Analysis of an Aggregation-based Algebraic Multigrid Method and its Parallelization

Abstract

The interests of this thesis are twofold. First, a two-grid convergence analysis based on the paper [ \textit{Algebraic analysis of aggregation-based multigrid } by A. Napov and Y. Notay, Numer. Lin. Alg. Appl. 18 (2011), pp. 539-564 ] is derived for various aggregation schemes applied to a finite element discretization of a rotated anisotropic diffusion equation. As expected, it is shown that the best aggregation scheme is one in which aggregates are aligned with the anisotropy. In practice, however, this is not what automatic aggregation procedures do. We suggest an approach for determining appropriate aggregates based on eigenvectors associated with small eigenvalues of a block splitting matrix. In the second part of the thesis several issues regarding the parallel implementation of aggregation-based multigrid methods are discussed. The coarsest grid solving stage of multigrid cycles has been a bottleneck for parallel multigrid algorithms to attain a good speedup. A comparison between a parallel linear system direct solver (MUMPS) and a few steps of preconditioned conjugate gradient (PCG) methods for solving the coarsest grid system is carried out and tested on TACC Lonestar multi-processor machine. Regarding the preconditioner of conjugate gradient iterations, a parallel sparse approximate inverse (SAI) algorithm is used to construct an approximate inverse of the original matrix in order to replace the preconditioner solving step, which is inherently sequential, by matrix-vector multiplications. The linear systems tested arise from discretization of 2D or 3D partial differential equations, which are symmetric positive definite. The results exhibit that using PCG on the coarsest grid attains better speedup and overall better performance than MUMPS when the number of processors is greater than about 100. The effects of different decompositions of the physical domain (rows/slab versus blocks/pencils) on the scaling and efficiency of aggregation-based algebraic multigrid are also studied and one sees that the blocks/pencils decomposition of the physical domain reduces the amount of communication and hence has better performance.