Load balancing for problems with good bisectors, and applications in finite element simulations

Abstract

This paper studies load balancing issues for classes of problems with certain bisection properties. A class of problems has a-bisectors if every problem in the class can be subdivided into two subproblems whose weight (i.e. workload) is not smaller than an a-fraction of the original problem. It is shown that the maximum weight of a subproblem produced by Algorithm HF, which partitions a given problem into N subproblems by always subdividing the problem with maximum weight, is at most a factor of [l/a] 9 (1 -a) (11~1-2 greater than the theoretical optimum (uniform partition). This bound is proved to be asymptotically tight. Two strategies to use Algorithm HF for load balancing distributed hierarchical finite element simulations are presented. For this purpose, a certain class of weighted binary trees representing the load of such applications is shown to have 1/4-bisectors. The maximum resulting load is at most a factor of 9/4 larger than in a perfectly uniform distribution in this case.

Parallel Load Balancing for Problems with Good Bisectors

“…Theorem 2.1 [4]. Let P be a class of problems with weight function w : P Ä R + that has :-bisectors.…”

“…As in [3,4], we study the following simplified model for dynamic load balancing. The parallel system consists of N processors, numbered from 1 to N. The number of a processor is referred to as its identifier.…”

“…In order to parallelize the main part of the computation, the FE-tree must be split into subtrees that can be distributed among the available processors. Useful bisection methods for FE-trees and the speed-up achieved by incorporating dynamic load balancing using bisections are reported in [4].…”

“…In Section 2 we introduce our load balancing model and briefly review the results obtained by the authors for a sequential load balancing algorithm, Algorithm HF, in [3,4]. Section 3 first presents and analyzes a parallel implementation of Algorithm HF and discusses certain drawbacks regarding the communication overhead of this parallelization (Section 3.1).…”

“…For example, this solver can be applied to problems from structural engineering in order to compute the response of objects under external load. A particular application (simulating a short cantilever that is attached to a wall on one side and uniformly loaded on its upper side) is explained in more detail in [13] and [4]. The recursive substructuring phase yields an unbalanced binary tree (called an FE-tree), and the main part of the computation then consists of a large number of iterations in each of which the FE-tree is traversed in top-down and bottom-up direction.…”