Progress in Sparse Matrix Methods for Large Linear Systems On Vector Supercomputers

AbstractModern astrometric techniques lead to large, linear systems solved by the precepts of least-squares. These systems are usually sparse, and one should take advantage of the sparsity to facilitate their solution. As long as the matrix A of the equations of condition possesses the weak Hall property, characteristic of linear systems derived from astrometric reductions, it is possible to find a sparse Cholesky factor. Before the equations of condition are accumulated, by use of the fast Givens transformation, a symbolic factorization of A using Tewarson’s length of intersection technique determines the ordering of the columns of A that result in low fill-in. The non-null elements are stored in a sparse, dynamic data structure by use of dynamic hashing. Numerical experimentation shows that this competes well with alternatives such as nested dissection, and large, but sparse, linear systems with several thousand unknowns can be solved in a reasonable amount of time, even on personal computers.

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Cluster Analysis and Lumpability of Large Linear Systems

10.23919/acc.1986.4789114 ◽

1986 ◽

Author(s):

Pamela G. Coxson ◽

Kenneth B. Bischoff

Keyword(s):

Cluster Analysis ◽

Linear Systems ◽

Large Linear Systems

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Amesos2 and Belos: Direct and Iterative Solvers for Large Sparse Linear Systems

Scientific Programming ◽

10.1155/2012/243875 ◽

2012 ◽

Vol 20 (3) ◽

pp. 241-255 ◽

Cited By ~ 18

Author(s):

Eric Bavier ◽

Mark Hoemmen ◽

Sivasankaran Rajamanickam ◽

Heidi Thornquist

Keyword(s):

Linear Systems ◽

Iterative Methods ◽

Sparse Matrix ◽

Sparse Matrices ◽

Direct Methods ◽

Iterative Solvers ◽

Software Project ◽

Sparse Linear Systems ◽

Scalar Type ◽

Mixed Precision

Solvers for large sparse linear systems come in two categories: direct and iterative. Amesos2, a package in the Trilinos software project, provides direct methods, and Belos, another Trilinos package, provides iterative methods. Amesos2 offers a common interface to many different sparse matrix factorization codes, and can handle any implementation of sparse matrices and vectors, via an easy-to-extend C++ traits interface. It can also factor matrices whose entries have arbitrary “Scalar” type, enabling extended-precision and mixed-precision algorithms. Belos includes many different iterative methods for solving large sparse linear systems and least-squares problems. Unlike competing iterative solver libraries, Belos completely decouples the algorithms from the implementations of the underlying linear algebra objects. This lets Belos exploit the latest hardware without changes to the code. Belos favors algorithms that solve higher-level problems, such as multiple simultaneous linear systems and sequences of related linear systems, faster than standard algorithms. The package also supports extended-precision and mixed-precision algorithms. Together, Amesos2 and Belos form a complete suite of sparse linear solvers.

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Sparse matrix methods for chemical process separation calculations on supercomputers

Proceedings Supercomputing '92 ◽

10.1109/superc.1992.236662 ◽

2003 ◽

Cited By ~ 3

Author(s):

S.E. Zitney

Keyword(s):

Chemical Process ◽

Sparse Matrix ◽

Matrix Methods

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MRILDU: An Improvement to ILUT Based on Incomplete LDU Factorization and Dropping in Multiple Rows

Journal of Applied Mathematics ◽

10.1155/2014/467672 ◽

2014 ◽

Vol 2014 ◽

pp. 1-9

Author(s):

Jian-Ping Wu ◽

Huai-Fa Ma

Keyword(s):

Computational Complexity ◽

Linear Systems ◽

Numerical Weather Prediction ◽

Sparse Matrix ◽

Weather Prediction ◽

Concrete Sample ◽

Storage Requirement ◽

Sparse Linear Systems ◽

Numerical Weather ◽

Do So

We provide an improvement MRILDU to ILUT for general sparse linear systems in the paper. The improvement is based on the consideration that relatively large elements should be kept down as much as possible. To do so, two schemes are used. Firstly, incomplete LDU factorization is used instead of incomplete LU. Besides, multiple rows are computed at a time, and then dropping is applied to these rows to extract the relatively large elements in magnitude. Incomplete LDU is not only fairer when there are large differences between the elements of factorsLandU, but also more natural for the latter dropping in multiple rows. And the dropping in multiple rows is more profitable, for there may be large differences between elements in different rows in each factor. The provided MRILDU is comparable to ILUT in storage requirement and computational complexity. And the experiments for spare linear systems from UF Sparse Matrix Collection, inertial constrained fusion simulation, numerical weather prediction, and concrete sample simulation show that it is more effective than ILUT in most cases and is not as sensitive as ILUT to the parameterp, the maximum number of nonzeros allowed in each row of a factor.

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