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].

scholarly journals Second-refinement of Gauss-Seidel iterative method for solving linear system of equations

2020 ◽  
Vol 13 (1) ◽  
pp. 1-15
Author(s):  
Tesfaye Kebede Enyew ◽  
Gurju Awgichew ◽  
Eshetu Haile ◽  
Gashaye Dessalew Abie
Keyword(s):  
Linear System ◽  
Positive Definite ◽  
System Of Equations ◽  
Linear System Of Equations ◽  
Symmetric Positive Definite ◽  
Modified Method ◽  
Seidel Method ◽  
Diagonally Dominant Matrix ◽  
Diagonally Dominant ◽  
Number Of Iterations

Although large and sparse linear systems can be solved using iterative methods, its number of iterations is relatively large. In this case, we need to modify the existing methods in order to get approximate solutions in a small number of iterations. In this paper, the modified method called second-refinement of Gauss-Seidel method for solving linear system of equations is proposed. The main aim of this study was to minimize the number of iterations, spectral radius and to increase rate of convergence. The method can also be used to solve differential equations where the problem is transformed to system of linear equations with coefficient matrices that are strictly diagonally dominant matrices, symmetric positive definite matrices or M-matrices by using finite difference method. As we have seen in theorem 1and we assured that, if A is strictly diagonally dominant matrix, then the modified method converges to the exact solution. Similarly, in theorem 2 and 3 we proved that, if the coefficient matrices are symmetric positive definite or M-matrices, then the modified method converges. And moreover in theorem 4 we observed that, the convergence of second-refinement of Gauss-Seidel method is faster than Gauss-Seidel and refinement of Gauss-Seidel methods. As indicated in the examples, we demonstrated the efficiency of second-refinement of Gauss-Seidel method better than Gauss-Seidel and refinement of Gauss-Seidel methods.

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 Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification

2021 ◽  
Vol 47 (5) ◽  
Author(s):  
I. G. Graham ◽  
O. R. Pembery ◽  
E. A. Spence
Keyword(s):  
Finite Element ◽  
Dirichlet Problem ◽  
Helmholtz Equation ◽  
Uncertainty Quantification ◽  
Numerical Experiments ◽  
Helmholtz Equations ◽  
Independent Number ◽  
Theoretical Evidence ◽  
Number Of Iterations ◽  
Quantities Of Interest

AbstractThis paper analyses the following question: let Aj, j = 1,2, be the Galerkin matrices corresponding to finite-element discretisations of the exterior Dirichlet problem for the heterogeneous Helmholtz equations ∇⋅ (Aj∇uj) + k2njuj = −f. How small must $\|A_{1} -A_{2}\|_{L^{q}}$ ∥ A 1 − A 2 ∥ L q and $\|{n_{1}} - {n_{2}}\|_{L^{q}}$ ∥ n 1 − n 2 ∥ L q be (in terms of k-dependence) for GMRES applied to either $(\mathbf {A}_1)^{-1}\mathbf {A}_2$ ( A 1 ) − 1 A 2 or A2(A1)− 1 to converge in a k-independent number of iterations for arbitrarily large k? (In other words, for A1 to be a good left or right preconditioner for A2?) We prove results answering this question, give theoretical evidence for their sharpness, and give numerical experiments supporting the estimates. Our motivation for tackling this question comes from calculating quantities of interest for the Helmholtz equation with random coefficients A and n. Such a calculation may require the solution of many deterministic Helmholtz problems, each with different A and n, and the answer to the question above dictates to what extent a previously calculated inverse of one of the Galerkin matrices can be used as a preconditioner for other Galerkin matrices.

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.

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.

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