An Approximate Minimal Elimination Ordering Scheme

2011 ◽  
Vol 268-270 ◽  
pp. 1533-1536
Author(s):  
Lu Yao ◽  
Zheng Hua Wang ◽  
Wei Cao ◽  
Zong Zhe Li ◽  
Yong Xian Wang
Keyword(s):  
Factorization Method ◽  
Positive Definite ◽  
Cholesky Factorization ◽  
Running Time ◽  
Symmetric Positive Definite ◽  
Matrix Ordering ◽  
Positive Definite System

Matrix ordering is a key technique when applying Cholesky factorization method to solving sparse symmetric positive definite system Ax = b. In view of some known minimal elimination ordering methods, an efficient heuristic approximate minimal elimination ordering scheme is proposed, which has the total running time of O(n+m). It is noteworthy that the algorithm can not only find a good ordering efficiently, but also achieve the result of symbolic factorization simultaneously.

A CONSTRAINT PRECONDITIONER FOR SOLVING SYMMETRIC POSITIVE DEFINITE SYSTEMS AND APPLICATION TO THE HELMHOLTZ EQUATIONS AND POISSON EQUATIONS

2010 ◽  
Vol 15 (3) ◽  
pp. 299-311 ◽  
Author(s):  
Zhuo-Hong Huang ◽  
Ting-Zhu Huang
Keyword(s):  
Numerical Experiments ◽  
Positive Definite ◽  
Cholesky Factorization ◽  
Helmholtz Equations ◽  
Poisson Equations ◽  
Symmetric Positive Definite ◽  
Incomplete Cholesky Factorization ◽  
Preconditioned Matrix ◽  
Factorization Methods ◽  
Number Of Iterations

In this paper, first, by using the diagonally compensated reduction and incomplete Cholesky factorization methods, we construct a constraint preconditioner for solving symmetric positive definite linear systems and then we apply the preconditioner to solve the Helmholtz equations and Poisson equations. Second, according to theoretical analysis, we prove that the preconditioned iteration method is convergent. Third, in numerical experiments, we plot the distribution of the spectrum of the preconditioned matrix M−1A and give the solution time and number of iterations comparing to the results of [5, 19].

GPU Accelerated Parallel Cholesky Factorization

2011 ◽  
Vol 148-149 ◽  
pp. 1370-1373 ◽  
Author(s):  
Liang Wang ◽  
Yi Sheng Zhang ◽  
Bin Zhu ◽  
Chi Xu ◽  
Xiao Wei Tian ◽  
...  
Keyword(s):  
Optimization Methods ◽  
Positive Definite ◽  
Cholesky Factorization ◽  
Positive Definite Matrices ◽  
Symmetric Positive Definite ◽  
Memory Scheduling ◽  
Parallel Factorization ◽  
Symmetric Positive Definite Matrices ◽  
Linear Equation Systems ◽  
Novel Algorithm

One of the fundamental problems in scientific computing is to find solutions for linear equation systems. For finite element problem, Cholesky factorization is often used to solve symmetric positive definite matrices. In this paper, Cholesky factorization is massively parallelized and three different optimization methods - highly parallel factorization, tile strategy and memory scheduling are used to accelerate Cholesky factorization effectively. A novel algorithm using OpenCL is implemented. Testing on GPU shows that performance of the algorithm increases with the dimension of matrix, reaching 785.41GFlops, about 50x times speedup. Cholesky factorization is remarkably improved with OpenCL on GPU.

scholarly journals Parallel Solution of Hierarchical Symmetric Positive Definite Linear Systems

2017 ◽  
Vol 2 (1) ◽  
pp. 201-212 ◽  
Author(s):  
José I. Aliaga ◽  
Rocío Carratalá-Sáez ◽  
Enrique S. Quintana-Ortí
Keyword(s):  
Linear Systems ◽  
Linear Algebra ◽  
High Performance ◽  
Parallel Implementation ◽  
Positive Definite ◽  
Cholesky Factorization ◽  
Task Parallelism ◽  
Parallel Solution ◽  
Symmetric Positive Definite ◽  
Performance Computing

AbstractWe present a prototype task-parallel algorithm for the solution of hierarchical symmetric positive definite linear systems via the ℋ-Cholesky factorization that builds upon the parallel programming standards and associated runtimes for OpenMP and OmpSs. In contrast with previous efforts, our proposal decouples the numerical aspects of the linear algebra operation from the complexities associated with high performance computing. Our experiments make an exhaustive analysis of the efficiency attained by different parallelization approaches that exploit either task-parallelism or loop-parallelism via a runtime. Alternatively, we also evaluate a solution that leverages multi-threaded parallelism via the parallel implementation of the Basic Linear Algebra Subroutines (BLAS) in Intel MKL.

Implementing Sparse Matrix Ordering Using Hypergraph Partitioning

2013 ◽  
Vol 706-708 ◽  
pp. 1890-1893
Author(s):  
Lu Yao ◽  
Yi Yang ◽  
Zheng Hua Wang ◽  
Wei Cao
Keyword(s):  
Sparse Matrix ◽  
Factorization Method ◽  
Cholesky Factorization ◽  
The Novel ◽  
Nested Dissection ◽  
Vertex Separator ◽  
Hypergraph Partitioning ◽  
Ordering Algorithms ◽  
Sparse Matrix Ordering ◽  
Matrix Ordering

Matrix ordering is a key technique when applying Cholesky factorization method to solving sparse symmetric positive definite system Ax = b. Much effort has been devoted to the development of powerful heuristic ordering algorithms. This paper implements a sparse matrix ordering scheme based on hypergraph partitioning. The novel nested dissection ordering scheme achieve the vertex separator by hypergraph partitioning. Experimental results show that the novel scheme produces results that are substantially better than METIS.

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