Algebraic analysis of aggregation-based multigrid

Abstract

A convergence analysis of two-grid methods based on coarsening by (unsmoothed) aggregation is presented. For diagonally dominant symmetric (M-)matrices, it is shown that the analysis can be conducted locally; that is, the convergence factor can be bounded above by computing separately for each aggregate a parameter, which in some sense measures its quality. The procedure is purely algebraic and can be used to control a posteriori the quality of automatic coarsening algorithms. Assuming the aggregation pattern is sufficiently regular, it is further shown that the resulting bound is asymptotically sharp for a large class of elliptic boundary value problems, including problems with variable and discontinuous coefficients. In particular, the analysis of typical examples shows that the convergence rate is insensitive to discontinuities under some reasonable assumptions on the aggregation scheme.A. NAPOV AND Y. NOTAY Aggregation schemes are not new and trace back to [4,5]. They did not receive much attention till recently because of the difficulty to obtain grid independent convergence on their basis [6, p. 522-524], see also [7, p. 663], where an accurate three grid analysis is presented for the model Poisson problem. This may be related to the fact that the piecewise constant prolongation does not correspond to an interpolation which is at least first-order accurate, as required by the standard multigrid theory [2, Sections 3.5 and 6.3.2].That is why aggregation is often associated with smoothed aggregation, a procedure in which a tentative piecewise constant prolongation operator is smoothed [8,9]. This allows to develop an appropriate convergence theory, but, at the same time, some of the attractive features of pure (unsmoothed) aggregation are lost. In particular, assuming the same aggregation pattern, the coarse grid matrices are less sparse and more costly to compute when using smoothed aggregation.In this paper, we investigate such pure aggregation schemes based on the piecewise constant prolongation. They may indeed lead to two-grid methods with grid-independent convergence properties, as recently shown in [10] for model constant coefficient discrete partial differential equations (PDE) problems. There is no contradiction with the above quoted results, whose focus is on the convergence properties of two-grid methods used recursively in so-called V-cycle scheme [1]. Indeed, aggregation-based multigrid methods tend to scale poorly with the number of levels when using simple V-or even W-cycles, even though the two-grid scheme converges nicely [10,11]. However, this may be cured using more sophisticated K-cycles, in which Krylov subspace acceleration is used at each level [12]. It is also possible to improve the scalability by increasing the number of smoothing steps on coarser levels [13]. Now, the (Fourier) analysis developed in [10] only addresses constant coefficient problems with artificial (periodic) boundary conditions. Although there are numerical evidences that aggregationbased methods can be robust in th...

A new algebraic multigrid approach for Stokes problems

“…A key ingredient is a splitting A = A b + A r of the matrix A such that both A b and A r are nonnegative definite whereas A b is block diagonal with respect to the partitioning in aggregates. Then, particularized to the present context, and using Definition 4.1, Theorem 3.2 of [21] essentially says that…”

“…Regardingκ A andκ C , we may apply the results in [21] to obtain bounds on the approximation property constant for aggregation-based prolongations. We first considerκ A .…”

Algebraic Multigrid for Moderate Order Finite Elements

“…In this section we consider aggregation-based AMG methods as developed in [15,16,20,22]. They determine the prolongation P through the agglomeration of the unknowns into n c nonempty disjoint sets G k , k = 1, .…”

“…(1) On the one hand, the standard theoretical analyzes (as developed in [5,25,28] for classical AMG and in [15,16] for aggregation-based AMG) mainly rely on the assumption that the system matrix is an M-matrix with nonnegative row-sum. Recall, an M-matrix has nonpositive offdiagonal entries, and hence M-matrices with nonnegative row-sum are also diagonally dominant.…”

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

“…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.…”