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.

A survey of direct methods for sparse linear systems

Acta Numerica ◽  
2016 ◽  
Vol 25 ◽  
pp. 383-566 ◽  
Author(s):  
Timothy A. Davis ◽  
Sivasankaran Rajamanickam ◽  
Wissam M. Sid-Lakhdar
Keyword(s):  
Linear Systems ◽  
Sparse Matrix ◽  
Direct Methods ◽  
Theory And Practice ◽  
Sparse Linear Systems ◽  
Least Squares Problems ◽  
Know How ◽  
Survey Article ◽  
Wide Range ◽  
Matrix Problems

Wilkinson defined a sparse matrix as one with enough zeros that it pays to take advantage of them.1This informal yet practical definition captures the essence of the goal of direct methods for solving sparse matrix problems. They exploit the sparsity of a matrix to solve problems economically: much faster and using far less memory than if all the entries of a matrix were stored and took part in explicit computations. These methods form the backbone of a wide range of problems in computational science. A glimpse of the breadth of applications relying on sparse solvers can be seen in the origins of matrices in published matrix benchmark collections (Duff and Reid 1979a, Duff, Grimes and Lewis 1989a, Davis and Hu 2011). The goal of this survey article is to impart a working knowledge of the underlying theory and practice of sparse direct methods for solving linear systems and least-squares problems, and to provide an overview of the algorithms, data structures, and software available to solve these problems, so that the reader can both understand the methods and know how best to use them.

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.

Residual Replacement in Mixed-Precision Iterative Refinement for Sparse Linear Systems

2018 ◽  
pp. 554-561
Author(s):  
Hartwig Anzt ◽  
Goran Flegar ◽  
Vedran Novaković ◽  
Enrique S. Quintana-Ortí ◽  
Andrés E. Tomás
Keyword(s):  
Linear Systems ◽  
Iterative Refinement ◽  
Sparse Linear Systems ◽  
Mixed Precision

scholarly journals Analysis of Monte Carlo accelerated iterative methods for sparse linear systems

2017 ◽  
Vol 24 (3) ◽  
pp. e2088 ◽  
Author(s):  
Michele Benzi ◽  
Thomas M. Evans ◽  
Steven P. Hamilton ◽  
Massimiliano Lupo Pasini ◽  
Stuart R. Slattery
Keyword(s):  
Monte Carlo ◽  
Linear Systems ◽  
Iterative Methods ◽  
Sparse Linear Systems

Parallel iterative solvers for sparse linear systems in circuit simulation

2005 ◽  
Vol 21 (8) ◽  
pp. 1275-1284 ◽  
Author(s):  
A. Basermann ◽  
U. Jaekel ◽  
M. Nordhausen ◽  
K. Hachiya
Keyword(s):  
Linear Systems ◽  
Circuit Simulation ◽  
Iterative Solvers ◽  
Sparse Linear Systems ◽  
Parallel Iterative
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