Multi-color incomplete Cholesky conjugate gradient methods for vector computers
This research is concerned with the solution on vector computers of linear systems of equations. Ax = b, where A is a large, sparse symmetric positive definite matrix with non-zero elements lying only along a few diagonals of the matrix. The system is solved using the incomplete Cholesky conjugate gradient method (ICCG). Multi-color orderings are used of the unknowns in the linear system to obtain p-color matrices for which a no-fill block ICCG method is implemented on the CYBER 205 with O(N/p) length vector operations in both the decomposition of A and, more importantly, in the forward and back solves necessary at each iteration of the method. (N is the number of unknowns and p is a small constant). A p-colored matrix is a matrix that can be partitioned into a p x p block matrix where the diagonal blocks are diagonal matrices. The matrix is stored by diagonals and matrix multiplication by diagonals is used to carry out the decomposition of A and the forward and back solves. Additionally, if the vectors across adjacent blocks line up, then some of the overhead associated with vector startups can be eliminated in the matrix vector multiplication necessary at each conjugate gradient iteration. Necessary and sufficient conditions are given to determine which multi-color orderings of the unknowns correspond to p-color matrices, and a process is indicated for choosing multi-color orderings.
- Research Organization:
- Virginia Univ., Charlottesville (USA)
- OSTI ID:
- 5180602
- Resource Relation:
- Other Information: Thesis (Ph. D.)
- Country of Publication:
- United States
- Language:
- English
Similar Records
Multi-grid solution of the pressure equation in reservoir simulation. [Incomplete Cholesky Conjugate Gradient (ICCG), Modified Incomplete Cholesky Conjugate Gradient (MICCG), ORTHOMIN]
Program generator for the Incomplete Cholesky Conjugate Gradient (ICCG) method with a symmetrizing preprocessor. [GENIC code package]