arXiv subjects

Hussam Al Daas

Publications and source records attributed to Hussam Al Daas.

At least 19 recordsLinked to original sources

arXiv2026

Communication Lower Bounds and Algorithms for Sketching with Random Dense Matrices

Sketching is widely used in randomized linear algebra for low-rank matrix approximation, column subset selection, and many other problems, and it has gained significant traction in machine learning applications. However, sketching large matrices often necessitates distributed memory algorithms, where communication overhead becomes a critical bottleneck on modern supercomputing clusters. Despite its growing relevance, distributed-memory parallel strategies for sketching remain largely unexplored. In this work, we establish communication lower bounds for sketching using dense matrices that determine how much data movement is required to perform it in parallel. One important observation of our lower bounds is that no communication is required for a small number of processors. We show that our lower bounds are tight by presenting communication optimal algorithms. Furthermore, we extend our approach to determine communication lower bounds for computations of Nyström approximation where sketching is applied twice. We also introduce novel parallel algorithms whose communication costs are close to the lower bounds. Finally, we implement our algorithms on modern state-of-the-art supercomputing infrastructures which have both CPU- and GPU-equipped systems and demonstrate their parallel scalability.

cs.DC↗

arXiv2026

Efficient Sketching-Based Summation of Tucker Tensors

We present efficient, sketching-based methods for the summation of tensors in Tucker format. Leveraging the algebraic structure of Khatri-Rao and Kronecker products, our approach enables compressed arithmetic on Tucker tensors while controlling rank growth and computational cost. The proposed sketching framework avoids the explicit formation of large intermediate tensors, instead operating directly on the factor matrices and core tensors to produce accurate low-rank approximations of tensor sums. Furthermore, we analyze the computational complexity and the theoretical approximation properties of the proposed methodology. Numerical experiments demonstrate the effectiveness of our approach on four problems: two synthetic test cases, a parameter-dependent elliptic equation (commonly referred to as the cookie problem) solved via GMRES, and a one-dimensional linear transport problem discretized via high-order discontinuous Galerkin methods, where repeated tensor summation arises as a core computational bottleneck. Across these examples, the sketching-based summation achieves substantial computational savings while preserving high accuracy relative to direct summation and re-compression.

math.NA↗

arXiv2026

Extended-Krylov-subspace methods for trust-region and norm-regularization subproblems

We consider an effective new method for solving trust-region and norm-regularization problems that arise as subproblems in many optimization applications. We show that the solutions to such subproblems effectively lie in a very-low-dimensional subspace as a function of their controlling parameters (trust-region radius or regularization weight). Based on this, we build a basis spanning these solutions using an efficient extended-Krylov-subspace iteration that involves a single matrix factorization. The problems within the subspace using such a basis may be solved at very low cost using effective high-order root-finding methods. This then provides an alternative to common methods using multiple factorizations or standard Krylov subspaces. We provide numerical results to illustrate the effectiveness of our TREK/NREK approach.

math.NA↗

arXiv2026

Reducing acquisition time and radiation damage: data-driven subsampling for spectro-microscopy

Spectro-microscopy is an experimental technique which can be used to observe spatial variations in chemical state and changes in chemical state over time or under experimental conditions. As a result it has broad applications across areas such as energy materials, catalysis, environmental science and biological samples. However, the technique is often limited by factors such as long acquisition times and radiation damage. We present two measurement strategies that allow for significantly shorter experiment times and total doses applied. The strategies are based on taking only a small subset of all the measurements (e.g. sparse acquisition or subsampling), and then computationally reconstructing all unobserved measurements using mathematical techniques. The methods are data-driven, using spectral and spatial importance subsampling distributions to identify important measurements. As a result, taking as little as 4-6\% of the measurements is sufficient to capture the same information as in a conventional scan.

physics.med-ph↗

arXiv2026

Minimal residual rational Krylov subspace method for sequences of shifted linear systems

The solution of sequences of shifted linear systems is a classic problem in numerical linear algebra, and a variety of efficient methods have been proposed over the years. Nevertheless, there still exist challenging scenarios witnessing a lack of performing solvers. For instance, state-of-the-art procedures struggle to handle nonsymmetric problems where the shifts are complex numbers that do not come as conjugate pairs. We design a novel projection strategy based on the rational Krylov subspace equipped with a minimal residual condition. We also devise a novel pole selection procedure, tailored to our problem, providing poles for the rational Krylov basis construction that yield faster convergence than those computed by available general-purpose schemes. A panel of diverse numerical experiments shows that our novel approach performs better than state-of-the-art techniques, especially on the very challenging problems mentioned above.

math.NA↗

arXiv2026

Matrix perturbation analysis of methods for extracting singular values from approximate singular subspaces

Given (orthonormal) approximations $\tilde{U}$ and $\tilde{V}$ to the left and right subspaces spanned by the leading singular vectors of a matrix $A$, we discuss methods to approximate the leading singular values of $A$ and study their accuracy. In particular, we focus our analysis on the generalized Nyström approximation, as surprisingly, it is able to obtain significantly better accuracy than classical methods, namely Rayleigh-Ritz and (one-sided) projected SVD. A key idea of the analysis is to view the methods as finding the exact singular values of a perturbation of $A$. In this context, we derive a matrix perturbation result that exploits the structure of such $2\times2$ block matrix perturbation. Furthermore, we extend it to block tridiagonal matrices. We then obtain bounds on the accuracy of the extracted singular values. This leads to sharp bounds that predict well the approximation error trends and explain the difference in the behavior of these methods. Finally, we present an approach to derive an a-posteriori version of those bounds, which are more amenable to computation in practice.

math.NA↗

arXiv2026

Algebraic Multigrid with Overlapping Schwarz Smoothers and Local Spectral Coarse Grids for Least Squares Problems

This paper develops a new algebraic multigrid (AMG) method for sparse least-squares systems of the form $A=G^TG$ motivated by challenging applications in scientific computing where classical AMG methods fail. First we review and relate the use of local spectral problems in distinct fields of literature on AMG, domain decomposition (DD), and multiscale finite elements. We then propose a new approach blending aggregation-based coarsening, overlapping Schwarz smoothers, and locally constructed spectral coarse spaces. By exploiting the factorized structure of $A$, we construct an inexpensive symmetric positive semidefinite splitting that yields local generalized eigenproblems whose solutions define sparse, nonoverlapping coarse basis functions. This enables a fully algebraic and naturally recursive multilevel hierarchy that can either coarsen slowly to achieve AMG-like operator complexities, or coarsen aggressively-with correspondingly larger local spectral problems-to ensure robustness on problems that cannot be solved by existing AMG methods. The method requires no geometric information, avoids global eigenvalue solves, and maintains efficient parallelizable setup through localized operations. Numerical experiments demonstrate that the proposed least-squares AMG-DD method achieves convergence rates independent of anisotropy on rotated diffusion problems and remains scalable with problem size, while for small amounts of anisotropy we obtain convergence and operator complexities comparable with classical AMG methods. Most notably, for extremely anisotropic heat conduction operators arising in magnetic confinement fusion, where AMG and smoothed aggregation fail to reduce the residual even marginally, our method provides robust and efficient convergence across many orders of magnitude in anisotropy strength.

math.NA↗

arXiv2025

Adaptive Randomized Tensor Train Rounding using Khatri-Rao Products

Approximating a tensor in the tensor train (TT) format has many important applications in scientific computing. Rounding a TT tensor involves further compressing a tensor that is already in the TT format. This paper proposes new randomized algorithms for TT-rounding that uses sketches based on Khatri-Rao products (KRP). When the TT-ranks are known in advance, the proposed methods are comparable in cost to the sketches that used a sketching matrix in the TT-format~\cite{al2023randomized}. However, the use of KRP sketches enables adaptive algorithms to round the tensor in the TT-format within a fixed user-specified tolerance. An important component of the adaptivity is the estimation of error using KRP sketching, for which we develop theoretical guarantees. We report numerical experiments on synthetic tensors, parametric low-rank kernel approximations, and the solution of parametric partial differential equations. The numerical experiments show that we obtain speed-ups of up to $50\times$ compared to deterministic TT-rounding. Both the computational cost analysis and numerical experiments verify that the adaptive algorithms are competitive with the fixed rank algorithms, suggesting the adaptivity introduces only a low overhead.

math.NA↗

arXiv2025

Minimizing Communication for Parallel Symmetric Tensor Times Same Vector Computation

In this article, we focus on the parallel communication cost of multiplying the same vector along two modes of a $3$-dimensional symmetric tensor. This is a key computation in the higher-order power method for determining eigenpairs of a $3$-dimensional symmetric tensor and in gradient-based methods for computing a symmetric CP decomposition. We establish communication lower bounds that determine how much data movement is required to perform the specified computation in parallel. The core idea of the proof relies on extending a key geometric inequality for $3$-dimensional symmetric computations. We demonstrate that the communication lower bounds are tight by presenting an optimal algorithm where the data distribution is a natural extension of the triangle block partition scheme for symmetric matrices to 3-dimensional symmetric tensors.

cs.DC↗

arXiv2024

Communication Lower Bounds and Optimal Algorithms for Symmetric Matrix Computations

In this article, we focus on the communication costs of three symmetric matrix computations: i) multiplying a matrix with its transpose, known as a symmetric rank-k update (SYRK) ii) adding the result of the multiplication of a matrix with the transpose of another matrix and the transpose of that result, known as a symmetric rank-2k update (SYR2K) iii) performing matrix multiplication with a symmetric input matrix (SYMM). All three computations appear in the Level 3 Basic Linear Algebra Subroutines (BLAS) and have wide use in applications involving symmetric matrices. We establish communication lower bounds for these kernels using sequential and distributed-memory parallel computational models, and we show that our bounds are tight by presenting communication-optimal algorithms for each setting. Our lower bound proofs rely on applying a geometric inequality for symmetric computations and analytically solving constrained nonlinear optimization problems. The symmetric matrix and its corresponding computations are accessed and performed according to a triangular block partitioning scheme in the optimal algorithms.

cs.DC↗

arXiv2024

A Robust Two-Level Schwarz Preconditioner For Sparse Matrices

This paper introduces a fully algebraic two-level additive Schwarz preconditioner for general sparse large-scale matrices. The preconditioner is analyzed for symmetric positive definite (SPD) matrices. For those matrices, the coarse space is constructed based on approximating two local subspaces in each subdomain. These subspaces are obtained by approximating a number of eigenvectors corresponding to dominant eigenvalues of two judiciously posed generalized eigenvalue problems. The number of eigenvectors can be chosen to control the condition number. For general sparse matrices, the coarse space is constructed by approximating the image of a local operator that can be defined from information in the coefficient matrix. The connection between the coarse spaces for SPD and general matrices is also discussed. Numerical experiments show the great effectiveness of the proposed preconditioners on matrices arising from a wide range of applications. The set of matrices includes SPD, symmetric indefinite, nonsymmetric, and saddle-point matrices. In addition, we compare the proposed preconditioners to the state-of-the-art domain decomposition preconditioners.

math.NA↗

arXiv2023

Communication Lower Bounds and Optimal Algorithms for Multiple Tensor-Times-Matrix Computation

Multiple Tensor-Times-Matrix (Multi-TTM) is a key computation in algorithms for computing and operating with the Tucker tensor decomposition, which is frequently used in multidimensional data analysis. We establish communication lower bounds that determine how much data movement is required to perform the Multi-TTM computation in parallel. The crux of the proof relies on analytically solving a constrained, nonlinear optimization problem. We also present a parallel algorithm to perform this computation that organizes the processors into a logical grid with twice as many modes as the input tensor. We show that with correct choices of grid dimensions, the communication cost of the algorithm attains the lower bounds and is therefore communication optimal. Finally, we show that our algorithm can significantly reduce communication compared to the straightforward approach of expressing the computation as a sequence of tensor-times-matrix operations.

cs.DC↗

arXiv2022

Tight Memory-Independent Parallel Matrix Multiplication Communication Lower Bounds

Communication lower bounds have long been established for matrix multiplication algorithms. However, most methods of asymptotic analysis have either ignored the constant factors or not obtained the tightest possible values. Recent work has demonstrated that more careful analysis improves the best known constants for some classical matrix multiplication lower bounds and helps to identify more efficient algorithms that match the leading-order terms in the lower bounds exactly and improve practical performance. The main result of this work is the establishment of memory-independent communication lower bounds with tight constants for parallel matrix multiplication. Our constants improve on previous work in each of three cases that depend on the relative sizes of the aspect ratios of the matrices.

cs.DC↗

arXiv2022

Efficient Algebraic Two-Level Schwarz Preconditioner For Sparse Matrices

Domain decomposition methods are among the most efficient for solving sparse linear systems of equations. Their effectiveness relies on a judiciously chosen coarse space. Originally introduced and theoretically proved to be efficient for self-adjoint operators, spectral coarse spaces have been proposed in the past few years for indefinite and non-self-adjoint operators. This paper presents a new spectral coarse space that can be constructed in a fully-algebraic way unlike most existing spectral coarse spaces. We present theoretical convergence result for Hermitian positive definite diagonally dominant matrices. Numerical experiments and comparisons against state-of-the-art preconditioners in the multigrid community show that the resulting two-level Schwarz preconditioner is efficient especially for non-self-adjoint operators. Furthermore, in this case, our proposed preconditioner outperforms state-of-the-art preconditioners.

math.NA↗

arXiv2022

A Robust Algebraic Domain Decomposition Preconditioner for Sparse Normal Equations

Solving the normal equations corresponding to large sparse linear least-squares problems is an important and challenging problem. For very large problems, an iterative solver is needed and, in general, a preconditioner is required to achieve good convergence. In recent years, a number of preconditioners have been proposed. These are largely serial and reported results demonstrate that none of the commonly used preconditioners for the normal equations matrix is capable of solving all sparse least-squares problems. Our interest is thus in designing new preconditioners for the normal equations that are efficient, robust, and can be implemented in parallel. Our proposed preconditioners can be constructed efficiently and algebraically without any knowledge of the problem and without any assumption on the least-squares matrix except that it is sparse. We exploit the structure of the symmetric positive definite normal equations matrix and use the concept of algebraic local symmetric positive semi-definite splittings to introduce two-level Schwarz preconditioners for least-squares problems. The condition number of the preconditioned normal equations is shown to be theoretically bounded independently of the number of subdomains in the splitting. This upper bound can be adjusted using a single parameter $τ$ that the user can specify. We discuss how the new preconditioners can be implemented on top of the PETSc library using only 150 lines of Fortran, C, or Python code. Problems arising from practical applications are used to compare the performance of the proposed new preconditioner with that of other preconditioners.

math.NA↗

arXiv2021

Randomized algorithms for rounding in the Tensor-Train format

The Tensor-Train (TT) format is a highly compact low-rank representation for high-dimensional tensors. TT is particularly useful when representing approximations to the solutions of certain types of parametrized partial differential equations. For many of these problems, computing the solution explicitly would require an infeasible amount of memory and computational time. While the TT format makes these problems tractable, iterative techniques for solving the PDEs must be adapted to perform arithmetic while maintaining the implicit structure. The fundamental operation used to maintain feasible memory and computational time is called rounding, which truncates the internal ranks of a tensor already in TT format. We propose several randomized algorithms for this task that are generalizations of randomized low-rank matrix approximation algorithms and provide significant reduction in computation compared to deterministic TT-rounding algorithms. Randomization is particularly effective in the case of rounding a sum of TT-tensors (where we observe 20x speedup), which is the bottleneck computation in the adaptation of GMRES to vectors in TT format. We present the randomized algorithms and compare their empirical accuracy and computational time with deterministic alternatives.

math.NA↗

arXiv2021

A Robust Algebraic Multilevel Domain Decomposition Preconditioner For Sparse Symmetric Positive Definite Matrices

Domain decomposition (DD) methods are widely used as preconditioner techniques. Their effectiveness relies on the choice of a locally constructed coarse space. Thus far, this construction was mostly achieved using non-assembled matrices from discretized partial differential equations (PDEs). Therefore, DD methods were mainly successful when solving systems stemming from PDEs. In this paper, we present a fully algebraic multilevel DD method where the coarse space can be constructed locally and efficiently without any information besides the coefficient matrix. The condition number of the preconditioned matrix can be bounded by a user-prescribed number. Numerical experiments illustrate the effectiveness of the preconditioner on a range of problems arising from different applications.

math.NA↗

arXiv2021

Parallel Algorithms for Tensor Train Arithmetic

We present efficient and scalable parallel algorithms for performing mathematical operations for low-rank tensors represented in the tensor train (TT) format. We consider algorithms for addition, elementwise multiplication, computing norms and inner products, orthogonalization, and rounding (rank truncation). These are the kernel operations for applications such as iterative Krylov solvers that exploit the TT structure. The parallel algorithms are designed for distributed-memory computation, and we use a data distribution and strategy that parallelizes computations for individual cores within the TT format. We analyze the computation and communication costs of the proposed algorithms to show their scalability, and we present numerical experiments that demonstrate their efficiency on both shared-memory and distributed-memory parallel systems. For example, we observe better single-core performance than the existing MATLAB TT-Toolbox in rounding a 2GB TT tensor, and our implementation achieves a $34\times$ speedup using all 40 cores of a single node. We also show nearly linear parallel scaling on larger TT tensors up to over 10,000 cores for all mathematical operations.

math.NA↗