An Aggregation-Based Algebraic Multigrid Method with Deflation Techniques and Modified Generic Factored Approximate Sparse Inverses

Abstract

In this paper, we examine deflation-based algebraic multigrid methods for solving large systems of linear equations. Aggregation of the unknown terms is applied for coarsening, while deflation techniques are proposed for improving the rate of convergence. More specifically, the V-cycle strategy is adopted, in which, at each iteration, the solution is computed by initially decomposing it utilizing two complementary subspaces. The approximate solution is formed by combining the solution obtained using multigrids and deflation. In order to improve performance and convergence behavior, the proposed scheme was coupled with the Modified Generic Factored Approximate Sparse Inverse preconditioner. Furthermore, a parallel version of the multigrid scheme is proposed for multicore parallel systems, improving the performance of the techniques. Finally, characteristic model problems are solved to demonstrate the applicability of the proposed schemes, while numerical results are given.

Approximation Properties of the Vector Weak Rescaled Pure Greedy Algorithm

“…Approximation using a sparse linear combination of elements from a fixed redundant family is actively used because of its concise representations and increased computational efficiency. It has been applied widely to signal processing, image compression, machine learning and PDE solvers (see [1][2][3][4][5][6][7][8][9][10]). Among others, simultaneous sparse approximation has been utilized in signal vector processing and multi-task learning (see [11][12][13][14]).…”