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On the solution of equality constrained quadratic programming problems arising . . .
, 1998
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Combinatorial Algorithms for Computing Column Space Bases That Have Sparse Inverses
 ETNA
"... Abstract. This paper presents a new combinatorial approach towards constructing a sparse, implicit basis for the null space of a sparse, underdetermined matrix. Our approach is to compute a column space basis of that has a sparse inverse, which could be used to represent a null space basis in impli ..."
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Abstract. This paper presents a new combinatorial approach towards constructing a sparse, implicit basis for the null space of a sparse, underdetermined matrix. Our approach is to compute a column space basis of that has a sparse inverse, which could be used to represent a null space basis in implicit form. We investigate three different algorithms for computing column space bases: two greedy algorithms implemented using graph matchings, and a third, which employs a divide and conquer strategy implemented with hypergraph partitioning followed by a matching. Our results show that for many matrices from linear programming, structural analysis, and circuit simulation, it is possible to compute column space bases having sparse inverses, contrary to conventional wisdom. The hypergraph partitioning method yields sparser basis inverses and has low computational time requirements, relative to the greedy approaches. We also discuss the complexity of selecting a column space basis when it is known that such a basis exists in block diagonal form with a given small block size. Key words. sparse column space basis, sparse null space basis, block angular matrix, block diagonal matrix, matching, hypergraph partitioning, inverse of a basis AMS subject classifications. 65F50, 68R10, 90C20 1. Introduction. Many
Dual variable methods for mixedhybrid finite element approximation of the potential fluid flow problem in porous media
, 2001
"... Abstract. Mixedhybrid finite element discretization of Darcy’s law and the continuity equation that describe the potential fluid flow problem in porous media leads to symmetric indefinite saddlepoint problems. In this paper we consider solution techniques based on the computation of a nullspace bas ..."
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Abstract. Mixedhybrid finite element discretization of Darcy’s law and the continuity equation that describe the potential fluid flow problem in porous media leads to symmetric indefinite saddlepoint problems. In this paper we consider solution techniques based on the computation of a nullspace basis of the whole or of a part of the left lower offdiagonal block in the system matrix and on the subsequent iterative solution of a projected system. This approach is mainly motivated by the need to solve a sequence of such systems with the same mesh but different material properties. A fundamental cycle nullspace basis of the whole offdiagonal block is constructed using the spanning tree of an associated graph. It is shown that such a basis may be theoretically rather illconditioned. Alternatively, the orthogonal nullspace basis of the subblock used to enforce continuity over faces can be easily constructed. In the former case, the resulting projected system is symmetric positive definite and so the conjugate gradient method can be applied. The projected system in the latter case remains indefinite and the preconditioned minimal residual method (or the smoothed conjugate gradient method) should be used. The theoretical rate of convergence for both algorithms is discussed and their efficiency is compared in numerical experiments. Key words. Saddlepoint problem, preconditioned iterative methods, sparse matrices, finite element method AMS subject classifications. 65F05, 65F50 1. Introduction. Let