2011/06/29 by Achi Brandt, James Brannick, Brandt, Achi +5
Computer Science · Engineering · #65 #Advanced Numerical Methods in Computational Mathematics #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1106.5990
openalex publication_date 2011/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Algebraic multigrid is an iterative method that is often optimal for solving the matrix equations that arise in a wide variety of applications, including discretized partial differential equations. It automatically constructs a sequence of increasingly smaller matrix problems that enable efficient resolution of all scales present in the solution. One of the main components of the method is an adequate choice of coarse grids. The current coarsening methodology is based on measuring how a so-called algebraically smooth error value at one point depends on the error values at its neighbors. Such a concept of strength of connection is well understood for operators whose principal part is an M-matrix; however, the strength concept for more general matrices is not yet clearly understood, and this lack of knowledge limits the scope of AMG applicability. The purpose of this paper is to motivate a general definition of strength of connection, based on the notion of algebraic distances, discuss its implementation, and present the results of initial numerical experiments. The algebraic distance measure, we propose, uses as its main tool a least squares functional, which is also applied to define interpolation.