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All sources NASA NTRS DOE OSTI

DOE OSTI2023

Fast truncated SVD of sparse and dense matrices on graphics processors

We investigate the solution of low-rank matrix approximation problems using the truncated singular value decomposition (SVD). For this purpose, we develop and optimize graphics processing unit (GPU) implementations for the randomized SVD and a blocked variant of the Lanczos approach. Our work takes advantage of the fact that the two methods are composed of very similar linear algebra building blocks, which can be assembled using numerical kernels from existing high-performance linear algebra libraries. Furthermore, the experiments with several sparse matrices arising in representative real-world applications and synthetic dense test matrices reveal a performance advantage of the block Lanczos algorithm when targeting the same approximation accuracy.

Computer Science↗

DOE OSTI2014

PyTrilinos

PyTrilinos is a Python interface to selected Trilinos packages. This makes Trilinos linear algebra classes, linear solvers, preconditioners, nonlinear solvers, eigensolvers, and tools available to Python programmers. This broadens the user base of Trilinos, and facilitates rapid prototyping of scientific codes and interactive manipulation of large, distributed data sets.

Spotz, William↗

DOE OSTI2015

PyTrilinos

Source record↗

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 OSTI2024

Explicit Quantum Circuits for Block Encodings of Certain Sparse Matrices

Many standard linear algebra problems can be solved on a quantum computer by using recently developed quantum linear algebra algorithms that make use of block encodings and quantum eigenvalue/singular value transformations. A block encoding embeds a properly scaled matrix of interest A in a larger unitary transformation U that can be decomposed into a product of simpler unitaries and implemented efficiently on a quantum computer. Although quantum algorithms can potentially achieve exponential speedup in solving linear algebra problems compared to the best classical algorithm, such a gain in efficiency ultimately hinges on our ability to construct an efficient quantum circuit for the block encoding of A, which is difficult in general, and not trivial even for well structured sparse matrices. Here, in this paper, we give a few examples on how efficient quantum circuits can be explicitly constructed for some well structured sparse matrices and discuss a few strategies used in these constructions. We also provide implementations of these quantum circuits in MATLAB.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Improving the Performance of DGEMM with MoA and Cache-Blocking: Preprint

The goal of this paper is to demonstrate performance enhancements of the high performance dense linear algebra matrix-matrix multiply DGEMM kernel, widely implemented by vendors in the basic linear algebra subroutine BLAS library. The mathematics of arrays (MoA) paradigm due to Mullin (1988) results in contiguous memory accesses in combination with Church-Rosser complete language constructs optimized for target processor architectures [3]. Our performance studies demonstrate that the MoA implementation of DGEMM combined with optimal cache-blocking strategies results in at least a 25% performance gain on both Intel Xeon Skylake and IBM Power-9 processors over the vendor supplied Intel MKL and IBM ESSL basic linear algebra libraries. Results are presented for the NREL Eagle and ORNL Summit supercomputers.

cache-blocking↗

DOE OSTI2022

Threaded Multi-Core GEMM with MoA and Cache-Blocking: Preprint

A threaded multi-core implementation of the high performance dense linear algebra matrix-matrix multiply GEMM kernel is described. This kernel is widely implemented by vendors in the basic linear algebra subroutine BLAS library. The mathematics of arrays (MoA) paradigm due to Mullin (1988) results in contiguous memory accesses by employing outer-product forms. Our performance studies demonstrate that the MoA implementation of double precision DGEMM combined with optimal cache-blocking strategies results in at least a 25% performance gain on the Intel Xeon Skylake processor over the vendor supplied Intel MKL basic linear algebra libraries. Results are presented for the NREL Eagle supercomputer. The multi-core DGEMM achieves over 100 GigaFlops/sec with eight openMP threads.

cache-blocking↗

DOE OSTI2022

High performance sparse multifrontal solvers on modern GPUs

Here, we have ported the numerical factorization and triangular solve phases of the sparse direct solver STRUMPACK to GPU. STRUMPACK implements sparse LU factorization using the multifrontal algorithm, which performs most of its operations in dense linear algebra operations on so-called frontal matrices of various sizes. Our GPU implementation off-loads these dense linear algebra operations, as well as the sparse scatter–gather operations between frontal matrices. For the larger frontal matrices, our GPU implementation relies on vendor libraries such as cuBLAS and cuSOLVER for NVIDIA GPUs and rocBLAS and rocSOLVER for AMD GPUs. For the smaller frontal matrices we developed custom CUDA and HIP kernels to reduce kernel launch overhead. Overall, high performance is achieved by identifying submatrix factorizations corresponding to sub-trees of the multifrontal assembly tree which fit entirely in GPU memory. The multi-GPU setting uses SLATE (Software for Linear Algebra Targeting Exascale) as a modern GPU-aware replacement for ScaLAPACK. On 4 nodes of SUMMIT the code runs ~10X faster when using all 24 V100 GPUs compared to when it only uses the 168 POWER9 cores. On 8 SUMMIT nodes, using 48 V100 GPUs, the sparse solver reaches over 50TFlop/s. Compared to SuperLU, on a single V100, for a set of 17 matrices our implementation is faster for all but one matrix, and is on average 5X (median 4X) faster

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2017

Spectral partitioning with blends of eigenvectors

We report many common methods for data analysis rely on linear algebra. We provide new results connecting data analysis error to numerical accuracy in the context of spectral graph partitioning. We provide pointwise convergence guarantees so that spectral blends (linear combinations of eigenvectors) can be employed to solve data analysis problems with confidence in their accuracy. We apply this theory to an accessible model problem, the ring of cliques, by deriving the relevant eigenpairs and finding necessary and sufficient solver tolerances. Analysis of the ring of cliques provides an upper bound on eigensolver tolerances for graph partitioning problems. These results bridge the gap between linear algebra based data analysis methods and the convergence theory of iterative approximation methods. These results explain how the combinatorial structure of a problem can be recovered much faster than numerically accurate solutions to the associated linear algebra problem.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Variational quantum and neural quantum states algorithms for the linear complementarity problem

Variational quantum algorithms (VQAs) are promising hybrid quantum-classical methods designed to leverage the computational advantages of quantum computing while mitigating the limitations of current noisy intermediate-scale quantum (NISQ) hardware. Although VQAs have been demonstrated as proofs of concept, their practical utility in solving real-world problems—and whether quantum-inspired classical algorithms can match their performance—remains an open question. We present a novel application of the variational quantum linear solver (VQLS) and its classical neural quantum states-based counterpart, the variational neural linear solver (VNLS), as key components within a minimum map Newton solver for a complementarity-based rigid-body contact model. We demonstrate using the VNLS that our solver accurately simulates the dynamics of rigid spherical bodies during collision events. These results suggest that quantum and quantum-inspired linear algebra algorithms can serve as viable alternatives to standard linear algebra solvers for modelling certain physical systems.

neural quantum states↗

DOE OSTI2014

An efficient algorithm for incompressible N-phase flows

We present an efficient algorithm within the phase field framework for simulating the motion of a mixture of N (N⩾2) immiscible incompressible fluids, with possibly very different physical properties such as densities, viscosities, and pairwise surface tensions. The algorithm employs a physical formulation for the N-phase system that honors the conservations of mass and momentum and the second law of thermodynamics. We present a method for uniquely determining the mixing energy density coefficients involved in the N-phase model based on the pairwise surface tensions among the N fluids. Our numerical algorithm has several attractive properties that make it computationally very efficient: (i) it has completely de-coupled the computations for different flow variables, and has also completely de-coupled the computations for the (N−1) phase field functions; (ii) the algorithm only requires the solution of linear algebraic systems after discretization, and no nonlinear algebraic solve is needed; (iii) for each flow variable the linear algebraic system involves only constant and time-independent coefficient matrices, which can be pre-computed during pre-processing, despite the variable density and variable viscosity of the N-phase mixture; (iv) within a time step the semi-discretized system involves only individual de-coupled Helmholtz-type (including Poisson) equations, despite the strongly-coupled phase–field system of fourth spatial order at the continuum level; (v) the algorithm is suitable for large density contrasts and large viscosity contrasts among the N fluids. Extensive numerical experiments have been presented for several problems involving multiple fluid phases, large density contrasts and large viscosity contrasts. In particular, we compare our simulations with the de Gennes theory, and demonstrate that our method produces physically accurate results for multiple fluid phases. We also demonstrate the significant and sometimes dramatic effects of the gravity, density ratios, pairwise surface tensions, and drop sizes on the N-phase configurations and dynamics. The numerical results show that the method developed herein is capable of dealing with N-phase systems with large density ratios, large viscosity ratios, and pairwise surface tensions, and that it can be a powerful tool for studying the interactions among multiple types of fluid interfaces.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2019

Implementation of Maxwell's equations in the reconstruction of the magnetic field in the g -2 storage ring

We present a method for implementing the constraints that are implied by Maxwell's equations in fits to measurements of the magnetic field in the muon storage ring of the $g - 2$ experiment. The method that we use makes use of toroidal-harmonic solutions of Laplace's equation. We point out that the fitting problem can be approximated well as a linear-algebra problem. We have devised an efficient algorithm for the linear-algebra problem that makes it possible to find a solution for 10 5 data points and 10 4 harmonics in less than an hour on a present-day desktop computer. We illustrate our method by applying it to some preliminary measurements of the magnetic field in the $g - 2$ storage ring.

43 PARTICLE ACCELERATORS↗

NASA NTRS1986

On recent advances and future research directions for computational fluid dynamics

This paper highlights some recent accomplishments regarding CFD numerical algorithm constructions for generation of discrete approximate solutions to classes of Reynolds-averaged Navier-Stokes equations. Following an overview of turbulent closure modeling, and development of appropriate conservation law systems, a Taylor weak-statement semi-discrete approximate solution algorithm is developed. Various forms for completion to the final linear algebra statement are cited, as are a range of candidate numerical linear algebra solution procedures. This development sequence emphasizes the key building blocks of a CFD RNS algorithm, including solution trial and test spaces, integration procedure and added numerical stability mechanisms. A range of numerical results are discussed focusing on key topics guiding future research directions.

Baker, A. J.↗

DOE OSTI2020

Toward a scalable robust security-constrained optimal power flow using a proximal projection bundle method

Robust security-constrained optimal power flow (rSCOPF) aims to find the worst-case contingencies of alternating current optimal power flow (ACOPF) in power systems. With the rise of GPU architectures on the upcoming supercomputer architectures, optimization algorithms that rely on sparse linear algebra and indefinite linear systems are becoming increasingly hard to solve efficiently (e.g. interior-point method). To address this we revisit a maximin optimization formulation of the rSCOPF and the single-level mixed-integer semidefinite programming (MISDP) reformulation, which is obtained by taking the Lagrangian relaxation of the inner minimization ACOPF problem. In this paper, we focus on the development of a proximal projection bundle method (PPBM) for solving continuous relaxation node subproblems of the MISDP problem, based primarily on the well-known alternating direction method of multipliers. Cutting planes reminiscent of bundle method ideas are also applied in coordination with updates of the proximal parameter. The cutting-plane method can generate a large number of linear inequalities, leading to a large scale but decomposable quadratic programming (QP) subproblem that is amenable to GPUs. We present the numerical results on the IEEE 30, 57, 118, and 300-bus systems by using our PBMM method. We discuss the main computational bottleneck of our method, which is the time taken to solve each iteration of a QP subproblem instance of the PPBM, and how GPU architectures can accelerate this solution process.

bundle method↗

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 OSTI2015

Fully-Implicit Orthogonal Reconstructed Discontinuous Petrov-Galerkin Method for Multiphysics Problems

This document is focussed on the development of reconstructed discontinuous Galerkin (RDG P n P m ) method, based on orthogonal basis functions. Orthogonality of basis functions is essential for enabling robust and efficient fully-implicit Newton-Krylov based time discretization. The method is designed for generic partial differential equations, including transient hyperbolic, parabolic and/or elliptic operators, which are attributed to many multiphysics problems. Our driving application of interest is additive manufacturing, which requires simulations of fluid-solid-gas systems, with compressibility effects and phase change (melting/ solidification). Thus, we demonstrate the method capabilities for solving nonlinear heat conduction equation coupled with multi-material Navier-Stokes based compressible fluid (in low Mach number limit) systems. We focus our attention to the method accuracy (in both time and space), as well as solvability of linear algebra involved in linear steps of the Newton-based solver. We demonstrate the accuracy and efficacy of the method, emphasizing the advantages from the orthogonality of basis functions, which makes better conditioning of underlying (approximate) Jacobian matrices, and rapid convergence of Krylov methods. iii

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2019

Approximate Green’s Function Coupled Cluster A Method Employing Effective Dimension Reduction

The Green’s function coupled cluster (GFCC) method, originally proposed in the early 1990s, is a powerful many-body tool for computing and analyzing the electronic structures of molecular structures and periodic systems, especially when electrons are strongly correlated. However, in order for the GFCC method to become an approach that can be routinely used in the electronic structure calculations, more robust linear algebraic techniques and approximations need to be employed to reduce its extremely high computational overhead. In our recent studies, we showed that the GFCC equation can be numerically solved by seeking solutions of multiple linear systems in the frequency regime of interest, which can be easily distributed in a massively parallel environment, and broaden the avenue for more state-of-the-art numerical linear algebraic tools associated with multiple linear systems to be kicked in to this field. In the present pursuit of this practice, we show a successful application of model order reduction (MOR) techniques in the GFCC framework. Briefly speaking, for a frequency regime of interest that requires a high resolution of spectral function, instead of solving GFCC linear equation in the original dimension for every single frequency point, an easily-solvable linear system model with reduced dimension can be built upon projecting the original GFCC linear system into a subspace, from which the interpolation and extrapolation can give a reasonable approximation to the results of the original GFCC linear equations. Here, we show that the subspace can be properly constructed in an iterative manner from the auxiliary vectors of the GFCC linear equations at some select frequencies inside the frequency regime. During the iterations, the quality of the subspace, as well as the linear system model, can be systematically improved. The approach is tested in this work in terms of the efficiency and accuracy of computing spectral functions for some typical molecular systems including carbon monoxide, 1,3-butadiene, benzene, and adenine molecules. As a byproduct, the high quality linear system model is also found to be able to provide high quality initial guess for the existing linear solver to improve the convergence performance.

Peng, Bo↗

DOE OSTI2024

Improved Evaluation of Large Network Matrices for Linear Power Flow Within Optimization Problems: Preprint

This work discusses methods for evaluating the Power Transfer Distribution Factor (PTDF) and Line Outage Distribution Factor (LODF) matrices by employing sparse linear algebra for large-scale computing applications. These matrices are critical in many power systems applications, such as the Unit Commitment Problem (UC), pre- and post-contingency power flow analysis, and transmission expansion. These matrices are typically dense, which means they require a significant amount of time and memory to be computed for large networks. However, by analyzing the structure of the matrices and their computation method, it is possible to use reduced memory methods based on sparse matrix operations. This paper shows that sparse linear algebra algorithms are faster and require less memory and time than traditional dense approaches. Additionally, we explore the effect of matrix sparsification by eliminating trailing digits on power flow calculations.

ENERGY PLANNING, POLICY, AND ECONOMY↗