In the past two years, these researchers have investigated multilevel recursive preconditioning techniques, sparse inverse approximation techniques, and eigenvalue shifting techniques. These methods focus on improving the robustness of standard preconditioners, which is where most difficulties lie with current software in real life applications. Preconditioners based on deflation and blocking were particularly effective.
Thierry Braconnier, Research Associate
Russ W. Burdick, Graduate Student Researcher
Paul Castillo, Graduate Student Researcher
James R. Chelikowsky, Faculty Collaborator
Francois Galilee, Equipe Parallelisme, Campus Universitaire, Grenoble, France
Ian Gates, Graduate Student Researcher
Lutz Grosz, Center for Mathematics and its Applications, The Australian National University, Canberra, Austrailia
Zhongze Li, Computer Science and Engineering Department, University of Minnesota, Minneapolis, Minnesota
Leigh J. Little, Research Associate
Irene Moulitsas, Graduate Student Researcher
Anand Nagarajan, Graduate Student Researcher
Yabo Peng, Graduate Student Researcher
Masha Sosonkina, Supercomputing Institute Research Scholar
Azzeddine Soulaimani, Departement de Genie Mecanique, Ecole de Technologie Superieure, Montreal, Quebec, Canada
Pryank Srivastava, Graduate Student Researcher
Brian Suchomel, Research Associate
Man-Chung Yeung, Research Associate
Jun Zhang, Research Associate
99/4 |
"Enhanced Parallel Multicolor Preconditioning Techniques for Linear Systems," Y. Saad and M. Sosonkina, University of Minnesota Supercomputing Institute Research Report UMSI 99/4, January 1999. |
99/92 |
"Parallel Methods and Tools for Predicting Material Properties," A. Stathopoulos, S. Ogut, Y. Saad, J.R. Chelikowsky, and H. Kim, University of Minnesota Supercomputing Institute Research Report UMSI 99/92, May 1999. Publication in press. |
99/104 |
"Block LU Preconditioners for Symmetric and Nonsymmetric Saddle Point Problems," L. Little and Y. Saad, University of Minnesota Supercomputing Institute Research Report UMSI 99/104, June 1999. |
99/107 |
"ARMS: An Algebraic Recursive Multilevel Solver for General Sparse Linear Systems," Y. Saad and B. Suchomel, University of Minnesota Supercomputing Institute Research Report UMSI 99/107, June 1999. |
99/152 |
"Iterative Solution of Linear Systems in the 20-th Century," Y. Saad and H.A. van der Vorst, University of Minnesota Supercomputing Institute Research Report UMSI 99/152, September 1999. |
99/193 |
"Electronic Structure Methods for Predicting the Properties of Materials: Grids in Space," J.R. Chelikowsky, Y. Saad, S. Ogut, I. Vasiliev, and A. Stathopoulos, Physica Status Solidi B, 217, p. 173 (2000). |
99/209 |
"Rational Approximation Preconditioners for General Sparse Linear Systems," P. Guillaume, Y. Saad, and M. Sosonkina, University of Minnesota Supercomputing Institute Research Report UMSI 99/209, November 1999. |
99/212 |
"Non-Standard Parallel Solution Strategies for Distributed Sparse Linear Systems," Y. Saad and M. Sosonkina, in Parallel Computation: Proceedings of ACPC'99 (Springer-Verlag, Berlin, 1999) p. 13. |
99/238 |
"Preconditioning Strategies for Linear Systems Arising in Tire Design," M. Sosonkina, J.T. Melson, Y. Saad, and L.T. Watson, University of Minnesota Supercomputing Institute Research Report UMSI 99/238, December 1999. |
Presently, these researchers are continuing their work on developing robust preconditioners for parallel platforms. They are simultaneously investigating parallel techniques that do not sacrafice robustness. Since they would like to test these approaches on some very large linear systems currently being viewed as challenging problems, it is mandatory that a supercomputer with a large memory is used. Some of these systems arise from the simulation of turbulent flow and others from a Tokamak application. Very large matrices from the tranair simulation code of Boeing are also being obtained.
The IBM SP2 supercomputer at the Supercomputing Institute is being used to examine preconditioning strategies for the linear systems that arise in a fluid-particle interaction code. In addition, the question of how to effectively partition the computational domain in the presence of particles is being addressed. A simplified version of the two-dimensional case has already been developed, but eventually, a fully three-dimensional code will be required.
A key element in the preconditioning studies is the use of PSPARSLIB, which is a library of parallel linear system solvers developed by Professor Saad. It is believed that PSPARSLIB can be a useful tool for solving the large systems that arise in the simulator. This is partly because the question of how to efficiently precondition the equations is still an open one, and PSPARSLIB contains a wide variety of general purpose preconditioners. In addition, it is felt that carefully examining the program and concentrating on the efficient parallel implementation of bottlenecks will result in a more effective code than could be obtained by automatic parallelization, either through parallel compilers or libraries such as PETSc.
Partitioning of the computational domain is challenging due to the interaction of the particles with the fluid. The simulations being studied generally contain hundreds or thousands of particles. Since the particles have a dramatic influence on the surrounding fluid, it is desired to have a partitioning that minimizes the communication between subdomains. Currently, a simple one-dimensional partitioning is implemented, but this may need to be replaced by a more sophisticated two-dimensional approach, particularly when the number of particles is large.
A parallel version of the ARMS (Algebraic Recursive Multilevel Solver) code is also being developed and tested on the SP and the Origin. The serial version of the code has been under development by Professor Saad and Brian Suchomel. It is a robust solver for general sparse linear systems. There is inherent parallelism in the algorithm in that a block factorization is performed based on independent sets that may be located on different processors.
Additionally, these researchers are working on enhancing the PSPARSLIB package with new distributed data structures to facilitate implementation of multilevel methods. This work is a part of the Parallel Algebraic Recursive Multilevel Solver project. Further work is investigating the potential of the mixed parallel programming environment, OpenMP and MPI, for solving large-scale linear systems. This work is well-suited for an implementation on the Origin 2000.
Investigation of parallel efficiency of iterative solvers and preconditioners is being continued. In particular, this research is focusing on the algorithms that adapt themselves at run-time. This is a challenging issue that requires a great deal of code development and experimental effort due to the great variety in algorithm choices and parameters and the differences in the performances of the IBM SP and Origin 2000 parallel architectures.
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