Interpolation Weights of Algebraic Multigrid

摘要

Algebraic multigrid (AMG) is a numerical method used to solve particularalgebraic systems, and interest in it has risen because of its multigrid-like efficiency. We have found that regular interpolation weight definitions are inadequate for solving certain discretized systems so an iterative approach to determine the weights will prove useful. This iterative weight definition must balance the requirement of keeping the interpolatory set of points small in order to reduce solver complexity while maintaining accurate interpolation to correctly represent the coarse-grid function on the fine grid. Furthermore, the weight definition process must be efficient enough to reduce setup phase costs. We present systems involving matrices where this iterative method significantly outperforms regular AMG weight definitions. Experimental results show that the iterative weight definition does not improve the convergence rate over standard AMG when applied to M-matrices; however, the improvement becomes significant when solving certain types of complicated, non-standard algebraic equations generated by irregular operators.