Fast Parallel Direct Solvers for Coarse Grid Problems

We have developed a fast direct solver for parallel solution of coarse grid problems, Ax=b, such as arise when domain decomposition or multigrid methods are applied to elliptic partial differential equations in d space dimensions. The approach is based on a (quasi-) sparse factorization of the inverse of A. If A is n×n and the number of processors is P, the algorithm requires O(n?logP) time for communication and O(n1+?/P) time for computation, where ??d?1d. The method is particularly suited to leading-edge multicomputer systems having thousands of processors. It achieves minimal message startup costs and substantially reduced message volume and arithmetic complexity compared with competing methods, which require O(nlogP) time for communication and O(n1+?) or O(n2/P) time for computation. Timings on the Intel Paragon and ASCI-Red machines reflect these complexity estimates.

[1]  Mitchell Luskin,et al.  Parallel Solution of Partial Differential Equations , 2000 .

[2]  PothenAlex,et al.  Partitioning sparse matrices with eigenvectors of graphs , 1990 .

[3]  YereminA. Yu.,et al.  Factorized sparse approximate inverse preconditionings I , 1993 .

[4]  M. Benzi,et al.  A Two-Level Parallel Preconditioner Based on Sparse Approximate Inverses , 1999 .

[5]  A. George.  Nested Dissection of a Regular Finite Element Mesh , 1973 .

[6]  H. D. Huskey,et al.  NOTES ON THE SOLUTION OF ALGEBRAIC LINEAR SIMULTANEOUS EQUATIONS , 1948 .

[7]  Robert A. van de Geijn,et al.  On Global Combine Operations , 1994, J. Parallel Distributed Comput..

[8]  David E. Keyes,et al.  Domain-Based Parallelism and Problem Decomposition Methods in Computational Science and Engineering , 1995 .

[9]  Charbel Farhat,et al.  Tailoring domain decomposition methods for efficient parallel coarse grid solution and for systems w , 1994 .

[10]  L. Kolotilina,et al.  Factorized Sparse Approximate Inverse Preconditionings I. Theory , 1993, SIAM J. Matrix Anal. Appl..

[11]  Robert E. Benner,et al.  Development of Parallel Methods for a $1024$-Processor Hypercube , 1988 .

[12]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[13]  P. F. Fischer,et al.  An overlapping Schwarz method for spectral element simulation of three-dimensional incompressible flows , 1998 .

[14]  H.M. Tufo,et al.  Terascale Spectral Element Algorithms and Implementations , 1999, ACM/IEEE SC 1999 Conference (SC'99).

[15]  Michele Benzi,et al.  A Sparse Approximate Inverse Preconditioner for the Conjugate Gradient Method , 1996, SIAM J. Sci. Comput..

[16]  R. Schreiber,et al.  Highly Parallel Sparse Triangular Solution , 1994 .

[17]  Tony F. Chan,et al.  Parallel Complexity of Domain Decomposition Methods and Optimal Coarse Grid Size , 1995, Parallel Comput..

[18]  Xiao-Chuan Cai.  The Use of Pointwise Interpolation in Domain Decomposition Methods with Nonnested Meshes , 1995, SIAM J. Sci. Comput..

[19]  Barry F. Smith,et al.  Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations , 1996 .

[20]  Alex Pothen,et al.  PARTITIONING SPARSE MATRICES WITH EIGENVECTORS OF GRAPHS* , 1990 .

[21]  William Gropp,et al.  Domain Decomposition with Local Mesh Refinement , 1992, SIAM J. Sci. Comput..

[22]  William Gropp,et al.  Parallel computing and domain decomposition , 1992 .