Coarsening, Sampling, and Smoothing: Elements of the Multilevel Method

Abstract

The multilevel method has emerged as one of the most effective methods for solving numerical and combinatorial problems. It has been used in multigrid, domain decomposition, geometric search structures, as well as optimization algorithms for problems such as partitioning and sparse-matrix ordering. This paper presents a systematic treatment of the fundamental elements of the multilevel method. We illustrate, using examples from several fields, the importance and effectiveness of coarsening, sampling, and smoothing (local optimization) in the application of the multilevel method.