Domain decomposition: parallel multilevel methods for elliptic partial differential equations
Domain decomposition: parallel multilevel methods for elliptic partial differential equations
A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs
SIAM Journal on Scientific Computing
The Design and Use of Algorithms for Permuting Large Entries to the Diagonal of Sparse Matrices
SIAM Journal on Matrix Analysis and Applications
Preconditioning Highly Indefinite and Nonsymmetric Matrices
SIAM Journal on Scientific Computing
On Algorithms For Permuting Large Entries to the Diagonal of a Sparse Matrix
SIAM Journal on Matrix Analysis and Applications
A new data-mapping scheme for latency-tolerant distributed sparse triangular solution
Proceedings of the 2002 ACM/IEEE conference on Supercomputing
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Crout Versions of ILU for General Sparse Matrices
SIAM Journal on Scientific Computing
Proceedings of the 2003 ACM/IEEE conference on Supercomputing
Multilevel ILU With Reorderings for Diagonal Dominance
SIAM Journal on Scientific Computing
Symmetric Permutations for I-matrices to Delay and Avoid Small Pivots During Factorization
SIAM Journal on Scientific Computing
Optimization of sparse matrix-vector multiplication on emerging multicore platforms
Proceedings of the 2007 ACM/IEEE conference on Supercomputing
Factors impacting performance of multithreaded sparse triangular solve
VECPAR'10 Proceedings of the 9th international conference on High performance computing for computational science
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In this article, we present two new algorithms for solving given triangular systems in parallel on a shared memory architecture. Multilevel incomplete LU factorization based preconditioners, which have been very successful for solving linear systems iteratively, require these triangular solves. Hence, the algorithms presented here can be seen as parallelizing the application of these preconditioners. The first algorithm solves the triangular matrix by block anti-diagonals. The drawback of this approach is that it can be difficult to choose an appropriate block structure. On the other hand, if a good block partition can be found, this algorithm can be quite effective. The second algorithm takes a hybrid approach by solving the triangular system by block columns and anti-diagonals. It is usually as effective as the first algorithm, but the block structure can be chosen in a nearly optimal manner. Although numerical results indicate that the speed-up can be fairly good, systems with matrices having a strong diagonal structure or narrow bandwidth cannot be solved effectively in parallel. Hence, for these matrices, the results are disappointing. On the other hand, the results are better for matrices having a more uniform distribution of non-zero elements. Although not discussed in this article, these algorithms can possibly be adapted for distributed memory architectures.