Adaptive data distribution for concurrent continuation

Abstract

Continuation methods compute paths of solutions of nonlinear equations that depend on a parameter. This paper examines some aspects of the multicomputer implementation of such methods. The computations are done on a mesh connected multicomputer with 64 nodes. One of the main issues in the development of concurrent programs is load balancing, achieved here by using appropriate data distributions. In the continuation process, many linear systems have to be solved. For nearby points along the solution path, the corresponding system matrices are closely related to each other. Therefore, pivots which are good for theLU-decomposition of one matrix are likely to be acceptable for a whole segment of the solution path. This suggests to choose certain data distributions that achieve good load balancing. In addition, if these distributions are used, the resulting code is easily vectorized. To test this technique, the invariant manifold of a system of two identical nonlinear oscillators is computed as a function of the coupling between them. This invariant manifold is determined by the solution of a system of nonlinear partial differential equations that depends on the coupling parameter. A symmetry in the problem reduces this system to one single equation, which is discretized by finite differences. The solution of the discrete nonlinear system is followed as the coupling parameter is changed.