Combinatorial Algorithms for Computing Column Space Bases ThatHave Sparse Inverses

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This paper presents a combinatorial study on the problem ofconstructing a sparse basis forthe null-space of a sparse, underdetermined, full rank matrix, A. Such a null-space is suitable forsolving solving many saddle point problems. Our approach is to form acolumn space basis of A that has a sparse inverse, by selecting suitablecolumns of A. This basis is then used to form a sparse null-space basisin fundamental form. We investigate three different algorithms forcomputing the column space basis: Two greedy approaches that rely onmatching, and a third employing a divide and conquer strategy implementedwith hypergraph partitioning followed by the greedy approach. ... continued below

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Pinar, Ali; Chow, Edmond & Pothen, Alex March 18, 2005.

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This paper presents a combinatorial study on the problem ofconstructing a sparse basis forthe null-space of a sparse, underdetermined, full rank matrix, A. Such a null-space is suitable forsolving solving many saddle point problems. Our approach is to form acolumn space basis of A that has a sparse inverse, by selecting suitablecolumns of A. This basis is then used to form a sparse null-space basisin fundamental form. We investigate three different algorithms forcomputing the column space basis: Two greedy approaches that rely onmatching, and a third employing a divide and conquer strategy implementedwith hypergraph partitioning followed by the greedy approach. We alsodiscuss the complexity of selecting a column basis when it is known thata block diagonal basis exists with a small given block size.

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  • Journal Name: Electronic Transactions on Numerical Analysis; Journal Volume: 22; Related Information: Journal Publication Date: 2006

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  • Report No.: LBNL--52482
  • Grant Number: DE-AC02-05CH11231
  • Office of Scientific & Technical Information Report Number: 918098
  • Archival Resource Key: ark:/67531/metadc880761

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  • March 18, 2005

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  • Sept. 22, 2016, 2:13 a.m.

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  • Sept. 29, 2016, 7:43 p.m.

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Pinar, Ali; Chow, Edmond & Pothen, Alex. Combinatorial Algorithms for Computing Column Space Bases ThatHave Sparse Inverses, article, March 18, 2005; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc880761/: accessed August 20, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.