An Algebraic Multigrid Method with Guaranteed Convergence Rate

Abstract

We consider the iterative solution of large sparse symmetric positive definite linear systems. We present an algebraic multigrid method which has a guaranteed convergence rate for the class of nonsingular symmetric M-matrices with nonnegative row sum. The coarsening is based on the aggregation of the unknowns. A key ingredient is an algorithm that builds the aggregates while ensuring that the corresponding two-grid convergence rate is bounded by a user-defined parameter. For a sensible choice of this parameter, it is shown that the recursive use of the two-grid procedure yields a convergence independent of the number of levels, provided that one uses a proper AMLIcycle. On the other hand, the computational cost per iteration step is of optimal order if the mean aggregate size is large enough. This cannot be guaranteed in all cases but is analytically shown to hold for the model Poisson problem. For more general problems, a wide range of experiments suggests that there are no complexity issues and further demonstrates the robustness of the method. The experiments are performed on systems obtained from low order finite difference or finite element discretizations of second order elliptic partial differential equations (PDEs). The set includes twoand three-dimensional problems, with both structured and unstructured grids, some of them with local refinement and/or reentering corner, and possible jumps or anisotropies in the PDE coefficients.

A divide-and-conquer bound for aggregate’s quality and algebraic connectivity

“…Eventually, we state the following lemma, which mainly relates the expression (2.1) of aggregate's quality to its more commonly used form (given, e.g., in [11]), as applicable beyond the context of graph Laplacian matrices. Simply stated, the result follows from the observation that the value of the right and left hand sides in (2.2) do not change when w is replaced by v = w −α1 ; the right hand side in (2.2) further reduces to the expression (2.3) if one chooses the value of α that satisfies w − α1 ⊥ Σ1 ; note that the resulting expression then corresponds to (2.1).…”

“…Note that left inequality (1.1) was first introduced in [11] ; the right inequality (1.1) is new to the best of our knowledge.…”

“…This motivation comes from aggregation-based multigrid methods, for which the aggregate's quality measures how well a given set of vertices, called aggregate, is represented by a single vertex. A popular aggregation approach is multiple pairwise aggregation [13,11], in which unknowns are merged into pairs, then the pairs are merged again into groups of size at most 4, and so on, until the average size of the aggregates reaches a prescribed value. The quality one wants to optimize is the quality of the final aggregate; however, for the reasons of cost the construction algorithm is based on the quality of the original pairs and the µ c parameter for the merged pairs of aggregates.…”

“…Aggregate's quality is used in the context of aggregation-based multigrid methods [2,3,13] as a tool for the design of robust multigrid solvers. Although initially introduced for discretized partial differential equations [10,11,12,14], aggregate's quality is now also used for graph Laplacian systems [9]. Aggregate's quality is defined on a set of vertices, also called aggregate, and measures the maximal impact on the multigrid convergence of representing this set of vertices by a single vertex.…”

CPU and GPU Performance of Large Scale Numerical Simulations in Geophysics

“…The method is however purely algebraic. The software package provides subroutines, written in FORTRAN, which implement the method described in [23], with further improvements from [20,24].…”