Comparison of Two-Level Preconditioners Derived from Deflation, Domain Decomposition and Multigrid Methods

Abstract

For various applications, it is well-known that a multi-level, in particular twolevel, preconditioned CG (PCG) method is an efficient method for solving large and sparse linear systems with a coefficient matrix that is symmetric positive definite. The corresponding two-level preconditioner combines traditional and projection-type preconditioners to get rid of the effect of both small and large eigenvalues of the coefficient matrix. In the literature, various two-level PCG methods are known, coming from the fields of deflation, domain decomposition and multigrid. Even though these two-level methods differ a lot in their specific components, it can be shown that from an abstract point of view they are closely related to each other. We investigate their equivalences, robustness, spectral and convergence properties, by accounting for their implementation, the effect of roundoff errors and their sensitivity to inexact coarse solves, severe termination criteria and perturbed starting vectors.

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

“…Moreover, from [30,48], the null space of P K never enters the iteration process, and the corresponding zero-eigenvalues do not influence the solution. The DPCG method [46] is given as Algorithm 1.…”

“…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.…”

Fast Deflation Methods with Applications to Two-Phase Flows

“…To stabilize the method, we can choose for the so-called adapted DPCG (ADPCG) method, see also e.g. [14], where we have to solve…”

Algebraic Theory of Two-Grid Methods

“…Alternatively, balancing Neumann-Neumann domain decomposition methods [27,28,49,54] symmetrize the preconditioner by repeating the coarse grid correction twice. Nevertheless, the results in Section 3.2 remain applicable in both cases because the eigenvalue distribution of the preconditioned matrix is identical or closely related to that of BA with B as in (2.7) and, say, only a single post-smoothing step (i.e., ν 1 = 0 , ν 2 = 1 and therefore Y = M 2 ); we refer to [53] for a comparative discussion of implementations and a detailed account on this matter.…”