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 multi- level hierarchical platforms. We note that we focus on multilevel QR factorization, and give a brief description of the multilevel LU factoriza- tion. 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 al- gorithms 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 perfor- mance predictions showing the need for such hierarchical algorithms on large platforms