Roger G. Grimes

Progress in Sparse Matrix Methods for Large Linear Systems On Vector Supercomputers

This paper summarizes progress in the use of direct methods for solving very large sparse symmetric positive definite systems of linear equations on vector supercomputers. Sparse di rect solvers based on the multifrontal method or the general sparse method now outperform band or envelope solvers on vector supercomputers such as the CRAY X-MP. This departure from conventional wisdom is due to several advances. The hardware gather/scatter feature or indirect address feature of some recent vector super computers permits vectorization of the general sparse factorization. Other advances are algo rithmic. The new multiple minimum degree algo rithm calculates a powerful ordering much faster than its predecessors. Exploiting the supernode structure of the factored matrix provides vectori zation over nested loops, giving greater speed in the factorization module for the multifrontal and general sparse methods. Out-of-core versions of both methods are now available. Numerical re sults on the CRAY X-MP for several structural engineering problems demonstrate the impact of these improvements.

On Vectorizing Incomplete Factorization and SSOR Preconditioners

We consider the problem of vectorizing the recursive calculations found in modified incomplete factorizations and SSOR preconditioners for the conjugate gradient method. We examine matrix problems derived from partial differential equations which are discretized on regular 2-D and 3-D grids, where the grid nodes are ordered in the natural ordering. By performing data dependency analyses of the computations, we show that there is concurrency in both the factorization and the forward and backsolves. The computations may be performed with an average vector length of

Sparse matrix test problems

O(n)

Multifrontal computations on GPUs and their multi-core hosts

on an

Evaluation of orderings for unsymmetric sparse matrics

n^2

Condition Number Estimation for Sparse Matrices

or

The solution of large dense generalized eigenvalue problems on the Cray X-MP/24 with SSD

n^3

Solving systems of large dense linear equations

grid in two and three dimensions. Numerical studies on four model problems show that the conjugate gradient method using these vectorized implementations of the modified incomplete factorizations and SSOR preconditioners achieves overall speeds approaching 100 megaflops on a Cray X-MP/24 vector computer. Furthermore, these methods require considerably less overall execution time than diagonal scaling and no-fill red-black incomplete factorization preconditioners, both of which allow full vectorization but are not as convergent.

Eigenvalue Problems and Algorithms in Structural Engineering

The development, analysis and production of algorithms in sparse linear algebra often requires the use of test problems to demonstrate the effectiveness and applicability of the algorithms. Many algorithms have been developed in the context of specific application areas and have been tested in the context of sets of test problems collected by the developers. Comparisons of algorithms across application areas and comparisons between algorithms has often been incomplete, due to the lack of a comprehensive set of test problems. Additionally we believe that a comprehensive set of test problems will lead to a better understanding of the range of structures in sparse matrix problems and thence to better classification and development of algorithms. We have agreed to sponsor and maintain a general library of sparse matrix test problems, available on request to anyone for a nominal fee to cover postal charges. Contributors to the library will, of course, receive a free copy.

Performance comparison of the CRAY X-MP/24 with SDD and the CRAY-2

The use of GPUs to accelerate the factoring of large sparse symmetric matrices shows the potential of yielding important benefits to a large group of widely used applications. This paper examines how a multifrontal sparse solver performs when exploiting both the GPU and its multi-core host. It demonstrates that the GPU can dramatically accelerate the solver relative to one host CPU. Furthermore, the solver can profitably exploit both the GPU to factor its larger frontal matrices and multiple threads on the host to handle the smaller frontal matrices.