The Connection Machine: PDE solution on 65536 processors

Abstract The Connection Machine is a massively parallel architecture with 65 536 single-bit processors and 32 Mbytes of memory, organized as a high-dimensional hypercube. A sophisticated router system provides efficient communication between remote processors. A rich software environment, including a parallel extension of COMMON LISP, provides access to the processors and network. Virtual processor capability extends the degree of fine-grained parallelism beyond 1 000 000. We describe the hardware and the parallel programming environment. We then present implementations of SOR, Multigrid and Conjugate Gradient algorithms for solving Partial Differential Equations on the Connection Machine. Measurements of computational efficiency are provided as well as an analysis of opportunities for achieving better performance. Despite the lack of floating-point hardware, computation rates above 100 Mflops have been achieved in PDE solution. Virtual processors prove to be a real advantage, easing the effort of software development while improving system performance significantly.

[1]  C. Lanczos.  An iteration method for the solution of the eigenvalue problem of linear differential and integral operators , 1950 .

[2]  Oliver A. McBryan,et al.  Elliptic equation algorithms on parallel computers , 1986 .

[3]  U. Trottenberg,et al.  On the construction of fast solvers for elliptic equations , 1982 .

[4]  W. Daniel Hillis,et al.  The connection machine , 1985 .

[5]  Achi Brandt,et al.  Multigrid solvers on parallel computers , 1981 .

[6]  D. Brandt,et al.  Multi-level adaptive solutions to boundary-value problems math comptr , 1977 .

[7]  Gene H. Golub,et al.  Matrix computations , 1983 .

[8]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

[9]  W. Hackbusch.  Convergence of multigrid iterations applied to difference equations , 1980 .

[10]  Oliver A. McBryan,et al.  Parallel algorithms for elliptic equations , 1985 .

[11]  Oliver A. McBryan.  State of the Art of Multi-Processors , 1988 .

[12]  Guy L. Steele,et al.  Common Lisp the Language , 1984 .

[13]  Oliver A. McBryan,et al.  Hypercube Algorithms and Implementations , 1985, PPSC.

[14]  Oliver A. McBryan,et al.  The multigrid method on parallel processors , 1986 .

[15]  E. Stiefel,et al.  Refined Iterative Methods for Computation of the Solution and the Eigenvalues of Self-Adjoint Boundary Value Problems , 1959 .

[16]  Oliver A. McBryan,et al.  New architectures: Performance highlights and new algorithms , 1988, Parallel Comput..

[17]  Oliver A. McBryan,et al.  Fluids, discontinuities and renormalization group methods , 1984 .

[18]  Paul O. Frederickson,et al.  Parallel Superconvergent Multigrid , 1987 .

[19]  Oliver A. McBryan,et al.  Matrix and vector operations on hypercube parallel processors , 1987, Parallel Comput..