An energy‐based AMG coarsening strategy

Abstract

SUMMARYAlgebraic multigrid (AMG) is an iterative method that is often optimal for solving the matrix equations that arise in a wide variety of applications, including discretized partial di erential equations. It automatically constructs a sequence of increasingly smaller matrix problems that hopefully enables e cient resolution of all scales present in the solution. The methodology is based on measuring how a so-called algebraically smooth error value at one point depends on its value at another. 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 deÿnition of strength of connection, discuss its implementation, and present the results of initial numerical experiments.

A Greedy Strategy for Coarse-Grid Selection

“…Typical relaxation schemes are the Jacobi and Gauss-Seidel iterations, but other schemes, including symmetric Gauss-Seidel and ILU factorizations, are also possible. Choice of the coarse grid may be done based on the classical AMG definitions of strength of connection [28], using independent set algorithms (possibly modified for parallel processing environments) [14,20], based on modified strength measures [7], or using the compatible relaxation method [5,8,15,22] discussed in more detail below. Once the coarse grid has been chosen, the dimensions of the intergrid transfer operators, R and P , are set, and their non-zero structure and values can be determined.…”

“…When the connection between two points is said to be strong, it is treated as an important connection to account for in interpolation, while a weak connection is to be discarded. The classical definition for the set of points that i strongly depends on is S i = {j : −a ij ≥ β max k =i {−a ik }} for a chosen strength threshold, β [10] (other possibile algebraic definitions of strong connections include those in [7,11]). The set C i is then defined by C i = C ∩ S i , while F i = Adj(i) = 0} \ C i .…”

A scaled thickness conditioning for solid- and solid-shell discretizations of thin-walled structures

“…This is due to the anisotropic nature of the mesh that is not recognized in the black box standard construction of AMG prolongation and restriction operators that has been utilized here, though more advanced AMG methods capable of anisotropy capturing do exist [5,6,12,21]. The improvement of the convergence behavior in this case results in a large speedup in time.…”

Smoothed aggregation for Helmholtz problems

“…For a given level, Algorithm 1 proceeds first by determining strong connections in the matrix graph of A k [1,34,36], resulting in the sparse matrix S k with non-zero entry (i, j) if the two degrees-of-freedom i and j are deemed strongly connected. From strength matrix S k , the n k vertices in the matrix graph are aggregated into a k+1 groups.…”