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Results for “numerical linear algebra”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Bringing randomized algorithms to mainstream numerical linear algebra

Numerical linear algebra (NLA) underpins huge swaths of computational science and engineering. For scientists and engineers to make the most of the DOE’s computing resources, it is essential that they have access to high-performance implementations of algorithms with best-in-class scalability and reliability. Despite this, prevailing NLA libraries have little to no support for breakthrough algorithms from the field of randomized numerical linear algebra (RandNLA) that have been developed over the past twenty years. The goal of this LDRD was to break a log-jam that had prevented broad adoption of RandNLA. Our work had two thrusts. The first was to develop RandBLAS: a trustworthy and high-performance C++ library for randomized dimension reduction (an operation widely known as sketching). The second was the development of a novel randomized algorithm for computing a challenging type of matrix decomposition known as Householder QR with column pivoting (Householder QRCP). In this one-year late-start LDRD we successfully delivered RandBLAS 1.0 and new CPU and GPU codes for Householder QRCP. RandBLAS has extensive documentation at https://randblas.readthedocs.io/en/stable/. Papers on RandBLAS and and our high-performance QRCP codes are forthcoming.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

A survey of numerical linear algebra methods utilizing mixed-precision arithmetic

The efficient utilization of mixed-precision numerical linear algebra algorithms can offer attractive acceleration to scientific computing applications. Especially with the hardware integration of low-precision special-function units designed for machine learning applications, the traditional numerical algorithms community urgently needs to reconsider the floating point formats used in the distinct operations to efficiently leverage the available compute power. In this study, we provide a comprehensive survey of mixed-precision numerical linear algebra routines, including the underlying concepts, theoretical background, and experimental results for both dense and sparse linear algebra problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Inexact iterative numerical linear algebra for neural network-based spectral estimation and rare-event prediction

Understanding dynamics in complex systems is challenging because there are many degrees of freedom, and those that are most important for describing events of interest are often not obvious. The leading eigenfunctions of the transition operator are useful for visualization, and they can provide an efficient basis for computing statistics, such as the likelihood and average time of events (predictions). Here, we develop inexact iterative linear algebra methods for computing these eigenfunctions (spectral estimation) and making predictions from a dataset of short trajectories sampled at finite intervals. We demonstrate the methods on a low-dimensional model that facilitates visualization and a high-dimensional model of a biomolecular system. Implications for the prediction problem in reinforcement learning are discussed.

Chemistry↗
DOE OSTI2021

Mixed Precision Numerical Linear Algebra (Final Report)

The objective of this subcontract was to identify opportunities for the use of mixed precision within iterative solvers and to explore these opportunities both theoretically and experimentally. All quarterly milestones were achieved. We summarize the achievements each quarter in sections below.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Mixed Precision Numerical Linear Algebra (Final Report)

The objective of this subcontract was to identify opportunities for the use of mixed precision within iterative solvers and to explore these opportunities both theoretically and experimentally. All quarterly milestones were achieved. We summarize the achievements each quarter in this report.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Mixed Precision Numerical Linear Algebra (Final Report)

The objective of this subcontract was to identify opportunities for the use of mixed precision within iterative solvers and to explore these opportunities both theoretically and experimentally. All quarterly milestones were achieved. Quarterly achievements are summarized in sections.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Mixed Precision Numerical Linear Algebra (Final Report)

The objective of this subcontract was to identify opportunities for the use of mixed precision within iterative solvers and to explore these opportunities both theoretically and experimentally. All quarterly milestones were achieved. We summarize the achievements each quarter in sections below.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Gradient Coding With Iterative Block Leverage Score Sampling

Gradient coding is a method for mitigating straggling servers in a centralized computing network that uses erasure-coding techniques to distributively carry out first-order optimization methods. Randomized numerical linear algebra uses randomization to develop improved algorithms for large-scale linear algebra computations. In this study, we propose a method for distributed optimization that combines gradient coding and randomized numerical linear algebra. The proposed method uses a randomized ℓ 2 -subspace embedding and a gradient coding technique to distribute blocks of data to the computational nodes of a centralized network, and at each iteration the central server only requires a small number of computations to obtain the steepest descent update. The novelty of our approach is that the data is replicated according to importance scores, called block leverage scores, in contrast to most gradient coding approaches that uniformly replicate the data blocks. Furthermore, we do not require a decoding step at each iteration, avoiding a bottleneck in previous gradient coding schemes. We show that our approach results in a valid ℓ 2 -subspace embedding, and that our resulting approximation converges to the optimal solution.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Yet Another NLA Library: T-LAPACK

In recent years, Randomized numerical linear algebra (RandNLA) proved to be more than a theoretical novelty: projects like RandLAPACK demonstrate its practical value across architectures, and projects like RandBLAS build trust in randomization as a tool for high-performance NLA. This BoF considers two main questions. First, what are the pressing issues in software standards and implementation that need to be resolved for RandNLA to become a core component of HPC? Second, how can we mobilize a community effort to make progress on these issues? The BoF will engage the audience to discuss the idea of growing the role of RandNLA in high-performance computing and what it would take to scale from niche prototypes to robust, production-quality software libraries.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Fast synchronization-free algorithms for parallel sparse triangular solves with multiple right-hand sides

The sparse triangular solve kernels, SpTRSV and SpTRSM, are important building blocks for a number of numerical linear algebra routines. Parallelizing SpTRSV and SpTRSM on today's many-core platforms, such as GPUs, is not an easy task since computing a component of the solution may depend on previously computed components, enforcing a degree of sequential processing. As a consequence, most existing work introduces a preprocessing stage to partition the components into a group of level-sets or colour-sets so that components within a set are independent and can be processed simultaneously during the subsequent solution stage. However, this class of methods requires a long preprocessing time as well as significant runtime synchronization over-heads between the sets. To address this, we

HPC↗
DOE OSTI2018

Diagonal and circulant or skew-circulant splitting preconditioners for spatial fractional diffusion equations

We propose three new preconditioners: diagonal and optimal-circulant splitting preconditioner, diagonal and skew-circulant splitting preconditioner, and diagonal and optimal-skew-circulant splitting preconditioner for solving the diagonal-plus-Toeplitz linear system discretized from the spatial fractional diffusion equations. Theoretical analysis shows that these three preconditioners can make the eigenvalues of the preconditioned matrices be clustered around 1, especially when the grids of the discretizations are refined. These results coincide with the one about the diagonal and circulant splitting preconditioner constructed recently by Bai et al. (Numer Linear Algebra Appl 24:e2093, 2017). Numerical experiments exhibit that the proposed preconditioners can significantly improve the convergence of the Krylov subspace iteration methods like GMRES and BiCGSTAB, and they outperform the diagonal and circulant splitting preconditioner as well.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Iterated Gauss-Seidel GMRES

The GMRES algorithm of Saad and Schultz [SIAM J. Sci. Stat. Comput., 7 (1986), pp. 856-869] is an iterative method for approximately solving linear systems Ax = b, with initial guess x0 and residual r0 = b Ax0. The algorithm employs the Arnoldi process to generate the Krylov basis vectors (the columns of Vk ). It is well known that this process can be viewed as a QR factorization of the matrix Bk = [r0, AVk] at each iteration. Despite an O (..epsilon..)..kappa.. (Bk ) loss of orthogonality, for unit roundoff ..epsilon..and condition number ..kappa.. , the modified Gram-Schmidt formulation was shown to be backward stable in the seminal paper by Paige et al. [SIAM J. Matrix Anal.Appl., 28 (2006), pp. 264-284]. We present an iterated Gauss-Seidel formulation of the GMRES algorithm (IGS-GMRES) based on the ideas of Ruhe [Linear Algebra Appl., 52 (1983), pp. 591-601] and Swirydowicz et al. [Numer. Linear Algebra Appl., 28 (2020), pp. 1-20]. IGS-GMRES maintains orthogonality to the level O (..epsilon..)..kappa.. (Bk ) or O (..epsilon..), depending on the choice of one or two iterations; for two Gauss-Seidel iterations, the computed Krylov basis vectors remain orthogonal to working accuracy and the smallest singular value of Vk remains close to one. The resulting GMRES method is thus backward stable. We show that IGS-GMRES can be implemented with only a single synchronization point per iteration, making it relevant to large-scale parallel computing environments. We also demonstrate that, unlike MGS-GMRES, in IGS-GMRES the relative Arnoldi residual corresponding to the computed approximate solution no longer stagnates above machine precision even for highly nonnormal systems.

Arnoldi-QR↗
DOE OSTI2024

A Contextually-Aware Sensitivity Analysis to Guide the Design of Randomized Least Squares Solvers in Applications

Our work on the DOE-sponsored project “A Contextually-Aware Sensitivity Analysis to Guide the Design of Randomized Least Squares Solvers in Applications,” was an effort to address critical challenges in nu merical computing and its applications to optimization. The increasing demand for robust and scalable solutions to large-scale linear algebra problems has highlighted the limitations of traditional approaches, particularly in heterogeneous and extreme-scale computing environments. Randomized Numerical Linear Algebra (RandNLA) offers a promising framework to address these challenges, and this proposal builds on this foundation by introducing innovations in sensitivity analysis and computational adaptability.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Replicated Computational Results (RCR) Report for "Adaptive Precision Block-Jacobi for High Performance Preconditioning in the Ginkgo Linear Algebra Software''

In, a practical implementation of a novel adaptive precision block-Jacobi preconditioner is introduced. In particular, the authors present a heavily-tuned GPU implementation of the adaptive precision block-Jacobi preconditioner within the Ginkgo numerical linear algebra library. The performance of the methodology and implementation is demonstrated using the proposed preconditioning scheme within Ginkgo’s high-performance Conjugate Gradient (CG) implementation on an NVIDIA Volta GPU. In this report, we replicate a subset of the computational results presented in. The focus is generating results from Fig. 9 to evaluate the performance of using Ginkgo’s CG solver integrated with either the full or the adaptive precision block-Jacobi preconditioner applied to a variety of test cases

97 MATHEMATICS AND COMPUTING↗