scholarly journals The Design of Data-Structure-Neutral Libraries for the Iterative Solution of Sparse Linear Systems

1996 ◽  
Vol 5 (4) ◽  
pp. 329-336 ◽  
Author(s):  
Barry F. Smith ◽  
William D. Gropp
Keyword(s):  
Data Structure ◽  
Linear Systems ◽  
Shared Memory ◽  
Distributed Memory ◽  
Parallel Machines ◽  
Sparse Matrix ◽  
Iterative Solution ◽  
Sparse Linear Systems ◽  
The Past ◽  
Using Data

Over the past few years several proposals have been made for the standardization of sparse matrix storage formats in order to allow for the development of portable matrix libraries for the iterative solution of linear systems. We believe that this is the wrong approach. Rather than define one standard (or a small number of standards) for matrix storage, the community should define an interface (i.e., the calling sequences) for the functions that act on the data. In addition, we cannot ignore the interface to the vector operations because, in many applications, vectors may not be stored as consecutive elements in memory. With the acceptance of shared memory, distributed memory, and cluster memory parallel machines, the flexibility of the distribution of the elements of vectors is also extremely important. This issue is ignored in most proposed standards. In this article we demonstrate how such libraries may be written using data encapsulation techniques.

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

2012 ◽  
Vol 20 (3) ◽  
pp. 241-255 ◽  
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.

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

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.

Local RBFs Based Collocation Methods for Unsteady Navier-Stokes Equations

2015 ◽  
Vol 7 (4) ◽  
pp. 430-440 ◽  
Author(s):  
Xueying Zhang ◽  
Xin An ◽  
C. S. Chen
Keyword(s):  
Linear Systems ◽  
Numerical Experiments ◽  
Stokes Equations ◽  
Sparse Matrix ◽  
High Accuracy ◽  
Collocation Methods ◽  
Navier Stokes ◽  
Sparse Linear Systems ◽  
Navier Stokes Equations ◽  
Weight Coefficients

AbstractThe local RBFs based collocation methods (LRBFCM) is presented to solve two-dimensional incompressible Navier-Stokes equations. In avoiding the ill-conditioned problem, the weight coefficients of linear combination with respect to the function values and its derivatives can be obtained by solving low-order linear systems within local supporting domain. Then, we reformulate local matrix in the global and sparse matrix. The obtained large sparse linear systems can be directly solved instead of using more complicated iterative method. The numerical experiments have shown that the developed LRBFCM is suitable for solving the incompressible Navier-Stokes equations with high accuracy and efficiency.

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