical journey Open

Yousef Saad · 2020

This paper presents a brief historical survey of iterative methods for solving linear systems of equations. The journey begins with Gauss who developed the first known method that can be termed iterative. The early 20th century saw good pr…

The computation of seismic normal modes with rotation as a quadratic eigenvalue problem Open

Jia Shi, Ruipeng Li, Yuanzhe Xi, Yousef Saad, Maarten V. de Hoop · 2019

A new approach is presented to compute the seismic normal modes of a fully heterogeneous, rotating planet. Special care is taken to separate out the essential spectrum in the presence of a fluid outer core. The relevant elastic-gravitation…

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