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At least 145 records · Page 8

All sources NASA NTRS DOE OSTI

DOE OSTI2021

Online Voltage Event Detection Using Synchrophasor Data with Structured Sparsity-Inducing Norms

This paper develops an accurate and computationally efficient data-driven framework to detect voltage events from PMU data streams. It develops an innovative Proximal Bilateral Random Projection (PBRP) algorithm to quickly decompose the PMU data matrix into a low-rank matrix, a row-sparse event-pattern matrix and a noise matrix. Here, the row-sparse pattern matrix significantly distinguishes events from normal behavior. These matrices are then fed into a clustering algorithm to separate voltage events from normal operating conditions. Large-scale numerical study results on real-world PMU data show that the proposed algorithm is computationally more efficient and achieves higher F scores than state-of-the-art benchmarks.

24 POWER TRANSMISSION AND DISTRIBUTION↗

NASA NTRS1976

An iterative decoupling solution method for large scale Lyapunov equations

A great deal of attention has been given to the numerical solution of the Lyapunov equation. A useful classification of the variety of solution techniques are the groupings of direct, transformation, and iterative methods. The paper summarizes those methods that are at least partly favorable numerically, giving special attention to two criteria: exploitation of a general sparse system matrix structure and efficiency in resolving the governing linear matrix equation for different matrices. An iterative decoupling solution method is proposed as a promising approach for solving large-scale Lyapunov equation when the system matrix exhibits a general sparse structure. A Fortran computer program that realizes the iterative decoupling algorithm is also discussed.

Athay, T. M.↗

NASA NTRS1993

System identification and model reduction using modulating function techniques

Weighted least squares (WLS) and adaptive weighted least squares (AWLS) algorithms are initiated for continuous-time system identification using Fourier type modulating function techniques. Two stochastic signal models are examined using the mean square properties of the stochastic calculus: an equation error signal model with white noise residuals, and a more realistic white measurement noise signal model. The covariance matrices in each model are shown to be banded and sparse, and a joint likelihood cost function is developed which links the real and imaginary parts of the modulated quantities. The superior performance of above algorithms is demonstrated by comparing them with the LS/MFT and popular predicting error method (PEM) through 200 Monte Carlo simulations. A model reduction problem is formulated with the AWLS/MFT algorithm, and comparisons are made via six examples with a variety of model reduction techniques, including the well-known balanced realization method. Here the AWLS/MFT algorithm manifests higher accuracy in almost all cases, and exhibits its unique flexibility and versatility. Armed with this model reduction, the AWLS/MFT algorithm is extended into MIMO transfer function system identification problems. The impact due to the discrepancy in bandwidths and gains among subsystem is explored through five examples. Finally, as a comprehensive application, the stability derivatives of the longitudinal and lateral dynamics of an F-18 aircraft are identified using physical flight data provided by NASA. A pole-constrained SIMO and MIMO AWLS/MFT algorithm is devised and analyzed. Monte Carlo simulations illustrate its high-noise rejecting properties. Utilizing the flight data, comparisons among different MFT algorithms are tabulated and the AWLS is found to be strongly favored in almost all facets.

Shen, Yan↗

NASA NTRS1988

Domain decomposition methods for the parallel computation of reacting flows

Domain decomposition is a natural route to parallel computing for partial differential equation solvers. Subdomains of which the original domain of definition is comprised are assigned to independent processors at the price of periodic coordination between processors to compute global parameters and maintain the requisite degree of continuity of the solution at the subdomain interfaces. In the domain-decomposed solution of steady multidimensional systems of PDEs by finite difference methods using a pseudo-transient version of Newton iteration, the only portion of the computation which generally stands in the way of efficient parallelization is the solution of the large, sparse linear systems arising at each Newton step. For some Jacobian matrices drawn from an actual two-dimensional reacting flow problem, comparisons are made between relaxation-based linear solvers and also preconditioned iterative methods of Conjugate Gradient and Chebyshev type, focusing attention on both iteration count and global inner product count. The generalized minimum residual method with block-ILU preconditioning is judged the best serial method among those considered, and parallel numerical experiments on the Encore Multimax demonstrate for it approximately 10-fold speedup on 16 processors.

Keyes, David E.↗

NASA NTRS1991

Krylov Subspace Methods for Complex Non-Hermitian Linear Systems

We consider Krylov subspace methods for the solution of large sparse linear systems Ax = b with complex non-Hermitian coefficient matrices. Such linear systems arise in important applications, such as inverse scattering, numerical solution of time-dependent Schrodinger equations, underwater acoustics, eddy current computations, numerical computations in quantum chromodynamics, and numerical conformal mapping. Typically, the resulting coefficient matrices A exhibit special structures, such as complex symmetry, or they are shifted Hermitian matrices. In this paper, we first describe a Krylov subspace approach with iterates defined by a quasi-minimal residual property, the QMR method, for solving general complex non-Hermitian linear systems. Then, we study special Krylov subspace methods designed for the two families of complex symmetric respectively shifted Hermitian linear systems. We also include some results concerning the obvious approach to general complex linear systems by solving equivalent real linear systems for the real and imaginary parts of x. Finally, numerical experiments for linear systems arising from the complex Helmholtz equation are reported.

Freund, Roland W.↗

DOE OSTI2023

Sparse Approximate Multifrontal Factorization with Composite Compression Methods

This article presents a fast and approximate multifrontal solver for large sparse linear systems. In a recent work by Liu et al., we showed the efficiency of a multifrontal solver leveraging the butterfly algorithm and its hierarchical matrix extension, HODBF (hierarchical off-diagonal butterfly) compression to compress large frontal matrices. The resulting multifrontal solver can attain quasi-linear computation and memory complexity when applied to sparse linear systems arising from spatial discretization of high-frequency wave equations. To further reduce the overall number of operations and especially the factorization memory usage to scale to larger problem sizes, in this article we develop a composite multifrontal solver that employs the HODBF format for large-sized fronts, a reduced-memory version of the nonhierarchical block low-rank format for medium-sized fronts, and a lossy compression format for small-sized fronts. This allows us to solve sparse linear systems of dimension up to 2.7 × larger than before and leads to a memory consumption that is reduced by 70% while ensuring the same execution time. The code is made publicly available in GitHub.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1983

Progress on a generalized coordinates tensor product finite element 3DPNS algorithm for subsonic

A generalized coordinates form of the penalty finite element algorithm for the 3-dimensional parabolic Navier-Stokes equations for turbulent subsonic flows was derived. This algorithm formulation requires only three distinct hypermatrices and is applicable using any boundary fitted coordinate transformation procedure. The tensor matrix product approximation to the Jacobian of the Newton linear algebra matrix statement was also derived. Tne Newton algorithm was restructured to replace large sparse matrix solution procedures with grid sweeping using alpha-block tridiagonal matrices, where alpha equals the number of dependent variables. Numerical experiments were conducted and the resultant data gives guidance on potentially preferred tensor product constructions for the penalty finite element 3DPNS algorithm.

Baker, A. J.↗

NASA NTRS1990

Constraint treatment techniques and parallel algorithms for multibody dynamic analysis

Computational procedures for kinematic and dynamic analysis of three-dimensional multibody dynamic (MBD) systems are developed from the differential-algebraic equations (DAE's) viewpoint. Constraint violations during the time integration process are minimized and penalty constraint stabilization techniques and partitioning schemes are developed. The governing equations of motion, a two-stage staggered explicit-implicit numerical algorithm, are treated which takes advantage of a partitioned solution procedure. A robust and parallelizable integration algorithm is developed. This algorithm uses a two-stage staggered central difference algorithm to integrate the translational coordinates and the angular velocities. The angular orientations of bodies in MBD systems are then obtained by using an implicit algorithm via the kinematic relationship between Euler parameters and angular velocities. It is shown that the combination of the present solution procedures yields a computationally more accurate solution. To speed up the computational procedures, parallel implementation of the present constraint treatment techniques, the two-stage staggered explicit-implicit numerical algorithm was efficiently carried out. The DAE's and the constraint treatment techniques were transformed into arrowhead matrices to which Schur complement form was derived. By fully exploiting the sparse matrix structural analysis techniques, a parallel preconditioned conjugate gradient numerical algorithm is used to solve the systems equations written in Schur complement form. A software testbed was designed and implemented in both sequential and parallel computers. This testbed was used to demonstrate the robustness and efficiency of the constraint treatment techniques, the accuracy of the two-stage staggered explicit-implicit numerical algorithm, and the speed up of the Schur-complement-based parallel preconditioned conjugate gradient algorithm on a parallel computer.

Chiou, Jin-Chern↗

NASA NTRS1988

On the constrained Chebyshev approximation problem on ellipses

Constrained Chebyshev approximation problems of the type with minimum (p is an element of Pi(sub n):p(c)=1) and maximum (z is an element of E) with /p(z)/ are considered. Here Pi(sub n) denotes the set of all complex polynomials of degree at most n, E is any ellipse in the complex plane, and c is an element of C/E. Such approximation problems arise in the context of optimizing semi-iterative methods for the solution of large, sparse systems of linear equations Ax=b with complex non-Hermitian coefficient matrices A. The problem of obtaining optimal polynomial preconditioners for conjugate gradient type methods for Ax=b also leads to problems of this type. A new family of polynomials -- q(sub n)(z;c), n is an element of N, and c is an element of C/E -- are introduced as the polynomials which are optimal for a modified version of the Chebyshev approximation problem with Pi(sub n) replaced by a certain subfamily. Some simple properties of q(sub n) are also listed. A necessary and sufficient condition for q(sub n) to be the extremal polynomial for the approximation problem is then derived. Finally, it is shown that q(sub n) is indeed optimal for the problem for all fixed n whenever the distance between c and E is sufficiently large. Results of some numerical tests are presented.

Freund, Roland↗

DOE OSTI2023

Improved quantum algorithms for linear and nonlinear differential equations

We present substantially generalized and improved quantum algorithms over prior work for inhomogeneous linear and nonlinear ordinary differential equations (ODE). Specifically, we show how the norm of the matrix exponential characterizes the run time of quantum algorithms for linear ODEs opening the door to an application to a wider class of linear and nonlinear ODEs. In [1], a quantum algorithm for a certain class of linear ODEs is given, where the matrix involved needs to be diagonalizable. The quantum algorithm for linear ODEs presented here extends to many classes of non-diagonalizable matrices including singular matrices. The algorithm here is also exponentially faster than the bounds derived in [1] for certain classes of diagonalizable matrices. Our linear ODE algorithm is then applied to nonlinear differential equations using Carleman linearization (an approach taken recently by us in [2]). The improvement over that result is two-fold. First, we obtain an exponentially better dependence on error. This kind of logarithmic dependence on error has also been achieved by [3], but only for homogeneous nonlinear equations. Second, the present algorithm can handle any sparse matrix (that models dissipation) if it has a negative log-norm (including non-diagonalizable matrices), whereas [2] and [3] additionally require normality.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1990

Implementation of two projection methods on a shared memory multiprocessor - DEC VAX 6240

The relative performance of two iterative schemes, based on projection techniques, is compared on a shared memory multiprocessor - VAX 6240. The CG accelerated Block-SSOR method and the CG accelerated Symmetric-Kaczmarz method are considered for the solution of large sparse nonsymmetric systems of linear equations. It is shown that the regular structure of many matrices can be exploited by the CG-accelerated Block-SSOR method to provide good speedup in a multiprocessing environment. However, the CG accelerated Symmetric-Kaczmarz method, while being a viable alternative on a scalar machine, is unable to benefit from multiprocessing.

Kamath, C.↗

DOE OSTI2021

True Load Balancing for Matricized Tensor Times Khatri-Rao Product

MTTKRP is the bottleneck operation in algorithms used to compute the CP tensor decomposition. For sparse tensors, utilizing the compressed sparse fibers (CSF) storage format and the CSF-oriented MTTKRP algorithms is important for both memory and computational efficiency on distributed-memory architectures. Existing intelligent tensor partitioning models assume the computational cost of MTTKRP to be proportional to the total number of nonzeros in the tensor. However, this is not the case for the CSF-oriented MTTKRP on distributed-memory architectures. We outline two deficiencies of nonzero-based intelligent partitioning models when CSF-oriented MTTKRP operations are performed locally: failure to encode processors' computational loads and increase in total computation due to fiber fragmentation. We focus on existing fine-grain hypergraph model and propose a novel vertex weighting scheme that enables this model encode correct computational loads of processors. We also propose to augment the fine-grain model by fiber nets for reducing the increase in total computational load via minimizing fiber fragmentation. In this way, the proposed model encodes minimizing the load of the bottleneck processor. In conclusion, parallel experiments with real-world sparse tensors on up to 1024 processors prove the validity of the outlined deficiencies and demonstrate the merit of our proposed improvements in terms of parallel runtimes.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2005

Parallel Finite Element Domain Decomposition for Structural/Acoustic Analysis

A domain decomposition (DD) formulation for solving sparse linear systems of equations resulting from finite element analysis is presented. The formulation incorporates mixed direct and iterative equation solving strategics and other novel algorithmic ideas that are optimized to take advantage of sparsity and exploit modern computer architecture, such as memory and parallel computing. The most time consuming part of the formulation is identified and the critical roles of direct sparse and iterative solvers within the framework of the formulation are discussed. Experiments on several computer platforms using several complex test matrices are conducted using software based on the formulation. Small-scale structural examples are used to validate thc steps in the formulation and large-scale (l,000,000+ unknowns) duct acoustic examples are used to evaluate the ORIGIN 2000 processors, and a duster of 6 PCs (running under the Windows environment). Statistics show that the formulation is efficient in both sequential and parallel computing environmental and that the formulation is significantly faster and consumes less memory than that based on one of the best available commercialized parallel sparse solvers.

Nguyen, Duc T.↗

NASA NTRS1992

Sparse distributed memory and related models

Described here is sparse distributed memory (SDM) as a neural-net associative memory. It is characterized by two weight matrices and by a large internal dimension - the number of hidden units is much larger than the number of input or output units. The first matrix, A, is fixed and possibly random, and the second matrix, C, is modifiable. The SDM is compared and contrasted to (1) computer memory, (2) correlation-matrix memory, (3) feet-forward artificial neural network, (4) cortex of the cerebellum, (5) Marr and Albus models of the cerebellum, and (6) Albus' cerebellar model arithmetic computer (CMAC). Several variations of the basic SDM design are discussed: the selected-coordinate and hyperplane designs of Jaeckel, the pseudorandom associative neural memory of Hassoun, and SDM with real-valued input variables by Prager and Fallside. SDM research conducted mainly at the Research Institute for Advanced Computer Science (RIACS) in 1986-1991 is highlighted.

Kanerva, Pentti↗

NASA NTRS1988

Solving large sparse eigenvalue problems on supercomputers

An important problem in scientific computing consists in finding a few eigenvalues and corresponding eigenvectors of a very large and sparse matrix. The most popular methods to solve these problems are based on projection techniques on appropriate subspaces. The main attraction of these methods is that they only require the use of the matrix in the form of matrix by vector multiplications. The implementations on supercomputers of two such methods for symmetric matrices, namely Lanczos' method and Davidson's method are compared. Since one of the most important operations in these two methods is the multiplication of vectors by the sparse matrix, methods of performing this operation efficiently are discussed. The advantages and the disadvantages of each method are compared and implementation aspects are discussed. Numerical experiments on a one processor CRAY 2 and CRAY X-MP are reported. Possible parallel implementations are also discussed.

Philippe, Bernard↗

DOE OSTI2024

Accelerated Constrained Sparse Tensor Factorization on Massively Parallel Architectures

This study presents the first constrained sparse tensor factorization (cSTF) framework that optimizes and fully offloads computation to massively parallel GPU architectures, and the first performance characterization of cSTF on GPU architectures. In contrast to prior work on tensor factorization, where the matricized tensor times Khatri-Rao product (MTTKRP) is the primary performance bottleneck, our systematic analysis of the cSTF algorithm on GPUs reveals that adding constraints creates an additional bottleneck in the update operation for many real-world sparse tensors. While executing the update operation on the GPU brings significant speedup over its CPU counterpart, it remains a significant bottleneck. To further accelerate the update operation, we propose cuADMM, a new update algorithm that leverages algorithmic and code optimization strategies to minimize both computation and data movement on GPUs. As a result, our framework delivers significantly improved performance compared to prior state-of-the-art. On 10 real-world sparse tensors, our framework achieves geometric mean speedup of 5.1 × (max 41.59 ×) and 7.01 × (max 58.05 ×) on the NIVIDA A100 and H100 GPUs, respectively, over the state-of-the-art SPLATT library running on a 26-core Intel Ice Lake Xeon CPU.

Soh, Yongseok↗

DOE OSTI2021

Linear solvers for power grid optimization problems: A review of GPU-accelerated linear solvers

The linear equations that arise in interior methods for constrained optimization are sparse symmetric indefinite, and they become extremely ill-conditioned as the interior method converges. These linear systems present a challenge for existing solver frameworks based on sparse LU or LDL T decompositions. Here, we benchmark five well known direct linear solver packages on CPU- and GPU-based hardware, using matrices extracted from power grid optimization problems. The achieved solution accuracy varies greatly among the packages. None of the tested packages delivers significant GPU acceleration for our test cases. For completeness of the comparison we include results for MA57, which is one of the most efficient and reliable CPU solvers for this class of problem.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Sparse Approximate Multifrontal Factorization with Butterfly Compression for High-Frequency Wave Equations

In this work, we present a fast and approximate multifrontal solver for large-scale sparse linear systems arising from finite-difference, finite-volume or finite-element discretization of high-frequency wave equations. The proposed solver leverages the butterfly algorithm and its hierarchical matrix extension for compressing and factorizing large frontal matrices via graph-distance guided entry evaluation or randomized matrix-vector multiplication-based schemes. Complexity analysis and numerical experiments demonstrate $\mathcal{O}(N\log^2 N)$ computation and $\mathcal{O}(N)$ memory complexity when applied to an $N\times N$ sparse system arising from 3D high-frequency Helmholtz and Maxwell problems.

97 MATHEMATICS AND COMPUTING↗