Limits on parallelism in the numerical solution of linear PDEs

Technical Report · Sat Oct 01 04:00:00 EDT 1988

OSTI ID:6716185

Worley, P H

We consider approximating the solution of a linear scalar partial differential equation (PDE) at a single location in its problem domain. In previous work we described a lower bound on the amount of data required to satisfy an error bound for the approximation. Using this bound, we derive a lower bound on the parallel complexity of algorithms that approximate the solution. The lower bound is a linear function of log/sub 2/epsilon/sup /minus/1/, where epsilon is an upper bound on the error. Thus, the parallel complexity increases as epsilon decreases, independent of the number of processors, the interconnection topology, and the algorithm used. We also describe how the lower bound changes when the interconnection network or the number of processors in specified. Recent research has established that it is often possible to use a large number of processors efficiently when calculating the numerical solution of a PDE if the problem is sufficiently large. We argue that increasing the size of such a problem will usually come at the cost of increasing the execution time. We describe two examples verifying this conclusion, an algorithm-independent analysis of an elliptic PDE and an analysis of a specific algorithm for the approximation of a hyperbolic PDE. 16 refs., 4 figs.

• Research Organization: Oak Ridge National Lab., TN (USA)
• DOE Contract Number: AC05-84OR21400
• OSTI ID: 6716185
• Report Number(s): ORNL/TM-10945; ON: DE89002403
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
990210* -- Supercomputers-- (1987-1989)
990230 -- Mathematics & Mathematical Models-- (1987-1989)
ALGORITHMS
ARRAY PROCESSORS
BOUNDARY CONDITIONS
DIFFERENTIAL EQUATIONS
EQUATIONS
MATHEMATICAL LOGIC
NUMERICAL SOLUTION
PARALLEL PROCESSING
PARTIAL DIFFERENTIAL EQUATIONS
PROGRAMMING