Performance Predictions of Multilevel Communication Optimal LU and QR Factorizations on Hierarchical Platforms

In this paper we study the performance of two classical dense linear algebra algorithms, the LU and the QR factorizations, on multilevel hierarchical platforms. We note that we focus on multilevel QR factorization, and give a brief description of the multilevel LU factorization. We first introduce a performance model called Hierarchical Cluster Platform (Hcp), encapsulating the characteristics of such platforms. The focus is set on reducing the communication requirements of studied algorithms at each level of the hierarchy. Lower bounds on communication are therefore extended with respect to the Hcp model. We then present a multilevel QR factorization algorithm tailored for those platforms, and provide a detailed performance analysis. We also provide a set of performance predictions showing the need for such hierarchical algorithms on large platforms.