Performance Analysis of MG Preconditioning on Intel Xeon Phi: Towards Scalability for Extreme Scale Problems with Fractional Laplacians

Abstract

The Intel Xeon Phi architecture is currently a popular choice for supercomputers, with many entries of the Top 500 list, using it either as main processors or as accelerators/coprocessors. In this paper, we explore the performance and scalability of the Intel Xeon Phi chips in the context of large sparse linear systems, commonly arising from the discretization of PDEs. At the first step, the PCG [1] is applied as a basic iterative solution method in the case of sparse SPD problems. The parallel multigrid (MG) implementation from Trilinos ML package is utilized as a preconditioner. A matrix free algebraic multilevel solver is used to reduce the memory requirements, thus allowing the cores to be more efficiently utilized. The second part of the paper is devoted to the fractional Laplacian, that is, we consider the equation \(-\varDelta ^\alpha {\mathbf {u}} = {\mathbf {f}}\), \(0<\alpha < 1\), Open image in new window . The related elliptic boundary value problem describes anomalous diffusion phenomena also referred to as super-diffusion. The implemented method approximates the solution of the nonlocal problem by a series of local elliptic problems. The currently available numerical methods for fractional diffusion Laplacian have computational complexity, comparable e.g., to the complexity of solving local elliptic problem in Open image in new window . The presented parallel results are for \(\varOmega =(0,1)^3\), including meshes of very large scale. The numerical experiments are run on the Avitohol computer at the Institute of Information and Communication Technologies, IICT-BAS. The presented results show very good scalability when the CPU-cores and MIC work together for a certain number of compute nodes.