scholarly journals MPI+OpenMP parallel implementation of conjugate gradient method with preconditioner of block partial inverse triangular decomposition of IC2S and IC1

2021 ◽  
pp. 1-32
Author(s):  
Olga Yurievna Milyukova
Keyword(s):  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Parallel Implementation ◽  
Sparse Matrix ◽  
Triangular Decomposition ◽  
Algebraic Equations ◽  
First Order ◽  
Partial Inverse ◽  
Linear Algebraic Equations

The paper proposes a new preconditioner for solving systems of linear algebraic equations with a symmetric positively defined matrix by the method of conjugate gradients – Block Incomplete Inverse Cholesky BIIC preconditioner in combination with a triangular first-order decomposition "by value" - BIIC-IC1. The algorithm based on MPI+OpenMP techniques is proposed for the construction and application of the BIIC preconditioner combined with stabilized triangular decomposition of the second order "by value" (BIIC-IS2S). In this case, the BIIC-IC2S preconditioner uses the number of blocks multiple of the number of processors used and the number of threads used. Two algorithms based on MPI+OpenMP techniques are proposed for the construction and application of the BIIC-IC1 preconditioner. Comparative timing results for the MPI+OpenMP and MPI implementations of the proposed preconditioning used with the conjugate gradient method for a model problem and the sparse matrix collections SuiteSparse are presented.

Tomographic inversion via the conjugate gradient method

Geophysics ◽  
1987 ◽  
Vol 52 (2) ◽  
pp. 179-185 ◽  
Author(s):  
John A. Scales
Keyword(s):  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Sparse Matrices ◽  
Conjugate Gradient Algorithm ◽  
Gradient Algorithm ◽  
Algebraic Equations ◽  
Tomographic Inversion ◽  
Linear Algebraic Equations ◽  
The Matrix

Tomographic inversion of seismic traveltime residuals is now an established and widely used technique for imaging the Earth’s interior. This inversion procedure results in large, but sparse, rectangular systems of linear algebraic equations; in practice there may be tens or even hundreds of thousands of simultaneous equations. This paper applies the classic conjugate gradient algorithm of Hestenes and Stiefel to the least‐squares solution of large, sparse systems of traveltime equations. The conjugate gradient method is fast, accurate, and easily adapted to take advantage of the sparsity of the matrix. The techniques necessary for manipulating sparse matrices are outlined in the Appendix. In addition, the results of the conjugate gradient algorithm are compared to results from two of the more widely used tomographic inversion algorithms.

Thermal Floorplan Base on Conjugate Gradient Solver in HotSpot

2014 ◽  
Vol 608-609 ◽  
pp. 908-912
Author(s):  
Zhong Hua Jiang ◽  
Ning Xu ◽  
Chun Xiang Wu
Keyword(s):  
Linear System ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Decomposition Method ◽  
Sparse Matrix ◽  
Linear Equations ◽  
Sparse Linear Systems ◽  
Vlsi Physical Design ◽  
Conjugate Gradient Solver

In this paper, we introduce an effective iterative method to solve the thermal linear system in HotSpot thermal floorplan, the iterative Conjugate Gradient Method is suitable to solve the traditional sparse matrix linear equations. We define a class of dummy sparse linear systems in iterative thermal floorplan algorithm, the iterative methods for linear system can be extended to apply to other iterative framework algorithm. We apply the conjugate gradient method to solve the thermal model in floorplan of VLSI physical design. The experiments' result shows that thermal floorplan using Conjugate gradient method is effective. The running time of our incremental conjugate gradient thermal solver with Jocabi Precondition is approximate 0.59 comparing with LU decomposition method.

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