Search NASA⌕ Search

SEARCH · Search NASA

Results for “Sparse matrix techniques”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 37 records · Page 2
All sourcesNASA NTRSDOE OSTI
NASA NTRS1996

Understanding and Improving High-Performance I/O Subsystems

This research program has been conducted in the framework of the NASA Earth and Space Science (ESS) evaluations led by Dr. Thomas Sterling. In addition to the many important research findings for NASA and the prestigious publications, the program has helped orienting the doctoral research program of two students towards parallel input/output in high-performance computing. Further, the experimental results in the case of the MasPar were very useful and helpful to MasPar with which the P.I. has had many interactions with the technical management. The contributions of this program are drawn from three experimental studies conducted on different high-performance computing testbeds/platforms, and therefore presented in 3 different segments as follows: 1. Evaluating the parallel input/output subsystem of a NASA high-performance computing testbeds, namely the MasPar MP- 1 and MP-2; 2. Characterizing the physical input/output request patterns for NASA ESS applications, which used the Beowulf platform; and 3. Dynamic scheduling techniques for hiding I/O latency in parallel applications such as sparse matrix computations. This study also has been conducted on the Intel Paragon and has also provided an experimental evaluation for the Parallel File System (PFS) and parallel input/output on the Paragon. This report is organized as follows. The summary of findings discusses the results of each of the aforementioned 3 studies. Three appendices, each containing a key scholarly research paper that details the work in one of the studies are included.

El-Ghazawi, Tarek A.↗
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↗
NASA NTRS2010

The OMPS Limb Profiler Instrument: Two-Dimensional Retrieval Algorithm

The upcoming Ozone Mapper and Profiler Suite (OMPS), which will be launched on the NPOESS Preparatory Project (NPP) platform in early 2011, will continue monitoring the global distribution of the Earth's middle atmosphere ozone and aerosol. OMPS is composed of three instruments, namely the Total Column Mapper (heritage: TOMS, OMI), the Nadir Profiler (heritage: SBUV) and the Limb Profiler (heritage: SOLSE/LORE, OSIRIS, SCIAMACHY, SAGE III). The ultimate goal of the mission is to better understand and quantify the rate of stratospheric ozone recovery. The focus of the paper will be on the Limb Profiler (LP) instrument. The LP instrument will measure the Earth's limb radiance (which is due to the scattering of solar photons by air molecules, aerosol and Earth surface) in the ultra-violet (UV), visible and near infrared, from 285 to 1000 nm. The LP simultaneously images the whole vertical extent of the Earth's limb through three vertical slits, each covering a vertical tangent height range of 100 km and each horizontally spaced by 250 km in the cross-track direction. Measurements are made every 19 seconds along the orbit track, which corresponds to a distance of about 150km. Several data analysis tools are presently being constructed and tested to retrieve ozone and aerosol vertical distribution from limb radiance measurements. The primary NASA algorithm is based on earlier algorithms developed for the SOLSE/LORE and SAGE III limb scatter missions. All the existing retrieval algorithms rely on a spherical symmetry assumption for the atmosphere structure. While this assumption is reasonable in most of the stratosphere, it is no longer valid in regions of prime scientific interest, such as polar vortex and UTLS regions. The paper will describe a two-dimensional retrieval algorithm whereby the ozone distribution is simultaneously retrieved vertically and horizontally for a whole orbit. The retrieval code relies on (1) a forward 2D Radiative Transfer code (to model limb radiances within a non-uniform atmosphere and evaluate 2D analytical partial derivatives) and (2) an optimal estimator inversion routine. The algorithm uses the typically sparse nature of the kernel matrices as well as fast matrix inversion techniques to allow for fast inversion of limb data with efficient memory management (as was done for MIPAS data processing). While the method has so far only been developed in the context of Single Scatter, the paper will show how the CPU intensive Multiple Scatter modeling can be implemented using parallel CPU processing. Initial results will be presented in terms of retrieved ozone profiles and code performance.

Rault, Didier F.↗
NASA NTRS1983

Preconditioners for the spectral multigrid method

The systems of algebraic equations which arise from spectral discretizations of elliptic equations are full and direct solutions of them are rarely feasible. Iterative methods are an attractive alternative because Fourier transform techniques enable the discrete matrix-vector products to be computed with nearly the same efficiency as is possible for corresponding but sparse finite difference discretizations. For realistic Dirichlet problems preconditioning is essential for acceptable convergence rates. A brief description of Chebyshev spectral approximations and spectral multigrid methods for elliptic problems is given. A survey of preconditioners for Dirichlet problems based on second-order finite difference methods is made. New preconditioning techniques based on higher order finite differences and on the spectral matrix itself are presented. The preconditioners are analyzed in terms of their spectra and numerical examples are presented.

Phillips, T. N.↗
NASA NTRS1986

Preconditioners for the spectral multigrid method

The systems of algebraic equations which arise from spectral discretizations of elliptic equations are full and direct solutions of them are rarely feasible. Iterative methods are an attractive alternative because Fourier transform techniques enable the discrete matrix-vector products to be computed with nearly the same efficiency as is possible for corresponding but sparse finite difference discretizations. For realistic Dirichlet problem preconditioning is essential for acceptable convergence rates. A brief description of Chebyshev spectral approximations and spectral multigrid methods for elliptic problems is given. A survey of preconditioners for Dirichlet problems based on second-order finite difference methods is made. New preconditioning techniques based on higher order finite differences and on the spectral matrix itself are presented. The preconditioners are analyzed in terms of their spectra and numerical examples are presented.

Phillips, T. N.↗
NASA NTRS2000

Parallel Conjugate Gradient: Effects of Ordering Strategies, Programming Paradigms, and Architectural Platforms

The Conjugate Gradient (CG) algorithm is perhaps the best-known iterative technique to solve sparse linear systems that are symmetric and positive definite. A sparse matrix-vector multiply (SPMV) usually accounts for most of the floating-point operations within a CG iteration. In this paper, we investigate the effects of various ordering and partitioning strategies on the performance of parallel CG and SPMV using different programming paradigms and architectures. Results show that for this class of applications, ordering significantly improves overall performance, that cache reuse may be more important than reducing communication, and that it is possible to achieve message passing performance using shared memory constructs through careful data ordering and distribution. However, a multi-threaded implementation of CG on the Tera MTA does not require special ordering or partitioning to obtain high efficiency and scalability.

Oliker, Leonid↗
NASA NTRS1995

Solution of the three-dimensional Helmholtz equation with nonlocal boundary conditions

The Helmholtz equation is solved within a three-dimensional rectangular duct with a nonlocal radiation boundary condition at the duct exit plane. This condition accurately models the acoustic admittance at an arbitrarily-located computational boundary plane. A linear system of equations is constructed with second-order central differences for the Helmholtz operator and second-order backward differences for both local admittance conditions and the gradient term in the nonlocal radiation boundary condition. The resulting matrix equation is large, sparse, and non-Hermitian. The size and structure of the matrix makes direct solution techniques impractical; as a result, a nonstationary iterative technique is used for its solution. The theory behind the nonstationary technique is reviewed, and numerical results are presented for radiation from both a point source and a planar acoustic source. The solutions with the nonlocal boundary conditions are invariant to the location of the computational boundary, and the same nonlocal conditions are valid for all solutions. The nonlocal conditions thus provide a means of minimizing the size of three-dimensional computational domains.

Hodge, Steve L.↗
NASA NTRS1993

Space station static and dynamic analyses using parallel methods

Algorithms for high-performance parallel computers are applied to perform static analyses of large-scale Space Station finite-element models (FEMs). Several parallel-vector algorithms under development at NASA Langley are assessed. Sparse matrix solvers were found to be more efficient than banded symmetric or iterative solvers for the static analysis of large-scale applications. In addition, new sparse and 'out-of-core' solvers were found superior to substructure (superelement) techniques which require significant additional cost and time to perform static condensation during global FEM matrix generation as well as the subsequent recovery and expansion. A method to extend the fast parallel static solution techniques to reduce the computation time for dynamic analysis is also described. The resulting static and dynamic algorithms offer design economy for preliminary multidisciplinary design optimization and FEM validation against test modes. The algorithms are being optimized for parallel computers to solve one-million degrees-of-freedom (DOF) FEMs. The high-performance computers at NASA afforded effective software development, testing, efficient and accurate solution with timely system response and graphical interpretation of results rarely found in industry. Based on the author's experience, similar cooperation between industry and government should be encouraged for similar large-scale projects in the future.

Gupta, V.↗
NASA NTRS1987

Relaxation schemes for spectral multigrid methods

The effectiveness of relaxation schemes for solving the systems of algebraic equations which arise from spectral discretizations of elliptic equations is examined. Iterative methods are an attractive alternative to direct methods because Fourier transform techniques enable the discrete matrix-vector products to be computed almost as efficiently as for corresponding but sparse finite difference discretizations. Preconditioning is found to be essential for acceptable rates of convergence. Preconditioners based on second-order finite difference methods are used. A comparison is made of the performance of different relaxation methods on model problems with a variety of conditions specified around the boundary. The investigations show that iterations based on incomplete LU decompositions provide the most efficient methods for solving these algebraic systems.

Phillips, Timothy N.↗
NASA NTRS2002

An Efficient Scheme for Updating Sparse Cholesky Factors

Raghavan had earlier developed the software package DCSPACK which can be used for solving sparse linear systems where the coefficient matrix is symmetric and positive definite (this project was not funded by NASA but by agencies such as NSF). DSCPACK-S is the serial code and DSCPACK-P is a parallel implementation suitable for multiprocessors or networks-of-workstations with message passing using MCI. The main algorithm used is the Cholesky factorization of a sparse symmetric positive positive definite matrix A = LL(T). The code can also compute the factorization A = LDL(T). The complexity of the software arises from several factors relating to the sparsity of the matrix A. A sparse N x N matrix A has typically less that cN nonzeroes where c is a small constant. If the matrix were dense, it would have O(N2) nonzeroes. The most complicated part of such sparse Cholesky factorization relates to fill-in, i.e., zeroes in the original matrix that become nonzeroes in the factor L. An efficient implementation depends to a large extent on complex data structures and on techniques from graph theory to reduce, identify, and manage fill. DSCPACK is based on an efficient multifrontal implementation with fill-managing algorithms and implementation arising from earlier research by Raghavan and others. Sparse Cholesky factorization is typically a four step process: (1) ordering to compute a fill-reducing numbering, (2) symbolic factorization to determine the nonzero structure of L, (3) numeric factorization to compute L, and, (4) triangular solution to solve L(T)x = y and Ly = b. The first two steps are symbolic and are performed using the graph of the matrix. The numeric factorization step is of dominant cost and there are several schemes for improving performance by exploiting the nested and dense structure of groups of columns in the factor. The latter are aimed at better utilization of the cache-memory hierarchy on modem processors to prevent cache-misses and provide execution rates (operations/second) that are close to the peak rates for dense matrix computations. Currently, EPISCOPACY is being used in an application at NASA directed by J. Newman and M. James. We propose the implementation of efficient schemes for updating the LL(T) or LDL(T) factors computed in DSCPACK-S to meet the computational requirements of their project. A brief description is provided in the next section.

Raghavan, Padma↗
NASA NTRS1989

Sparse Gaussian elimination with controlled fill-in on a shared memory multiprocessor

It is shown that in sparse matrices arising from electronic circuits, it is possible to do computations on many diagonal elements simultaneously. A technique for obtaining an ordered compatible set directly from the ordered incompatible table is given. The ordering is based on the Markowitz number of the pivot candidates. This technique generates a set of compatible pivots with the property of generating few fills. A novel heuristic algorithm is presented that combines the idea of an order-compatible set with a limited binary tree search to generate several sets of compatible pivots in linear time. An elimination set for reducing the matrix is generated and selected on the basis of a minimum Markowitz sum number. The parallel pivoting technique presented is a stepwise algorithm and can be applied to any submatrix of the original matrix. Thus, it is not a preordering of the sparse matrix and is applied dynamically as the decomposition proceeds. Parameters are suggested to obtain a balance between parallelism and fill-ins. Results of applying the proposed algorithms on several large application matrices using the HEP multiprocessor (Kowalik, 1985) are presented and analyzed.

Alaghband, Gita↗
NASA NTRSDate not supplied

Performance Optimization Methods for a Memory-Bound, Unstructured-Grid CFD Application on Massively Parallel GPU Platforms

Computational performance of the FUN3D unstructured-grid computational fluid dynamics (CFD) application on massively parallel GPU environments is memory-bound and highly dependent upon efficient reads from and atomic updates to the irregular cell-, edge-, and node-based data structures. In this talk, we present recent efforts into optimizing select performance-critical kernels on NVIDIA Tesla V100 and A100 GPUs and AMD CDNA MI100 GPUs. A novel use of L2 cache residency controls and asynchronous loads into on-chip shared memory are explored on the A100 GPU for the sparse iterative solver, which is dominated by mixed-precision, sparse matrix vector multiplication. Demonstrations show that these methods improve global memory bandwidth utilization by 13.5% on the A100 GPU. Several techniques are also presented that use registers and/or shared memory to facilitate array transposition and aggregation which combine to reduce the frequency and increase the cache efficiency of floating-point atomic updates to the irregular data structures. These methods are demonstrated to improve the kernel throughput by nearly 500% on select kernels on the AMD MI100 over atomic updates directly to global memory. Overall, both V100 and A100 GPUs outperformed the MI100 GPU on kernels dominated by double-precision atomic updates; however, the techniques demonstrated here reduced the performance gap and improved the MI100 performance.

GPU CPU unstructured CFD memory↗
NASA NTRS1978

Substructuring techniques - Status and projections

Substructuring techniques are examined in terms of their application to structural analysis. Attention is given to multilevel substructuring algorithms, hypermatrix and other sparse matrix schemes, automated design systems, and elasto-plastic problems. Applications include use with computing hardware such as CDC STAR-100 and minicomputer systems.

Noor, A. K.↗
NASA NTRS1989

A block-corrected subdomain solution procedure for recirculating flow calculations

This paper describes a robust and efficient subdomain solution procedure for two-dimensional recirculating flows. The solution domain is divided into a number of overlapping subdomains, and a direct fully coupled solution is obtained for each subdomain using a sparse matrix form of LU decomposition. An effective parabolic block correction procedure, which calculates global corrections to the tentative solution by a marching technique similar to that used for boundary layer flows, is used to accelerate the convergence of the basic procedure. The use of effective block correction is found to be essential for the success of the subdomain approach on strongly recirculating flows. In a number of laminar two-dimensional flows, the new block-corrected method performed extremely well, rivaling the best direct methods in execution time, while requiring substantially less computer storage. The new method proved to be from two to ten times faster than conventional iterative methods, while requiring only a moderate increase in storage.

Braaten, M. E.↗
NASA NTRS2011

Efficient Kriging Algorithms

More efficient versions of an interpolation method, called kriging, have been introduced in order to reduce its traditionally high computational cost. Written in C++, these approaches were tested on both synthetic and real data. Kriging is a best unbiased linear estimator and suitable for interpolation of scattered data points. Kriging has long been used in the geostatistic and mining communities, but is now being researched for use in the image fusion of remotely sensed data. This allows a combination of data from various locations to be used to fill in any missing data from any single location. To arrive at the faster algorithms, sparse SYMMLQ iterative solver, covariance tapering, Fast Multipole Methods (FMM), and nearest neighbor searching techniques were used. These implementations were used when the coefficient matrix in the linear system is symmetric, but not necessarily positive-definite.

Memarsadeghi, Nargess↗
NASA NTRS1989

Low-order design and high-order simulation of active closed-loop control for aerospace structures under construction

Partially constructed/assembled structures in space are complicated enough but their dynamics will also be operating in closed-loop with feedback controllers. The dynamics of such structures are modeled by large-scale finite element models. The model dimension L is extremely large (approximately 10,000) while the numbers of actuators (M) and sensors (P) are small. The model parameters M(sub m) mass matrix, D(sub o) damping matrix, and K(sub o) stiffness matrix, are all symmetric and sparse (banded). Thus simulation of open-loop structure models of very large dimension can be accomplished by special integration techniques for sparse matrices. The problem of simulation of closed-loop control of such structures is complicated by the addition of controllers. Simulation of closed-loop controlled structures is an essential part of the controller design and evaluation process. Current research in the following areas is presented: high-order simulation of actively controlled aerospace structures; low-order controller design and SCI compensation for unmodeled dynamics; prediction of closed-loop stability using asymptotic eigenvalue series; and flexible robot manipulator control experiment.

Balas, Mark J.↗
NASA NTRS2000

Ordering Unstructured Meshes for Sparse Matrix Computations on Leading Parallel Systems

The ability of computers to solve hitherto intractable problems and simulate complex processes using mathematical models makes them an indispensable part of modern science and engineering. Computer simulations of large-scale realistic applications usually require solving a set of non-linear partial differential equations (PDES) over a finite region. For example, one thrust area in the DOE Grand Challenge projects is to design future accelerators such as the SpaHation Neutron Source (SNS). Our colleagues at SLAC need to model complex RFQ cavities with large aspect ratios. Unstructured grids are currently used to resolve the small features in a large computational domain; dynamic mesh adaptation will be added in the future for additional efficiency. The PDEs for electromagnetics are discretized by the FEM method, which leads to a generalized eigenvalue problem Kx = AMx, where K and M are the stiffness and mass matrices, and are very sparse. In a typical cavity model, the number of degrees of freedom is about one million. For such large eigenproblems, direct solution techniques quickly reach the memory limits. Instead, the most widely-used methods are Krylov subspace methods, such as Lanczos or Jacobi-Davidson. In all the Krylov-based algorithms, sparse matrix-vector multiplication (SPMV) must be performed repeatedly. Therefore, the efficiency of SPMV usually determines the eigensolver speed. SPMV is also one of the most heavily used kernels in large-scale numerical simulations.

Oliker, Leonid↗
NASA NTRS2012

A Sparse Matrix Approach for Simultaneous Quantification of Nystagmus and Saccade

The vestibulo-ocular reflex (VOR) consists of two intermingled non-linear subsystems; namely, nystagmus and saccade. Typically, nystagmus is analysed using a single sufficiently long signal or a concatenation of them. Saccade information is not analysed and discarded due to insufficient data length to provide consistent and minimum variance estimates. This paper presents a novel sparse matrix approach to system identification of the VOR. It allows for the simultaneous estimation of both nystagmus and saccade signals. We show via simulation of the VOR that our technique provides consistent and unbiased estimates in the presence of output additive noise.

Kukreja, Sunil L.↗