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Ph.D de KHABOU Amal
Group :

Dense matrix computations: communication cost and numerical stability

Starts on 01/10/2009
Advisor : GRIGORI, Laura

Funding : contrat doctoral du Ministère
Affiliation : Université Paris-Sud
Laboratory : LRI - GRAND LARGE

Defended on 11/02/2013, committee :
Nicholas Higham (Rapporteur), Professeur, School of Mathematics, the University of Manchester
Yves Robert (Rapporteur), Professeur, Ecole Normale Supérieure de Lyon

Iain Duff (Examinateur), Professeur, the University of Strathclde
Yannis Manoussakis (Examinateur), Professeur, Université Paris Sud
Jean-Louis Roch (Examinateur), Maître de Conférences, IMAG

Laura Grigori (Directeur de thèse), Directeur de Recherche, INRIA

Research activities :

Abstract :
This dissertation focuses on a widely used linear algebra kernel to solve linear systems, that is the LU
decomposition. Usually, to perform such a computation one uses the Gaussian elimination with partial
pivoting (GEPP). The backward stability of GEPP depends on a quantity which is referred to as the
growth factor, it is known that in general GEPP leads to modest element growth in practice. However
its parallel version does not attain the communication lower bounds. Indeed the panel factorization represents
a bottleneck in terms of communication. To overcome this communication bottleneck, Grigori
et al [60] have developed a communication avoiding LU factorization (CALU), which is asymptotically
optimal in terms of communication cost at the cost of some redundant computation. In theory, the upper
bound of the growth factor is larger than that of Gaussian elimination with partial pivoting, however
CALU is stable in practice. To improve the upper bound of the growth factor, we study a new pivoting
strategy based on strong rank revealing QR factorization. Thus we develop a new block algorithm for
the LU factorization. This algorithm has a smaller growth factor upper bound compared to Gaussian
elimination with partial pivoting. The strong rank revealing pivoting is then combined with tournament
pivoting strategy to produce a communication avoiding LU factorization that is more stable than CALU.
For hierarchical systems, multiple levels of parallelism are available. However, none of the previously
cited methods fully exploit these hierarchical systems. We propose and study two recursive algorithms
based on the communication avoiding LU algorithm, which are more suitable for architectures with
multiple levels of parallelism. For an accurate and realistic cost analysis of these hierarchical algorithms,
we introduce a hierarchical parallel performance model that takes into account processor and
network hierarchies. This analysis enables us to accurately predict the performance of the hierarchical
LU factorization on an exascale platform.

Ph.D. dissertations & Faculty habilitations
The original manuscript conceptualizes the recent rise of digital platforms along three main dimensions: their nature of coordination devices fueled by data, the ensuing transformations of labor, and the accompanying promises of societal innovation. The overall ambition is to unpack the coordination role of the platform and where it stands in the horizon of the classical firm – market duality. It is also to precisely understand how it uses data to do so, where it drives labor, and how it accommodates socially innovative projects. I extend this analysis to show continuity between today’s society dominated by platforms and the “organizational society”, claiming that platforms are organized structures that distribute resources, produce asymmetries of wealth and power, and push social innovation to the periphery of the system. I discuss the policy implications of these tendencies and propose avenues for follow-up research.