Sparse Gaussian elimination with partial pivoting computes the factorization PAQ = LU of a sparse matrix A, where the row ordering P is selected during factorization using standard partial pivoting with row interchanges. The goal is to select a column preordering, Q, based solely on the nonzero pattern of A such that the factorization remains as sparse as possible, regardless of the subsequent choice of P. The choice of Q can have a dramatic impact on the number of nonzeros in L and U. One scheme for determining a good column ordering for A is to compute a symmetric ordering that reduces fill-in in the Cholesky factorization of ATA. This approach, which requires the sparsity structure of ATA to be computed, can be expensive both in

We present the main algorithmic features in the software package SuperLU DIST, a distributedmemory sparse direct solver for large sets of linear equations. We give in detail our parallelization strategies, with a focus on scalability issues, and demonstrate the software’s parallel performance and scalability on current machines. The solver is based on sparse Gaussian elimination, with an innovative static pivoting strategy proposed earlier by the authors. The main advantage of static pivoting over classical partial pivoting is that it permits a priori determination of data structures and communication patterns, which lets us exploit techniques used in parallel sparse Cholesky algorithms to better parallelize both LU decomposition and triangular solution on large-scale distributed machines.

A new method for sparse LU factorization is presented that combines a column pre-ordering strategy with a right-looking unsymmetric-pattern multifrontal numerical factorization. The column ordering is selected to give a good a priori upper bound on fill-in and then refined during numerical factorization (while preserving the bound). Pivot rows are selected to maintain numerical stability and to preserve sparsity. The method analyzes the matrix and automatically selects one of three pre-ordering and pivoting strategies. The number of nonzeros in the LU factors computed by the method is typically less than or equal to those found by a wide range of unsymmetric sparse LU factorization methods, including left-looking methods and prior multifrontal methods.

This dissertation presents new techniques for solving large sparse unsymmetric linear systems on high performance computers, using Gaussian elimination with partial pivoting. The efficiencies of the new algorithms are demonstrated for matrices from various fields and for a variety of high performance machines. In the first part we discuss optimizations of a sequential algorithm to exploit the memory hierarchies that exist in most RISC-based superscalar computers. We begin with the left-looking supernode-column algorithm by Eisenstat, Gilbert and Liu, which includes Eisenstat and Liu's symmetric structural reduction for fast symbolic factorization. Our key contribution is to develop both numeric and symbolic schemes to perform supernodepanel updates to achieve better data reuse in cache and floating-point register...

The classical formulation of large displacement visco-elasticity requires the geometrically nonlinear Green tensor. Keeping track of the rotational part of strain permits alternative formulations, that allow the tensor to stay linear, and at the same time maintaining rotational invariance. We replace a recently proposed heuristical warping technique by the application of the polar decomposition. The polar decomposition exactly extracts rotations, thus enhances stability and accuracy. We combine it with a hierarchical finite element basis, which allows us to compute accurate rotations from a coarse level nonlinear simulation and use them with corotated tensors for finer detail.

This report has been submitted for publication outside of IBM and will probably be copyrighted if accepted for publication. It has been issued as a Research Report for early dissemination of its contents. In view of the transfer of copyright to the outside publisher, its distribution outside of IBM prior to publication should be limited to peer communications and specific requests. After outside publication, requests should be filled only by reprints or legally obtained copies of the article (e.g., payment of royalties). IBM Research Division Almaden \Delta Austin \Delta China \Delta Delhi \Delta Haifa \Delta Tokyo \Delta Watson \Delta Zurich Recent Advances in Direct Methods for Solving Unsymmetric Sparse Systems of Linear Equations Anshul Gupta IBM T.J. Watson Research Center During the past few years, algorithmic improve

The rational Krylov sequence (RKS) method is a generalization of Arnoldi's method. It constructs an orthogonal reduction of a matrix pencil into an upper Hessenberg pencil. The RKS method is useful when the matrix pencil may be efficiently factored. This article considers approximately solving the resulting linear systems with iterative methods. We show that a Cayley transformation leads to a more efficient and robust eigensolver than the usual shift-invert transformation when the linear systems are solved inexactly within the RKS method. A relationship with the recently introduced Jacobi-Davidson method is also established.

Most of the current techniques for the direct solution of linear equations are based on supernodal or multifrontal approaches. An important feature of these methods is that arithmetic is performed on dense submatrices and Level 2 and Level 3 BLAS (matrixvector and matrix-matrix kernels) can be used. Both sparse LU and QR factorizations can be implemented within this framework. Partitioning and ordering techniques have seen major activity in recent years. We discuss bisection and multisection techniques, extensions to orderings to block triangular form, and recent improvements and modifications to standard orderings such as minimum degree. We also study advances in the solution of indefinite systems and sparse least-squares problems. The desire to exploit parallelism has been responsible for many of the developments in direct methods for sparse matrices over the last ten years. We examine this aspect in some detail, illustrating how current techniques have been developed or ...

A well known approach to compute the LU factorization of a general unsymmetric matrix A is to build the elimination tree associated with the pattern of the symmetric matrix A+A T and use it as a computational graph to drive the numerical factorization. This approach, although very efficient on a large range of unsymmetric matrices, does not capture the unsymmetric structure of the matrices. We introduce a new algorithm which detects and exploits the structural asymmetry of the submatrices involved during the processing of the elimination tree. We show that, with the new algorithm, significant gains both in memory and in time to perform the factorization can be obtained.

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