6 research outputs found
Graph partitioning using matrix values for preconditioning symmetric positive definite systems
Prior to the parallel solution of a large linear system, it is required to
perform a partitioning of its equations/unknowns. Standard partitioning
algorithms are designed using the considerations of the efficiency of the
parallel matrix-vector multiplication, and typically disregard the information
on the coefficients of the matrix. This information, however, may have a
significant impact on the quality of the preconditioning procedure used within
the chosen iterative scheme. In the present paper, we suggest a spectral
partitioning algorithm, which takes into account the information on the matrix
coefficients and constructs partitions with respect to the objective of
enhancing the quality of the nonoverlapping additive Schwarz (block Jacobi)
preconditioning for symmetric positive definite linear systems. For a set of
test problems with large variations in magnitudes of matrix coefficients, our
numerical experiments demonstrate a noticeable improvement in the convergence
of the resulting solution scheme when using the new partitioning approach
Combinatorial problems in solving linear systems
42 pages, available as LIP research report RR-2009-15Numerical linear algebra and combinatorial optimization are vast subjects; as is their interaction. In virtually all cases there should be a notion of sparsity for a combinatorial problem to arise. Sparse matrices therefore form the basis of the interaction of these two seemingly disparate subjects. As the core of many of today's numerical linear algebra computations consists of the solution of sparse linear system by direct or iterative methods, we survey some combinatorial problems, ideas, and algorithms relating to these computations. On the direct methods side, we discuss issues such as matrix ordering; bipartite matching and matrix scaling for better pivoting; task assignment and scheduling for parallel multifrontal solvers. On the iterative method side, we discuss preconditioning techniques including incomplete factorization preconditioners, support graph preconditioners, and algebraic multigrid. In a separate part, we discuss the block triangular form of sparse matrices
EXTENSIONS OF CERTAIN GRAPH-BASED ALGORITHMS FOR PRECONDITIONING ∗
Abstract. The original TPABLO algorithms are a collection of algorithms which compute a symmetric permutation of a linear system such that the permuted system has a relatively full block diagonal with relatively large nonzero entries. This block diagonal can then be used as a preconditioner. We propose and analyze three extensions of this approach: We incorporate a nonsymmetric permutation to obtain a large diagonal, we use a more general parametrization for TPABLO, and we use a block Gauss–Seidel preconditioner which can be implemented to have the same execution time as the corresponding block Jacobi preconditioner. Experiments are presented showing that for certain classes of matrices, the block Gauss–Seidel preconditioner used with the system permuted with the new algorithm can outperform the best ILUT preconditioners in a large set of experiments