BibTeX
@MISC{Schreiber_scalabilityof,
author = {Robert Schreiber and Robert Schreibert},
title = {SCALABILITY OF SPARSE DIRECT SOLVERS "},
year = {}
}
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Abstract
Abstract. We shall say that a scalable algorithm achieves efficiency that is bounded away from zero as the number of processors and the problem size increase in such a way that the size of the data structures increases linearly with the number of processors. In this paper we show that the column-oriented approach to sparse Cholesky for distributed-memory machines is not scalable. By considering message volume, node contention, and bisection width, one may obtain lower bounds on the time required for communication in a distributed algorithm. Applying this technique to distributed, column-oriented, full Cholesky leads to the conclusion that N (the order of the matrix) must scale with P (the number of processors) so that storage grows like p2. So the algorithm is not scalable. Identical conclusions have previously been obtained by consideration of communication and computation latency on the critical path in the algorithm; these results complement and reinforce that conclusion. For the sparse case, we have experimental measurements that make the same point: for column-oriented distributed methods, the number of gridpoints (which is O(N)) must grow as P _ in order to maintain parallel efficiency bounded above zero. Our sparse matrix results employ the "fan-in " distributed scheme, implemented on machines with either a grid or a fat-tree interconnect using a subtree-to-submachine mapping of the columns.
Keyphrases
scalability sparse direct solver quot distributed algorithm experimental measurement result complement critical path column-oriented distributed method subtree-to-submachine mapping message volume column-oriented approach computation latency bisection width parallel efficiency node contention scalable algorithm achieves efficiency sparse case sparse matrix result problem size increase fan-in quot storage grows identical conclusion fat-tree interconnect distributed-memory machine full cholesky
