# Problem size, parallel architecture and optimal speedup

## Abstract

The communication and synchronization overhead inherent in parallel processing can lead to situations where adding processors to the solution method actually increases execution time. Problem type, problem size, and architecture type all affect the optimal number of processors to employ. The numerical solution of an elliptic partial differential equation is examined in order to study the relationship between problem size and architecture. The equation's domain is discretized into n sup 2 grid points which are divided into partitions and mapped onto the individual processor memories. The relationships between grid size, stencil type, partitioning strategy, processor execution time, and communication network type are analytically quantified. In so doing, the optimal number of processors was determined to assign to the solution, and identified (1) the smallest grid size which fully benefits from using all available processors, (2) the leverage on performance given by increasing processor speed or communication network speed, and (3) the suitability of various architectures for large numerical problems.

- Authors:

- Publication Date:

- Research Org.:
- National Aeronautics and Space Administration, Hampton, VA (USA). Langley Research Center

- OSTI Identifier:
- 6157506

- Report Number(s):
- N-87-22444; NASA-CR-178282; ICASE-87-7; NAS-1.26:178282

- Resource Type:
- Conference

- Resource Relation:
- Other Information: Final Report

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; ARRAY PROCESSORS; CONFIGURATION; EFFICIENCY; PERFORMANCE TESTING; PARALLEL PROCESSING; COMPUTER NETWORKS; NUMERICAL SOLUTION; PARTIAL DIFFERENTIAL EQUATIONS; DIFFERENTIAL EQUATIONS; EQUATIONS; PROGRAMMING; TESTING; 990210* - Supercomputers- (1987-1989)

### Citation Formats

```
Nicol, D M, and Willard, F H.
```*Problem size, parallel architecture and optimal speedup*. United States: N. p., 1987.
Web.

```
Nicol, D M, & Willard, F H.
```*Problem size, parallel architecture and optimal speedup*. United States.

```
Nicol, D M, and Willard, F H. Wed .
"Problem size, parallel architecture and optimal speedup". United States.
```

```
@article{osti_6157506,
```

title = {Problem size, parallel architecture and optimal speedup},

author = {Nicol, D M and Willard, F H},

abstractNote = {The communication and synchronization overhead inherent in parallel processing can lead to situations where adding processors to the solution method actually increases execution time. Problem type, problem size, and architecture type all affect the optimal number of processors to employ. The numerical solution of an elliptic partial differential equation is examined in order to study the relationship between problem size and architecture. The equation's domain is discretized into n sup 2 grid points which are divided into partitions and mapped onto the individual processor memories. The relationships between grid size, stencil type, partitioning strategy, processor execution time, and communication network type are analytically quantified. In so doing, the optimal number of processors was determined to assign to the solution, and identified (1) the smallest grid size which fully benefits from using all available processors, (2) the leverage on performance given by increasing processor speed or communication network speed, and (3) the suitability of various architectures for large numerical problems.},

doi = {},

journal = {},

number = ,

volume = ,

place = {United States},

year = {1987},

month = {4}

}