The Divide-and-Conquer Framework: A Suitable Setting for the DDM of the Future

TL;DR

This paper demonstrates that the traditional ideal speedup assumption in domain decomposition methods (DDM) underestimates potential performance gains, proposing a Divide-and-Conquer framework that better captures achievable speedups and guides future research.

Contribution

It introduces a new theoretical framework based on Divide-and-Conquer principles that explains larger-than-expected speedups in DDM algorithms, challenging the standard ideal speedup assumption.

Findings

Algorithms achieved speedups over seventy times the number of processors.

Standard ideal speedup assumptions limit DDM performance goals.

Divide-and-Conquer framework better explains observed speedups.

Abstract

This paper was prompted by numerical experiments we performed, in which algorithms already available in the literature (DVS-BDDM) yielded accelerations (or speedups) many times larger (more than seventy in some examples already treated, but probably often much larger) than the number of processors used. Based on these outstanding results, here it is shown that believing in the standard ideal speedup, which is taken to be equal to the number of processors, has limited much the performance goal sought by research on domain decomposition methods (DDM) and has hindered much its development, thus far. Hence, an improved theory in which the speedup goal is based on the Divide and Conquer algorithmic paradigm, frequently considered as the leitmotiv of domain decomposition methods, is proposed as a suitable setting for the DDM of the future.