Current Numerical Projects
Iterative Methods for Sparse Systems
U. M. Yang,
K. Gallivan,
X. Wang, J. Bordner, Z. Zlatev
The hybrid direct-iterative algorithm in the parallel Y12M sparse solver has been extended to include
CGS, BiCGSTAB, and transpose-free QMR. A new nonsymmetric ordering (LORA) has been
developed that exposes the structure in nonsymmetric matrices that can be exploited to compute a
factorization in parallel. The iterative solvers in the collection of sparse software SPLIB have been
extended and ported to the Alliant. A new multicluster version of the row projection solver has been
developed for Cedar. The code is intended for unstructured sparse systems and allows the user to choose
the view of Cedar used and the algorithmic level at which multicluster parallelism will be used. Finally, a
family of preconditioners for sparse least-squares problems based on incomplete Gram-Schmidt
orthogonalization has been developed and analyzed.
Conjugate Gradient Algorithms on Cedar
U. M. Yang,
K. Gallivan
Multicluster versions of several conjugate gradient schemes, the classical CG, a block ICCG
preconditioned CG, and the reduced system CG for well-structured sparse symmetric positive definte
linear systems, arising from two- and three-dimensional elliptical self-adjoint PDEs, have been
implemented on Cedar. The effect on performance of the use of different views of the Cedar architecture
and sparse kernel design has been considered.
Numerical Handling of Large Linear Systems
E. Gallopoulos, C. Baldwin, V. Simoncini
Several applications (time-dependent partial differential equations, computational electromagnetics)
demand the solution of large linear systems. The corresponding matrices may be complex symmetric or
nonsymmetric. Furthermore, the solution of systems with multiple right-hand sides may be required. We
have designed two new iterative methods for solving such problems, one for nonsymmetric systems
based on a combination of the QMR and BiCGStab solvers and the other for nonsymmetric systems for
multiple right-hand sides, based on a hybrid GMRES formulation combined with a projection scheme.
We have analyzed the convergence properties of block GMRES using a matrix-valued polynomial
formulation. The tools developed in this will be useful for other block methods as well.
Numerical Handling of Partial Differential Equations
E. Gallopoulos, C. Baldwin, V. Simoncini
Research in this area involves the design and implementation of parallel algorithsm for the solution of
time-dependent problems. Parallelism is introduced into the problem using partial fraction decomposition
while large systems are handled using Krylov-based methods. In particular we are analyzing the parallel
implementation of the multishift QMR technique for the evaluation of the matrix exponential applied on a
vector. Current work explores several issues related to the solution of time-dependent problems (e. g. ,
preconditioning, approximation techniques) and the stability of the partial fraction approximation.
Direct Methods
K. Gallivan, B. Marsolf, H. Wijshoff
Efforts continue on the development of single- and multiple-cluster codes for the direct solution of
sparse linear systems on Cedar. Work on schemes that expose structure suitable for multicluster
parallelism has produced a novel hybrid reordering algorithm. Significant performance improvements have
been made to the H* reordering scheme via parallelism and algorithm improvements. The factorization
phase of Mcsparse has also been improved, and a detailed set of multicluster performance tests has been
done. A distributed memory version of Mcsparse has also been implemented.
Circuit Simulation --- Relaxation Methods
K. Gallivan, G. G. Hung, R. Saleh, Y. C. Wen
Circuit simulation techniques that use nonlinear or waveform relaxation methods are the focus of this
research. The performance/algorithm trade-offs of a multicluster implementation of a hybrid
waveform/nonlinear relaxation algorithm are under investigation. The multicluster performance of the
main configurations of the code has improved significantly. An initial version of the code, which is
portable to Cedar, an Alliant FX/80, a hypercube, and a network of Sun workstations, has also been
developed.
Ocean Circulation Simulation with Virtual Memory
L. DeRose, K. Gallivan, E. Gallopoulos
The simulation of ocean circulation is a problem of great importance in the
geophysical sciences. Realistic simulations impose great demands on existing
computer systems. Apart from the computational requirements, such simulations
are I/O intensive. This research deals with issues arising from the mapping
of an ocean circulation code on the Alliant and Cedar multiprocessors, in
particular the virtual memory organizations as combined with vector
multiprocessing and their effect on performance. The performance of the
multicluster code has been considered, and significant improvement has been
made by a reorganization of the data transfers and the addition of a more
effective land-avoidance strategy.
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Last Modified: February 14, 1996
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