arXiv subjects

Yousef Saad

Publications and source records attributed to Yousef Saad.

At least 19 recordsLinked to original sources

arXiv2026

Graph Neural Multilevel Preconditioners for Iterative Solvers

Solving large, sparse linear systems is a core task in scientific computing, and efficient iterative solvers rely critically on effective and robust preconditioning. While classical methods such as algebraic multigrid (AMG) are highly scalable, their robustness can degrade on indefinite or nonsymmetric systems where heuristics originally developed for elliptic PDEs are less reliable. Recently, Graph Neural Networks (GNNs) have emerged as data-driven preconditioners; yet, the practical impact of imposing an AMG-style hierarchy remains underexplored for general sparse matrices. In this work, we propose a Graph Neural Multilevel Preconditioner (GMP) that adopts an AMG hierarchy as a structural prior and learns smoothing, restriction, and interpolation operators in a unified framework. Our method targets general sparse systems and is instantiated as a drop-in preconditioner for standard Krylov solvers. On a benchmark of over 800 sparse matrices, we compare against classical AMG, single-level ILUT, and state-of-the-art GNN preconditioners, and characterize the regimes where multilevel graph neural preconditioning improves convergence or, conversely, introduces overhead relative to strong single-level baselines. These results highlight both the promise and the limitations of enforcing AMG-style multilevel structure in learned preconditioners for large-scale scientific simulations.

math.NA↗

arXiv2026

Hierarchical Muon: Tiled Newton-Schulz Updates for Efficient Muon Optimization

Muon-type optimizers construct update directions for dense neural-network weights by applying a finite Newton-Schulz map to momentum-gradient matrices. For an $H \times W$ matrix, with $r=\min\{H,W\}$ and $s=\max\{H,W\}$, $K$ steps of the full-matrix Newton-Schulz update require $O(r^2 s K)$ work and couple all rows and columns through repeated Gram matrix products. We introduce Hierarchical Muon (HiMuon), a tiled Newton-Schulz scheme for Muon-type optimization. HiMuon partitions each momentum-gradient matrix into $T \times T$ tiles, applies the same finite Newton-Schulz map independently to each tile, and reassembles the results. For finite $T$ below the matrix dimensions, HiMuon defines a local matrix-function map rather than a convergent approximation to the full-matrix update: spectral interactions are preserved within tiles and discarded across tile boundaries. For fixed finite $T$, the leading Newton-Schulz work decreases to $O(H W T K)$, and the computation decomposes into independent small dense matrix operations. This structure enables tile-size-dependent GPU kernels, cross-layer batching, memory-bounded chunking, and runtime tile-size schedules. Experiments on transformer training and controlled matrix-function diagnostics show that HiMuon improves optimizer-step efficiency while keeping training behavior close to full-matrix Muon in the tested regimes.

math.NA↗

arXiv2026

Eigenvector-based acceleration strategies for gradient-type methods

Several strategies are described and analyzed to speed-up gradient-type methods when applied to the minimization of strictly convex quadratics and strictly convex functions. The proposed techniques focus on relaxing the traditional optimal step length associated with gradient methods, including the steepest descent (SD) and the minimal residual (MR) methods. Such a relaxation avoids the well-known negative zigzag effect and allows the iterates to move in the entire space which in turn implies that every so often the search direction approaches some eigenvector of the underlying Hessian matrix. The proposed speedups then rely on taking advantage of the properties of the Lanczos method once a search direction that approaches an eigenvector has been identified in order to accelerate the convergence towards the global minimizer. After analyzing the proposed strategies, we illustrate them on the global minimization of strictly convex functions.

math.NA↗

arXiv2025

Design Criteria for SGD Preconditioners: Local Conditioning, Noise Floors, and Basin Stability

Stochastic Gradient Descent (SGD) often slows in the late stage of training due to anisotropic curvature and gradient noise. We analyze preconditioned SGD in the geometry induced by a symmetric positive definite matrix $\mathbf{M}$, deriving bounds in which both the convergence rate and the stochastic noise floor are governed by $\mathbf{M}$-dependent quantities: the rate through an effective condition number in the $\mathbf{M}$-metric, and the floor through the product of that condition number and the preconditioned noise level. For nonconvex objectives, we establish a preconditioner-dependent basin-stability guarantee: when smoothness and basin size are measured in the $\mathbf{M}$-norm, the probability that the iterates remain in a well-behaved local region admits an explicit lower bound. This perspective is particularly relevant in Scientific Machine Learning (SciML), where achieving small training loss under stochastic updates is closely tied to physical fidelity, numerical stability, and constraint satisfaction. The framework applies to both diagonal/adaptive and curvature-aware preconditioners and yields a simple design principle: choose $\mathbf{M}$ to improve local conditioning while attenuating noise. Experiments on a quadratic diagnostic and three SciML benchmarks validate the predicted rate-floor behavior.

math.NA↗

arXiv2025

Acceleration methods for fixed point iterations

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 procedures and this led practitioners to seek faster alternatives to reach the limit. ``Acceleration techniques'' comprise a broad array of methods specifically designed with this goal in mind. They started as a means of improving the convergence of general scalar sequences by various forms of ``extrapolation to the limit'', i.e., by extrapolating the most recent iterates to the limit via linear combinations. Extrapolation methods of this type, the best known example of which is Aitken's Delta-squared process, require only the sequence of vectors as input. However, limiting methods to only use the iterates is too restrictive. Accelerating sequences generated by fixed-point iterations by utilizing both the iterates and the fixed-point mapping itself has proven highly successful across various areas of physics. A notable example of these Fixed-Point accelerators (FP-Accelerators) is a method developed by D. Anderson in 1965 and now widely known as Anderson Acceleration (AA). Furthermore, Quasi-Newton and Inexact Newton methods can also be placed in this category as well. This paper presents an overview of these methods -- with an emphasis on those, such as AA, that are geared toward accelerating fixed point iterations.

math.NA↗

arXiv2025

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

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. This paper introduces an approach that eliminates orthogonalization requirements in traditional Rayleigh-Ritz projection methods. The proposed method employs non-orthogonal bases computed at reduced precision, resulting in bases computed without inner-products. A primary focus is on maintaining the linear independence of the basis vectors. Through extensive evaluation with both synthetic test cases and real-world applications, we demonstrate that the proposed approach achieves the desired accuracy while fully taking advantage of mixed-precision arithmetic.

math.NA↗

arXiv2024

Cucheb: A GPU implementation of the filtered Lanczos procedure

This paper describes the software package Cucheb, a GPU implementation of the filtered Lanczos procedure for the solution of large sparse symmetric eigenvalue problems. The filtered Lanczos procedure uses a carefully chosen polynomial spectral transformation to accelerate convergence of the Lanczos method when computing eigenvalues within a desired interval. This method has proven particularly effective for eigenvalue problems that arise in electronic structure calculations and density functional theory. We compare our implementation against an equivalent CPU implementation and show that using the GPU can reduce the computation time by more than a factor of 10.

math.NA↗

arXiv2024

Joint Approximate Partial Diagonalization of Large Matrices

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 matrix. We argue that when the matrices are of large dimension, then the natural generalization of this problem is to seek an orthonormal basis of a certain subspace that is a near eigenspace for all the matrices in the set. We refer to this as the problem of ``partial joint diagonalization of matrices.'' The approach proposed first finds this approximate common near eigenspace and then proceeds to a joint diagonalization of the restrictions of the input matrices in this subspace. A few solution methods for this problem are proposed and illustrations of its potential applications are provided.

math.NA↗

arXiv2024

Anderson Acceleration with Truncated Gram-Schmidt

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 the classical AA. In particular, an attractive feature of AATGS is that its iterates obey a three-term recurrence in the situation when it is applied to solving symmetric linear problems and this can lead to a considerable reduction of memory and computational costs. We analyze the convergence of AATGS in both full-depth and limited-depth scenarios and establish its equivalence to the classical AA in the linear case. We also report on the effectiveness of AATGS through a set of numerical experiments, ranging from solving nonlinear partial differential equations to tackling nonlinear optimization problems. In particular, the performance of the method is compared with that of the classical AA algorithms.

math.NA↗

arXiv2024

Straggler-tolerant stationary methods for linear systems

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 zeros. We assume that both the number of computed entries and their associated row index set are random variables, with the row index set sampled uniformly given the number of computed entries. This model of computations is realized in hybrid cloud computing architectures following the controller-worker distributed model under the influence of straggling workers. We propose straggler-tolerant Richardson iteration scheme and Chebyshev semi-iterative schemes, and prove sufficient conditions for their convergence in expectation. Numerical experiments verify the presented theoretical results as well as the effectiveness of the proposed schemes on a few sparse matrix problems.

math.NA↗

arXiv2024

Gradient-type subspace iteration methods for the symmetric eigenvalue problem

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 Grassmann manifold viewpoint are developed. A gradient method as well as a nonlinear conjugate gradient technique are described. Convergence of the gradient-based algorithm is analyzed and a few numerical experiments are reported, indicating that the proposed algorithms are sometimes superior to standard algorithms. This includes the Chebyshev-based subspace iteration and the locally optimal block conjugate gradient method, when compared in terms of number of matrix vector products and computational time, resp. The new methods, on the other hand, do not require estimating optimal parameters. An important contribution of this paper to achieve this good performance is the accurate and efficient implementation of an exact line search. In addition, new convergence proofs are presented for the non-accelerated gradient method that includes a locally exponential convergence if started in a $\mathcal{O(\sqrtδ)}$ neighbourhood of the dominant subspace with spectral gap $δ$.

math.NA↗

arXiv2024

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

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 with inexact Newton methods - depending on which variant is implemented. We prove theoretically and verify experimentally, on a variety of problems from simulation experiments to deep learning applications, that our method is a powerful accelerated iterative algorithm.

math.NA↗

arXiv2023

On the tubular eigenvalues of third-order tensors

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 alternative definitions of eigenvalue for third-order tensors that are known in the literature, namely eigentuple and T-eigenvalue. In addition, it establishes a few results on tubular spectra of tensors which can be exploited to analyze the convergence of tubular versions of iterative methods for solving tensor equations.

math.NA↗

arXiv2022

An Efficient Nonlinear Acceleration method that Exploits Symmetry of the Hessian

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 are limited. In this paper, we propose a nonlinear Truncated Generalized Conjugate Residual method (nlTGCR) whose goal is to exploit the symmetry of the Hessian to reduce memory usage. The proposed method can be interpreted as either an inexact Newton or a quasi-Newton method. We show that, with the help of global strategies like residual check techniques, nlTGCR can converge globally for general nonlinear problems and that under mild conditions, nlTGCR is able to achieve superlinear convergence. We further analyze the convergence of nlTGCR in a stochastic setting. Numerical results demonstrate the superiority of nlTGCR when compared with several other competitive baseline approaches on a few problems. Our code will be available in the future.

cs.LG↗

arXiv2022

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

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 simplicity. However, GDA can converge to non-optimal minimax points. We propose a new minimax optimization framework, GDA-AM, that views the GDAdynamics as a fixed-point iteration and solves it using Anderson Mixing to con-verge to the local minimax. It addresses the diverging issue of simultaneous GDAand accelerates the convergence of alternating GDA. We show theoretically that the algorithm can achieve global convergence for bilinear problems under mild conditions. We also empirically show that GDA-AMsolves a variety of minimax problems and improves GAN training on several datasets

cs.LG↗

arXiv2022

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

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 implemented in parGeMSLR is based on algebraic domain decomposition and partitions the symmetrized adjacency graph recursively into several non-overlapping partitions via a p-way vertex separator, where p is an integer multiple of the total number of MPI processes. From a numerical perspective, parGeMSLR builds a Schur complement approximate inverse preconditioner as the sum between the matrix inverse of the interface coupling matrix and a low-rank correction term. To reduce the cost associated with the computation of the approximate inverse matrices, parGeMSLR exploits a multilevel partitioning of the algebraic domain. The parGeMSLR library is implemented on top of the Message Passing Interface and can solve both real and complex linear systems. Furthermore, parGeMSLR can take advantage of hybrid computing environments with in-node access to one or more Graphics Processing Units. Finally, the parallel efficiency (weak and strong scaling) of parGeMSLR is demonstrated on a few model problems arising from discretizations of 3D Partial Differential Equations.

cs.MS↗

arXiv2021

A non-perturbative approach to computing seismic normal modes in rotating planets

A Continuous Galerkin method-based approach is presented to compute the seismic normal modes of rotating planets. Special care is taken to separate out the essential spectrum in the presence of a fluid outer core using a polynomial filtering eigensolver. The relevant elastic-gravitational system of equations, including the Coriolis force, is subjected to a mixed finite-element method, while self-gravitation is accounted for with the fast multipole method. Our discretization utilizes fully unstructured tetrahedral meshes for both solid and fluid regions. The relevant eigenvalue problem is solved by a combination of several highly parallel and computationally efficient methods. We validate our three-dimensional results in the non-rotating case using analytical results for constant elastic balls, as well as numerical results for an isotropic Earth model from standard ``radial" algorithms. We also validate the computations in the rotating case, but only in the slowly-rotating regime where perturbation theory applies, because no other independent algorithms are available in the general case. The algorithm and code are used to compute the point spectra of eigenfrequencies in several Earth and Mars models studying the effects of heterogeneity on a large range of scales.

physics.comp-ph↗

arXiv2021

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

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-gravitational system of equations, including the Coriolis force, is subjected to a mixed finite-element method, while self-gravitation is accounted for with the fast multipole method (FMM). To solve the resulting quadratic eigenvalue problem (QEP), the approach utilizes extended Lanczos vectors forming a subspace computed from a non-rotating planet -- with the shape of boundaries of a rotating planet and accounting for the centrifugal potential -- to reduce the dimension of the original problem significantly. The subspace is guaranteed to be contained in the space of functions to which the seismic normal modes belong. The reduced system can further be solved with a standard eigensolver. The computational accuracy is illustrated using all the modes with relative small meshes and also tested against standard perturbation calculations relative to a standard Earth model. The algorithm and code are used to compute the point spectra of eigenfrequencies in several Mars models studying the effects of heterogeneity on a large range of scales.

physics.comp-ph↗