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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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NASA NTRS1992

A performance study of sparse Cholesky factorization on INTEL iPSC/860

The problem of Cholesky factorization of a sparse matrix has been very well investigated on sequential machines. A number of efficient codes exist for factorizing large unstructured sparse matrices. However, there is a lack of such efficient codes on parallel machines in general, and distributed machines in particular. Some of the issues that are critical to the implementation of sparse Cholesky factorization on a distributed memory parallel machine are ordering, partitioning and mapping, load balancing, and ordering of various tasks within a processor. Here, we focus on the effect of various partitioning schemes on the performance of sparse Cholesky factorization on the Intel iPSC/860. Also, a new partitioning heuristic for structured as well as unstructured sparse matrices is proposed, and its performance is compared with other schemes.

Zubair, M.↗
NASA NTRS1990

Highly parallel sparse Cholesky factorization

Several fine grained parallel algorithms were developed and compared to compute the Cholesky factorization of a sparse matrix. The experimental implementations are on the Connection Machine, a distributed memory SIMD machine whose programming model conceptually supplies one processor per data element. In contrast to special purpose algorithms in which the matrix structure conforms to the connection structure of the machine, the focus is on matrices with arbitrary sparsity structure. The most promising algorithm is one whose inner loop performs several dense factorizations simultaneously on a 2-D grid of processors. Virtually any massively parallel dense factorization algorithm can be used as the key subroutine. The sparse code attains execution rates comparable to those of the dense subroutine. Although at present architectural limitations prevent the dense factorization from realizing its potential efficiency, it is concluded that a regular data parallel architecture can be used efficiently to solve arbitrarily structured sparse problems. A performance model is also presented and it is used to analyze the algorithms.

Gilbert, John R.↗
NASA NTRS1989

Communication requirements of sparse Cholesky factorization with nested dissection ordering

Load distribution schemes for minimizing the communication requirements of the Cholesky factorization of dense and sparse, symmetric, positive definite matrices on multiprocessor systems are presented. The total data traffic in factoring an n x n sparse symmetric positive definite matrix representing an n-vertex regular two-dimensional grid graph using n exp alpha, alpha not greater than 1, processors are shown to be O(n exp 1 + alpha/2). It is O(n), when n exp alpha, alpha not smaller than 1, processors are used. Under the conditions of uniform load distribution, these results are shown to be asymptotically optimal.

Naik, Vijay K.↗
NASA NTRS1988

Data traffic reduction schemes for sparse Cholesky factorizations

Load distribution schemes are presented which minimize the total data traffic in the Cholesky factorization of dense and sparse, symmetric, positive definite matrices on multiprocessor systems with local and shared memory. The total data traffic in factoring an n x n sparse, symmetric, positive definite matrix representing an n-vertex regular 2-D grid graph using n (sup alpha), alpha is equal to or less than 1, processors are shown to be O(n(sup 1 + alpha/2)). It is O(n(sup 3/2)), when n (sup alpha), alpha is equal to or greater than 1, processors are used. Under the conditions of uniform load distribution, these results are shown to be asymptotically optimal. The schemes allow efficient use of up to O(n) processors before the total data traffic reaches the maximum value of O(n(sup 3/2)). The partitioning employed within the scheme, allows a better utilization of the data accessed from shared memory than those of previously published methods.

Naik, Vijay K.↗
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 NTRS1994

Improved load distribution in parallel sparse Cholesky factorization

Compared to the customary column-oriented approaches, block-oriented, distributed-memory sparse Cholesky factorization benefits from an asymptotic reduction in interprocessor communication volume and an asymptotic increase in the amount of concurrency that is exposed in the problem. Unfortunately, block-oriented approaches (specifically, the block fan-out method) have suffered from poor balance of the computational load. As a result, achieved performance can be quite low. This paper investigates the reasons for this load imbalance and proposes simple block mapping heuristics that dramatically improve it. The result is a roughly 20% increase in realized parallel factorization performance, as demonstrated by performance results from an Intel Paragon system. We have achieved performance of nearly 3.2 billion floating point operations per second with this technique on a 196-node Paragon system.

Rothberg, Edward↗
NASA NTRS1991

Effects of partitioning and scheduling sparse matrix factorization on communication and load balance

A block based, automatic partitioning and scheduling methodology is presented for sparse matrix factorization on distributed memory systems. Using experimental results, this technique is analyzed for communication and load imbalance overhead. To study the performance effects, these overheads were compared with those obtained from a straightforward 'wrap mapped' column assignment scheme. All experimental results were obtained using test sparse matrices from the Harwell-Boeing data set. The results show that there is a communication and load balance tradeoff. The block based method results in lower communication cost whereas the wrap mapped scheme gives better load balance.

Venugopal, Sesh↗
NASA NTRS2006

Quantifying Global Uncertainties in a Simple Microwave Rainfall Algorithm

While a large number of methods exist in the literature for retrieving rainfall from passive microwave brightness temperatures, little has been written about the quantitative assessment of the expected uncertainties in these rainfall products at various time and space scales. The latter is the result of two factors: sparse validation sites over most of the world's oceans, and algorithm sensitivities to rainfall regimes that cause inconsistencies against validation data collected at different locations. To make progress in this area, a simple probabilistic algorithm is developed. The algorithm uses an a priori database constructed from the Tropical Rainfall Measuring Mission (TRMM) radar data coupled with radiative transfer computations. Unlike efforts designed to improve rainfall products, this algorithm takes a step backward in order to focus on uncertainties. In addition to inversion uncertainties, the construction of the algorithm allows errors resulting from incorrect databases, incomplete databases, and time- and space-varying databases to be examined. These are quantified. Results show that the simple algorithm reduces errors introduced by imperfect knowledge of precipitation radar (PR) rain by a factor of 4 relative to an algorithm that is tuned to the PR rainfall. Database completeness does not introduce any additional uncertainty at the global scale, while climatologically distinct space/time domains add approximately 25% uncertainty that cannot be detected by a radiometer alone. Of this value, 20% is attributed to changes in cloud morphology and microphysics, while 5% is a result of changes in the rain/no-rain thresholds. All but 2%-3% of this variability can be accounted for by considering the implicit assumptions in the algorithm. Additional uncertainties introduced by the details of the algorithm formulation are not quantified in this study because of the need for independent measurements that are beyond the scope of this paper. A validation strategy for these errors is outlined.

Kummerow, Christian↗
NASA NTRS1977

Preserving sparseness in multivariate polynominal factorization

Attempts were made to factor these ten polynomials on MACSYMA. However it did not get very far with any of the larger polynomials. At that time, MACSYMA used an algorithm created by Wang and Rothschild. This factoring algorithm was also implemented for the symbolic manipulation system, SCRATCHPAD of IBM. A closer look at this old factoring algorithm revealed three problem areas, each of which contribute to losing sparseness and intermediate expression growth. This study led to effective ways of avoiding these problems and actually to a new factoring algorithm. The three problems are known as the extraneous factor problem, the leading coefficient problem, and the bad zero problem. These problems are examined separately. Their causes and effects are set forth in detail; the ways to avoid or lessen these problems are described.

Wang, P. S.↗
NASA NTRS1990

A Shifted Block Lanczos Algorithm 1: The Block Recurrence

In this paper we describe a block Lanczos algorithm that is used as the key building block of a software package for the extraction of eigenvalues and eigenvectors of large sparse symmetric generalized eigenproblems. The software package comprises: a version of the block Lanczos algorithm specialized for spectrally transformed eigenproblems; an adaptive strategy for choosing shifts, and efficient codes for factoring large sparse symmetric indefinite matrices. This paper describes the algorithmic details of our block Lanczos recurrence. This uses a novel combination of block generalizations of several features that have only been investigated independently in the past. In particular new forms of partial reorthogonalization, selective reorthogonalization and local reorthogonalization are used, as is a new algorithm for obtaining the M-orthogonal factorization of a matrix. The heuristic shifting strategy, the integration with sparse linear equation solvers and numerical experience with the code are described in a companion paper.

Grimes, Roger G.↗
NASA NTRS1990

Partitioning sparse matrices with eigenvectors of graphs

The problem of computing a small vertex separator in a graph arises in the context of computing a good ordering for the parallel factorization of sparse, symmetric matrices. An algebraic approach for computing vertex separators is considered in this paper. It is shown that lower bounds on separator sizes can be obtained in terms of the eigenvalues of the Laplacian matrix associated with a graph. The Laplacian eigenvectors of grid graphs can be computed from Kronecker products involving the eigenvectors of path graphs, and these eigenvectors can be used to compute good separators in grid graphs. A heuristic algorithm is designed to compute a vertex separator in a general graph by first computing an edge separator in the graph from an eigenvector of the Laplacian matrix, and then using a maximum matching in a subgraph to compute the vertex separator. Results on the quality of the separators computed by the spectral algorithm are presented, and these are compared with separators obtained from other algorithms for computing separators. Finally, the time required to compute the Laplacian eigenvector is reported, and the accuracy with which the eigenvector must be computed to obtain good separators is considered. The spectral algorithm has the advantage that it can be implemented on a medium-size multiprocessor in a straightforward manner.

Pothen, Alex↗
NASA NTRS1990

Using a multifrontal sparse solver in a high performance, finite element code

We consider the performance of the finite element method on a vector supercomputer. The computationally intensive parts of the finite element method are typically the individual element forms and the solution of the global stiffness matrix both of which are vectorized in high performance codes. To further increase throughput, new algorithms are needed. We compare a multifrontal sparse solver to a traditional skyline solver in a finite element code on a vector supercomputer. The multifrontal solver uses the Multiple-Minimum Degree reordering heuristic to reduce the number of operations required to factor a sparse matrix and full matrix computational kernels (e.g., BLAS3) to enhance vector performance. The net result in an order-of-magnitude reduction in run time for a finite element application on one processor of a Cray X-MP.

King, Scott D.↗
NASA NTRS1992

Towards a fast implementation of spectral nested dissection

We describe the spectral nested dissection (SND) algorithm, a new algorithm for computing orderings appropriate for parallel factorization of sparse, symmetric matrices. The algorithm makes use of spectral properties of the Laplacian matrix associated with the given matrix to compute separators. We evaluate the quality of the spectral orderings with respect to several measures: fill, elimination tree height, height and weight balances of elimination trees, and clique tree heights. We use some very large structural analysis problems as test cases and demonstrate on these real applications (such as the Space Shuttle Solid Rocket Booster) that spectral orderings compare quite favorably with commonly used orderings, outperforming them by a wide margin for some of these measures. The only disadvantage of SND is its relatively long execution time. We will present some recent efforts to improve the execution time using both a multilevel and a hybrid approach. We use SND in computing a multifrontal numerical factorization with the different orderings on an eight processor Cray Y-MP and show its effectiveness. We believe that spectral nested dissection is a major breakthrough in terms of generating efficient sparse orderings for parallel machines.

Pothen, Alex↗
NASA NTRS1990

Optimal parallel solution of sparse triangular systems

A method for the parallel solution of triangular sets of equations is described that is appropriate when there are many right-handed sides. By preprocessing, the method can reduce the number of parallel steps required to solve Lx = b compared to parallel forward or backsolve. Applications are to iterative solvers with triangular preconditioners, to structural analysis, or to power systems applications, where there may be many right-handed sides (not all available a priori). The inverse of L is represented as a product of sparse triangular factors. The problem is to find a factored representation of this inverse of L with the smallest number of factors (or partitions), subject to the requirement that no new nonzero elements be created in the formation of these inverse factors. A method from an earlier reference is shown to solve this problem. This method is improved upon by constructing a permutation of the rows and columns of L that preserves triangularity and allow for the best possible such partition. A number of practical examples and algorithmic details are presented. The parallelism attainable is illustrated by means of elimination trees and clique trees.

Alvarado, Fernando L.↗
NASA NTRS1990

Automatic Management of Parallel and Distributed System Resources

Viewgraphs on automatic management of parallel and distributed system resources are presented. Topics covered include: parallel applications; intelligent management of multiprocessing systems; performance evaluation of parallel architecture; dynamic concurrent programs; compiler-directed system approach; lattice gaseous cellular automata; and sparse matrix Cholesky factorization.

Yan, Jerry↗
NASA NTRS1992

Multivariable frequency domain identification via 2-norm minimization

The author develops a computational approach to multivariable frequency domain identification, based on 2-norm minimization. In particular, a Gauss-Newton (GN) iteration is developed to minimize the 2-norm of the error between frequency domain data and a matrix fraction transfer function estimate. To improve the global performance of the optimization algorithm, the GN iteration is initialized using the solution to a particular sequentially reweighted least squares problem, denoted as the SK iteration. The least squares problems which arise from both the SK and GN iterations are shown to involve sparse matrices with identical block structure. A sparse matrix QR factorization method is developed to exploit the special block structure, and to efficiently compute the least squares solution. A numerical example involving the identification of a multiple-input multiple-output (MIMO) plant having 286 unknown parameters is given to illustrate the effectiveness of the algorithm.

Bayard, David S.↗
NASA NTRS2004

Finite Element Analysis in Concurrent Processing: Computational Issues

The purpose of this research is to investigate the potential application of new methods for solving large-scale static structural problems on concurrent computers. It is well known that traditional single-processor computational speed will be limited by inherent physical limits. The only path to achieve higher computational speeds lies through concurrent processing. Traditional factorization solution methods for sparse matrices are ill suited for concurrent processing because the null entries get filled, leading to high communication and memory requirements. The research reported herein investigates alternatives to factorization that promise a greater potential to achieve high concurrent computing efficiency. Two methods, and their variants, based on direct energy minimization are studied: a) minimization of the strain energy using the displacement method formulation; b) constrained minimization of the complementary strain energy using the force method formulation. Initial results indicated that in the context of the direct energy minimization the displacement formulation experienced convergence and accuracy difficulties while the force formulation showed promising potential.

Sobieszczanski-Sobieski, Jaroslaw↗