Joseph W. H. Liu

A Supernodal Approach to Sparse Partial Pivoting

We investigate several ways to improve the performance of sparse LU factorization with partial pivoting, as used to solve unsymmetric linear systems. We introduce the notion of unsymmetric supernodes to perform most of the numerical computation in dense matrix kernels. We introduce unsymmetric supernode-panel updates and two-dimensional data partitioning to better exploit the memory hierarchy. We use Gilbert and Peierlss depth-first search with Eisenstat and Lius symmetric structural reductions to speed up symbolic factorization. We have developed a sparse LU code using all these ideas. We present experiments demonstrating that it is significantly faster than earlier partial pivoting codes. We also compare its performance with UMFPACK, which uses a multifrontal approach; our code is very competitive in time and storage requirements, especially for large problems.

The role of elimination trees in sparse factorization

In this paper, the role of elimination trees in the direct solution of large sparse linear systems is examined. The notion of elimination trees is described and its relation to sparse Cholesky factorization is discussed. The use of elimination trees in the various phases of direct factorization are surveyed: in reordering, sparse storage schemes, symbolic factorization, numeric factorization, and different computing environments.

The multifrontal method for sparse matrix solution: theory and practice

This paper presents an overview of the multifrontal method for the solution of large sparse symmetric positive definite linear systems. The method is formulated in terms of frontal matrices, update matrices, and an assembly tree. Formal definitions of these notions are given based on the sparse matrix structure. Various advances to the basic method are surveyed. They include the role of matrix reorderings, the use of supernodes, and other implementatjon techniques. The use of the method in different computational environments is also described.

Modification of the minimum-degree algorithm by multiple elimination

The most widely used ordering scheme to reduce fills and operations in sparse matrix computation is the minimum-degree algorithm. The notion of multiple elimination is introduced here as a modification to the conventional scheme. The motivation is discussed using the k-by-k grid model problem. Experimental results indicate that the modified version retains the fill-reducing property of (and is often better than) the original ordering algorithm and yet requires less computer time. The reduction in ordering time is problem dependent, and for some problems the modified algorithm can run a few times faster than existing implementations of the minimum-degree algorithm. The use of external degree in the algorithm is also introduced.

Graph theory and sparse matrix computation

When reality is modelled by computation, matrices are often the connection between the continuous physical world and the finite algorithmic one. Usually, the more detailed the model, the bigger the matrix; however, efficiency demands that every possible advantage be exploited. The articles in this volume are based on recent research on sparse matrix computations. They examine graph theory as it connects to linear algebra, parallel computing, data structures, geometry and both numerical and discrete algorithms. The articles are grouped into three general categories: graph models of symmetric matrices and factorizations; graph models of algorithms on nonsymmetric matrices; and parallel sparse matrix algorithms.

Sparse Cholesky factorization on a local-memory multiprocessor

This article deals with the problem of factoring a large sparse positive definite matrix on a multiprocessor system. The processors are assumed to have substantial local memory but no globally shared memory. They communicate among themselves and with a host processor through message passing. Our primary interest is in designing an algorithm which exploits parallelism, rather than in exploiting features of the underlying topology of the hardware. However, part of our study is aimed at determining, for certain sparse matrix problems, whether hardware based on the binary hypercube topology adequately supports the communication requirements for such problems. Numerical results from experiments conducted on a hypercube multiprocessor are included.

Communication results for parallel sparse Cholesky factorization on a hypercube

Abstract We consider the problem of reducing data traffic among processor nodes during the parallel factorization of a sparse matrix on a hypercube multiprocessor. A task assignment strategy based on the structure of an elimination tree is presented. This assignment is aimed at achieving load balancing among the processors and also reducing the amount of processor-to-processor data communication. An analysis of regular grid problems is presented, providing a bound on communication volume generated by the new strategy, and showing that the allocation scheme is optimal in the asymptotic sense. Some experimental results on the performance of this scheme are presented.

A compact row storage scheme for Cholesky factors using elimination trees

For a given sparse symmetric positive definite matrix, a compact row-oriented storage scheme for its Cholesky factor is introduced. The scheme is based on the structure of an elimination tree defined for the given matrix. This new storage scheme has the distinct advantage of having the amount of overhead storage required for indexing always bounded by the number of nonzeros in the original matrix. The structural representation may be viewed as storing the minimal structure of the given matrix that will preserve the symbolic Cholesky factor. Experimental results on practical problems indicate that the amount of savings in overhead storage can be substantial when compared with Shermans compressed column storage scheme.

Computational models and task scheduling for parallel sparse Cholesky factorization

Abstract In this paper, a systematic and unified treatment of computational task models for parallel sparse Cholesky factorization is presented. They are classified as fine-, medium-, and large-grained graph models. In particular, a new medium-grained model based on column-oriented tasks is introduced, and it is shown to correspond structurally to the filled graph of the given sparse matrix. The task scheduling problem for the various task graphs is also discussed. A practical algorithm to schedule the column tasks of the medium-grained model for multiple processors is described. It is based on a heuristic critical path scheduling method. This will give an overall scheme for parallel sparse Cholesky factorization, appropriate for parallel machines with shared-memory architecture like the Denelcor HEP.

On the storage requirement in the out-of-core multifrontal method for sparse factorization

Two techniques are introduced to reduce the working storage requirement for the recent multifrontal method of Duff and Reid used in the sparse out-of-core factorization of symmetric matrices. For a given core size, the reduction in working storage allows some large problems to be solved without having to use auxiliary storage for the working arrays. Even if the working arrays exceed the core size, it will reduce the amount of input-output traffic necessary to manipulate the working vectors. Experimental results are provided to demonstrate significant storage reduction on practical problems using the proposed techniques.