Limits on parallelism in the numerical solution of linear partial differential equations

Journal Article · Tue Jan 01 04:00:00 EST 1991 · SIAM Journal on Scientific and Statistical Computing (Society for Industrial and Applied Mathematics); (United States)

DOI:https://doi.org/10.1137/0912001· OSTI ID:7152813

Worley, P H ^[1]

---

1. Oak Ridge National Lab., TN (United States)

The problem considered is that of approximating the solution of a linear scalar partial differential equation (PDE) at one or more locations in its domain. A lower bound on the amount of data required to satisfy a given error tolerance in the approximation is described. Using this bound, a lower found on the execution time of parallel algorithms that approximate the solution is derived. The lower bound on the execution time has the form [alpha][center dot] f[sub (+)][center dot]log[sub 2] [epsilon][sup [minus]1], where [alpha] is a problem-dependent constant, f[sub (+)] is a measure of the speed of floating point arithmetic, and [epsilon] is an upper bound on the error. Thus, when [alpha] > 0, the execution time increases as [epsilon] decreases, independent of the number of processors, the interconnection topology, and the algorithm used. Lower bounds on the execution time are also given for the cases where the interconnection network or the number of processors is 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. In this paper, it is shown that increasing the size of such a problem will usually come at the cost of increasing the execution time. Two examples are described that verify this conclusion, an algorithm-independent analysis of an elliptic PDE and an analysis of a specific algorithm for the approximation of a hyperbolic PDE.

• DOE Contract Number: AC05-84OR21400
• OSTI ID: 7152813
• Journal Information: SIAM Journal on Scientific and Statistical Computing (Society for Industrial and Applied Mathematics); (United States), Journal Name: SIAM Journal on Scientific and Statistical Computing (Society for Industrial and Applied Mathematics); (United States) Vol. 12:1; ISSN 0196-5204; ISSN SIJCD4
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
990200* -- Mathematics & Computers
ALGORITHMS
ARRAY PROCESSORS
COMPUTER ARCHITECTURE
DIFFERENTIAL EQUATIONS
EQUATIONS
MATHEMATICAL LOGIC
NUMERICAL SOLUTION
PARALLEL PROCESSING
PARTIAL DIFFERENTIAL EQUATIONS
PROGRAMMING