On the Scalability of the Schwarz Method

Abstract

Introduction and main resultsAn algorithm is said to be weakly scalable if it can solve progressively larger problems with an increasing number of processors in a fixed amount of time. According to classical Schwarz theory, the parallel Schwarz method (PSM) is not scalable (see, e.g., [2,7]). Recent results in computational chemistry, however, have shed more light on the scalability of the PSM: surprisingly, in contrast with classical Schwarz theory, the authors in [1] provide numerical evidence that in some cases the one-level PSM converges to a given tolerance within the same number of iterations independently of the number N of subdomains. This behaviour is observed if fixed-sized subdomains form a "chain-like" domain such that the intersection of the boundary of each subdomain with the boundary of the global domain is non-empty. This result was subsequently rigorously proved in [3,4,5] for the PSM and in [2] for other one-level methods. On the other hand, this weak scalability is lost if the fixed-sized subdomains form a "globular-type" domain Ω, where the boundaries of many subdomains lie in the interior of Ω. The following question therefore arises: is it possible to quantify the lack of scalability of the PSM for cases where individual subdomains are entirely embedded inside the global domain? To do so, for increasing N one would need to estimate the number of iterations necessary to achieve a given tolerance.Some isolated results in this direction do exist in the literature. For instance, in [2] a heuristic argument is used to explain why in the case of the PSM for the solution of a 1D Laplace problem an unfortunate initialisation leads to a contraction in the infinity norm being observed only after a number of iterations proportional to N.

Analysis of parallel Schwarz algorithms for time-harmonic problems using block Toeplitz matrices

“…Weak scalability results for the Laplace problem have been proven for more general chain-type geometries using various techniques, such as the maximum principle in [8] and a fully variational analysis in [9]. The most recent work on the topic without restrictive assumptions can be found in [10], where a propagation-tracking analysis based on graph theory and the maximum principle permitted a scalability analysis for very general decompositions. To our knowledge, there is no such analysis for Schwarz methods for time-harmonic wave propagation problems, where previous techniques no longer extend to, as the nature of the underlying equations is very different.…”

On the effect of boundary conditions on the scalability of Schwarz methods

“…= e 2λδ +e λL e 2λδ+λL +1 . Theorem 2 gives the same bound (8) for the convergence factors of PSM and OSM. This fact is not surprising.…”

“…which is known to be not scalable; see, e.g., [3,8]. The scalability of PSM and OSM for different external conditions applied at the top and at the bottom of the domain is summarized in Table The behavior of the bounds ρ RR (δ, q), ρ DR (δ, q) and ρ NR (δ, q) with respect to q is depicted in Fig.…”

Analysis of the Schwarz Domain Decomposition Method for the Conductor-like Screening Continuum Model

“…It turns out that for a convergence analysis in the latter case it is not obvious how results and tools available in classical analyses can be applied. Therefore, in recent papers [2,3,4,5,6] this topic has been addressed and new results on the convergence of the Schwarz domain decomposition iterative method on a family of domains Ω M , M = 2, 3, . .…”

“…no balls are allowed that are contained inside Ω M . A first step towards a more general analysis can be found in [6], which analyzes how the error propagates and contracts in the maximum norm in a general geometry. It is shown that for a molecule with N -layers, it takes N + 1 iterations until the first contraction in the maximum norm is obtained.…”