A Comparison of Two-Level Preconditioners Based on Multigrid and Deflation

Abstract

Abstract. It is well known that two-level and multilevel preconditioned conjugate gradient (PCG) methods provide efficient techniques for solving large and sparse linear systems whose coefficient matrices are symmetric and positive definite. A two-level PCG method combines a traditional (one-level) preconditioner, such as incomplete Cholesky, with a projection-type preconditioner to get rid of the effect of both small and large eigenvalues of the coefficient matrix; multilevel approaches arise by recursively applying the two-level technique within the projection step. In the literature, various such preconditioners are known, coming from the fields of deflation, domain decomposition, and multigrid (MG). At first glance, these methods seem to be quite distinct; however, from an abstract point of view, they are closely related. The aim of this paper is to relate two-level PCG methods with symmetric two-grid (V(1,1)-cycle) preconditioners (derived from MG approaches), in their abstract form, to deflation methods and a two-level domain-decomposition approach inspired by the balancing Neumann-Neumann method. The MG-based preconditioner is often expected to be more effective than these other two-level preconditioners, but this is shown to be not always true. For common choices of the parameters, MG leads to larger error reductions in each iteration, but the work per iteration is more expensive, which makes this comparison unfair. We show that, for special choices of the underlying one-level preconditioners in the deflation or domain-decomposition methods, the work per iteration of these preconditioners is approximately the same as that for the MG preconditioner, and the convergence properties of the resulting two-level PCG methods will also be (approximately) the same. This means that, in this respect, the particular choice of the twolevel preconditioner is less important than the choice of the parameters. Numerical experiments are presented to emphasize the theoretical results.

Fast Linear Solver for Pressure Computation in Layered Domains

“…), we have found that damping makes no difference for the CG convergence, so the outcomes for ω < 1 are not displayed. Such a result has also been observed theoretically by Tang et al (2010) for an alternative deflation variant (known as 'DEF'). For the preconditioning variant (Prec.…”

“…The latter has been observed theoretically by Tang et al (2010) for the so-called 'DEF' variant (recall that we study the alternative ADEF2 variant). For multigrid methods with smoother M = I, a "typical choice of [ω] is close to 1 ||A||2 ", although a "better choice of [ω] is possible if we make further assumptions on how the eigenvectors of A associated with small eigenvalues are treated by coarse-grid correction" -- Tang et al (2010). In that reference, the latter is established for a coarse space that is based on a set of orthonormal eigenvectors of A.…”

“…Deflation is related to multigrid in the sense that it also makes use of a coarse space that is combined with a smoothing operator at the fine level. This relation has been considered from an abstract point of view by Tang et al (2009Tang et al ( , 2010.…”

Comparison of the deflated preconditioned conjugate gradient method and algebraic multigrid for composite materials

“…Errors associated with small-energy modes are corrected through a coarse-grid correction process, in which the problem is projected onto a low-dimensional subspace (the coarse grid), and these errors are resolved through a recursive approach. This decomposition is, in many ways, the same as that in deflation, u = (I − P T )u + P T u; the relationship between deflation and multigrid has been explored in [46,45]. For homogeneous PDEs discretized on structured grids, the separation into large-energy and smallenergy errors is well-understood, leading to efficient geometric multigrid schemes that offer both optimal algorithmic and parallel scalability.…”

Algebraic Theory of Two-Grid Methods

“…More insight can of course be gained when using additional assumptions. In the Section 4 of [52], one can find an analysis of the particular case where the columns of P coincide with the eigenvectors of M [52] clearly shows that, with proper scaling (e.g., ω such that λ max (M −1 A) = 1), the condition number with two smoothing steps (noted κ MG in [52]) is always strictly smaller than with just one smoothing step (noted κ DEF ).…”