Permuting Sparse Rectangular Matrices into Block-Diagonal Form

@article{Aykanat2004PermutingSR,
  title={Permuting Sparse Rectangular Matrices into Block-Diagonal Form},
  author={Cevdet Aykanat and Ali Pinar and {\"U}mit V. Çataly{\"u}rek},
  journal={SIAM J. Sci. Comput.},
  year={2004},
  volume={25},
  pages={1860-1879}
}
We investigate the problem of permuting a sparse rectangular matrix into block-diagonal form. Block-diagonal form of a matrix grants an inherent parallelism for solving the deriving problem, as recently investigated in the context of mathematical programming, LU factorization, and QR factorization. To represent the nonzero structure of a matrix, we propose bipartite graph and hypergraph models that reduce the permutation problem to those of graph partitioning by vertex separator and hypergraph… 

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References

SHOWING 1-10 OF 104 REFERENCES

Computing the block triangular form of a sparse matrix

TLDR
This work considers the problem of permuting the rows and columns of a rectangular or square, unsymmetric sparse matrix to compute its block triangular form, a canonical decomposition of bipartite graphs induced by a maximum matching that was discovered by Dulmage and Mendelsohn.

Decomposing Irregularly Sparse Matrices for Parallel Matrix-Vector Multiplication

TLDR
Experimental results on sparse matrices confirm both the validity of the proposed hypergraph models and appropriateness of the multilevel approach to hypergraph partitioning.

Partitioning Rectangular and Structurally Unsymmetric Sparse Matrices for Parallel Processing

TLDR
This paper shows that the efficient parallelization of matrix-transpose-vector product operations can be expressed in terms of partitioning bipartite graphs and introduces several algorithms for this partitioning problem and compares their performance on a set of test matrices.

Partitioning Sparse Rectangular Matrices for Parallel Processing ?

TLDR
The rectangular matrix partitioning problem is formalized and several methods for solving it are discussed, including the spectral partitioning method for symmetric matrices to the rectangular case and this method is compared to three new methods | the alternating partitions method and two hybrid methods.

Hypergraph-Partitioning-Based Decomposition for Parallel Sparse-Matrix Vector Multiplication

In this work, we show that the standard graph-partitioning-based decomposition of sparse matrices does not reflect the actual communication volume requirement for parallel matrix-vector

Ordering Unsymmetric Matrices into Bordered Block Diagonal Form for Parallel Processing

TLDR
A multilevel ordering algorithm for reordering the unsymmetric matrix into a bordered block-diagonal (BBD) form to speed up the solution of linear systems of linear equations.

Decomposing Linear Programs for Parallel Solution

TLDR
This paper considers decomposing LP constraint matrices to obtain block angular structures with specified number of blocks for scalable parallelization and proposes hypergraph models to represent LP constraint Matrices for decomposition.

A Heuristic for Reducing Fill-In in Sparse Matrix Factorization

We present a heuristic that helps to improve the quality of the bisection returned by the Kernighan-Lin and greedy graph bisection algorithms. This in turns helps to reduce the amount of fill-in

PARTITIONING SPARSE MATRICES WITH EIGENVECTORS OF GRAPHS*

TLDR
It is shown that lower bounds on separator sizes can be obtained in terms of the eigenvalues of the Laplacian matrix associated with a graph, which can be used to compute good separators in grid graphs.

A multilevel algorithm for partitioning graphs

TLDR
A multilevel algorithm for graph partitioning in which the graph is approximated by a sequence of increasingly smaller graphs, and the smallest graph is then partitioned using a spectral method, and this partition is propagated back through the hierarchy of graphs.
...