Nonlinear Schwarz iterations with reduced rank extrapolation

Abstract

Abstract. Extrapolation methods can be a very effective technique used for accelerating the convergence of vector sequences. In this paper, these methods are used to accelerate the convergence of Schwarz iterative methods for nonlinear problems. A new implementation of the the reducedrank-extrapolation (RRE) method is introduced. Some convergence analysis is presented, and it is shown numerically that certain extrapolation methods can indeed be very effective in accelerating the convergence of Schwarz methods.Key words. Vector extrapolation, Schwarz methods, domain decomposition, iterative methods, nonlinear problems AMS subject classifications. 65B05, 65J15, 35J651. Introduction. A useful iterative method for solving large elliptic problems is to subdivide the domain into many (overlapping) subdomains and solve smaller elliptic problems on each subdomain in parallel. For example, the restricted additive Schwarz iterative method (RAS) solves a boundary value problem for a partial differential equation approximately by dividing it into boundary value problems on the smaller domains and adding the results corresponding to the subdomains without the overlap; see section 2 for a full description. Such methods are well-known to provide very efficient preconditioners as well. Different boundary conditions on the artificial interfaces, such as Dirichlet or Robin conditions, have been developed; see, e.g., [7,8,18,20,29].Extrapolation methods are used for accelerating the convergence of large class of vector sequences; see, e.g., the review article by Smith, Ford and Sidi [28], or the book by Brezinski and Redivo-Zaglia [2]. In particular, these methods are employed to accelerate the convergence of fixed point iterative techniques for linear and nonlinear systems of equations; see, e.g [2,9,14,28]. Vector extrapolation methods can be divided into two families, polynomial methods and epsilon algorithms. We consider in this paper the first family which include the minimal polynomial extrapolation (MPE) method of Cabay and Jackson [4] [25,28], and references therein. The convergence and stability analysis of the methods MPE, RRE, MMPE was treated by Sidi and his co-authors in [22,24,27]. We mention the recent paper by , in which a very efficient epsilon algoetithm is presented.In this paper, we apply the reduced-rank-extrapolation method to the sequence of vector iterates produced by RAS, and by its multiplicative counterpart. We show experimentally that this extrapolation can indeed provide a substantial acceleration,

Vector extrapolation method for non-overlapping Schwarz iterations

“…The main idea of the acceleration algorithm that we develop bellow was applied for ordinary differential equations ODEs for example in Duminil et al (2015).…”

“…Several works have treated the acceleration of Schwarz methods, for example, in Duminil et al (2015) the authors use the reduced rank extrapolation to accelerate Schwarz iterations, another idea was described in Linel (2011), to accelerate domain decomposition methods for the ordinary differential equations ODEs.…”

New approach for accelerating the nonlinear Schwarz iterations

“…There exist different variants of the Epsilon algorithm that can be used with vector sequences: the vector Epsilon algorithm, or the scalar Epsilon algorithm applied to each component of the vector sequences [7,10]. There have many works that have treated the acceleration of domain decomposition methods, for example in [19], the authors accelerate the nonlinear Schwarz iterations by reduced rank extrapolation method. Another idea was described in [16], to accelerate Schwarz iterations for ordinary differential equations ODEs.…”

Shanks Sequence Transformations and Anderson Acceleration

“…Their application to the solution of systems of linear and nonlinear equations will be discussed throughout the paper. For details on these methods, which are widely used, and their many applications, see, for example, [6,20,33,34,39,51,69,76].…”