Robust Preconditioners via Generalized Eigenproblems for Hybrid Sparse Linear Solvers

Abstract

The solution of large sparse linear systems is one of the most time consuming kernels in many numerical simulations. The domain decomposition community has developed many efficient and robust methods in the last decades. While many of these solvers fall into the abstract Schwarz (aS) framework, their robustness has originally been demonstrated on a case-by-case basis. In this paper, we propose a bound for the condition number of all deflated aS methods provided that the coarse grid consists of the assembly of local components that contain the kernel of some local operators. We show that classical results from the literature on particular instances of aS methods can be retrieved from this bound. We then show that such a coarse grid correction can be explicitly obtained algebraically via generalized eigenproblems, leading to a condition number independent of the number of domains. This result can be readily applied to retrieve or improve the bounds previously obtained via generalized eigenproblems in the particular cases of Neumann-Neumann (NN), Additive Schwarz (AS) and optimized Robin but also generalizes them when applied with approximate local solvers. Interestingly, the proposed methodology turns out to be a comparison of the considered particular aS method with generalized versions of both NN and AS for tackling the lower and upper part of the spectrum, respectively. We furthermore show that the application of the considered grid corrections in an additive fashion is robust in the AS case although it is not robust for aS methods in general. In particular, the proposed framework allows for ensuring the robustness of the AS method applied on the Schur complement (AS/S), either with deflation or additively, and with the freedom of relying on an approximate local Schur complement. Numerical experiments illustrate these statements.

An abstract theory of domain decomposition methods with coarse spaces of the GenEO family

“…A remarkable feature is that the coarse space V 0 NN (τ ) is the same as the alternate coarse space V 0 AS (1/τ ) for Additive Schwarz from Remark 7: there is a set of coarse vectors that fixes both the Neumann-Neumann preconditioners and the Additive Schwarz preconditioners. This was already pointed out in [1].…”

“…The amount of notation has been kept to the minimum. It is fair to mention that the article [1] proposes a setting similar to the one here. One difference is that here all abstract results are proved from scratch to fit exactly the framework that is considered.…”

“…Throughout this article we consider the problem of finding x * ∈ R n that is the solution of the following linear system Ax * = b, where A ∈ R n×n is symmetric positive and definite (spd), (1) for a given right hand side b ∈ R n . The applications to bare in mind are ones for which A is typically sparse and the number n of unknowns is very large.…”

Overlapping Schwarz methods with GenEO coarse spaces for indefinite and non-self-adjoint problems

“…In [37], it was rigorously proved that (when combined with a one-level method and using preconditioned conjugate gradients as the iterative solver), the resulting algorithm enjoys not only scalability with respect to the number of subdomains, but also robustness with respect to coefficient variation. Recent contributions in this very active field include [1,25,3] and [36], while more complete lists of contributions can be found in the literature surveys in, e.g., [37] and [15].…”

Fully algebraic domain decomposition preconditioners with adaptive spectral bounds

“…Throughout this article we consider the problem of finding x * ∈ R n that is the solution of the following linear system Ax * = b, where A ∈ R n×n is symmetric positive and definite (spd), (1) for a given right hand side b ∈ R n . The applications we have in mind are ones for which A is sparse and the number n of unknowns is very large.…”