Multiplicative Schwarz Method Research Articles

Convergence analysis of space-time domain decomposition method for parabolic equations

Space-time domain decomposition methods have been successfully applied to solve linear systems that arise from the discretization of time-dependent problems. In this paper, we present a space-time pure multiplicative Schwarz method for solving linear parabolic equations. With this method, the systems of equations are first coupled together and then solved by preconditioned GMRES, so the solutions at arbitrary time steps can be obtained simultaneously at a time. Under some mild assumptions, we develop an optimal convergence theory and show that the convergence rate is bounded independently of the mesh sizes, the subdomain partition and the window size. Some numerical results are reported to illustrate the optimality and scalability of the proposed method. Moreover, the numerical comparison also shows that our method has a faster convergence rate.

A Localized Version of the Method of Fundamental Solutions in a Multi-level Context

The Method of Fundamental Solutions is applied to the Laplace equation. Instead of using the traditional approach with external source points and boundary collocation points, the original domain decomposed into a lot of smaller, overlapping subdomains, and the Method of Fundamental Solutions is used to the individual local subdomains. After eliminating the local source points, local schemes are obtained. Instead of constructing a global scheme, the local subproblems are solved sequentially, in an iterative way. This mimics a multiplicative Schwarz method with overlapping subdomains, which assures the convergence of the method. Combining the iteration with a simple Seidel-type method, the resulting iteration is used as a smoothing procedure of a multi-level method. The points belonging to the coarse and fine levels are defined by a quadtree-generated cell system controlled by the boundary of the original domain. The multi-level character of the obtained method makes it possible to reduce the necessary number of iterations, that is, the overall computational cost can be significantly reduced. Moreover, the solution of large and ill-conditioned systems is completely avoided. The method is illustrated through several numerical test examples.

A two-level method for isogeometric discretizations based on multiplicative Schwarz iterations

Isogeometric Analysis (IGA) is a computational technique for the numerical approximation of partial differential equations (PDEs). This technique is based on the use of spline-type basis functions, that are able to hold a global smoothness and allow to exactly capture a wide set of common geometries. The current rise of this approach has encouraged the search of fast solvers for isogeometric discretizations and nowadays this topic is receiving a lot of attention. In this framework, a desired property of the solvers is the robustness with respect to both the polynomial degree p and the mesh size h. For this task, in this paper we propose a two-level method such that a discretization of order p is considered in the first level whereas the second level consists of a linear or quadratic discretization. On the first level, we suggest to apply one single iteration of a multiplicative Schwarz method. The choice of the block-size of such an iteration depends on the spline degree p, and is supported by a local Fourier analysis (LFA). At the second level one is free to apply any given strategy to solve the problem exactly. However, it is also possible to get an approximation of the solution at this level by using an h-multigrid method. The resulting solver is efficient and robust with respect to the spline degree p. Finally, some numerical experiments are given in order to demonstrate the good performance of the proposed solver.

Multilevel space‐time multiplicative Schwarz preconditioner for parabolic equations

AbstractSpace‐time methods are able to solve time‐dependent problems faster by exploiting the full power of high‐performance computer. In this article, a multilevel space‐time multiplicative Schwarz method is presented for solving parabolic equations in parallel on both spatial and temporal directions. In the implementation, the proposed Schwarz method is treated as preconditioner for GMRES, that is, a coupled system arising from the discretization of the parabolic equation is solved by using a multiplicative Schwarz preconditioned GMRES algorithm. We develop an optimal convergence theory to show that the convergence rate is bounded and independent of the spatial mesh sizes, the time step size, the number of subdomains, the number of levels, and the window size. Some numerical results obtained on a parallel computer with thousands of processors are presented to confirm the theory in terms of optimality and scalability. Moreover, numerical comparisons with traditional time‐stepping algorithms show that the proposed space‐time method earns lots of benefits when the number of processor cores is large.

Parallel multiplicative Schwarz preconditioner for solving nonselfadjoint elliptic problems

In this paper, a new multiplicative Schwarz method is presented for solving a system arising from the discretization of the nonselfadjoint elliptic equations. In the implementation, we apply the proposed Schwarz method as a Field-of-Value (FOV) equivalent preconditioner which is accelerated with the GMRES iterative solver. By employing a strengthened Cauchy–Schwarz inequality and a stable multilevel decomposition under a new norm, we obtain the optimal convergence theory by choosing the parameters in the Schwarz operator appropriately. It shows that the lower and upper bounds for the spectrum of the preconditioned Schwarz operator are bounded independently of the fine mesh size, the number of subdomains and mesh levels. Some numerical results are reported to verify the theory in terms of optimality and scalability. Moreover, numerical comparisons show that the proposed method is competitive with the classical multiplicative Schwarz algorithm for solving the convection–diffusion equations.

Analysis of the multiplicative Schwarz method for matrices with a special block structure

We analyze the convergence of the (algebraic) multiplicative Schwarz method applied to linear algebraic systems with matrices having a special block structure that arises, for example, when a (partial) differential equation is posed and discretized on a two-dimensional domain that consists of two subdomains with an overlap. This is a basic situation in the context of domain decomposition methods. Our analysis is based on the algebraic structure of the Schwarz iteration matrices, and we derive error bounds that are based on the block diagonal dominance of the given system matrix. Our analysis does not assume that the system matrix is symmetric (positive definite), or has the M- or H-matrix property. Our approach is motivated by, and significantly generalizes, an analysis for a special one-dimensional model problem of Echeverría et al. given in [Electron. Trans. Numer. Anal., 48 (2018), pp. 40–62].

Domain decomposition based SIC detectors for massive MIMO

Recently, domain decomposition techniques were used for the first time to develop a novel block-wise parallel Interference Cancellation (PIC) detector for communication systems (Bentrcia, 2017). The author simulated and tested a novel block-wise PIC detector to cancel Inter-Antenna-Interference (IAI) in a massive Multiple Input Multiple Output (MIMO) uplink system. In this work, we complement the work in Bentrcia (2017) by developing the serial version of the proposed block-wise PIC detector; specifically, we devise a new block-wise successive interference cancellation (SIC) detector using Multiplicative Schwarz (MS) method. As far as we know, no such work has been proposed before. Compared to the block-wise PIC detector of Bentrcia (2017), the proposed block-wise SIC detector exhibits several advantages such as lower computational complexity, lower detection delay and faster convergence. After deriving the update equation of the new block-wise SIC detector, we study its conditions of convergence and the impact of some design parameters on the performance of the novel block-wise SIC detector. Finally, extensive numerical simulations are conducted to validate our theoretical findings.

Convergence of the multiplicative Schwarz method for singularly perturbed convection-diffusion problems discretized on a Shishkin mesh

We analyze the convergence of the multiplicative Schwarz method applied to nonsymmetric linear algebraic systems obtained from discretizations of one-dimensional singularly perturbed convection-diffusion equations by upwind and central finite differences on a Shishkin mesh. Using the algebraic structure of the Schwarz iteration matrices we derive bounds on the infinity norm of the error that are valid from the first step of the iteration. Our bounds for the upwind scheme prove rapid convergence of the multiplicative Schwarz method for all relevant choices of parameters in the problem. The analysis for the central difference is more complicated, since the submatrices that occur are nonsymmetric and sometimes even fail to be M-matrices. Our bounds still prove the convergence of the method for certain parameter choices.

Domain decomposition adaptivity for the Richards equation model

This paper presents a study on efficient and economical domain decomposition adaptivity for Richards equation problems. Many real world applications of the Richards equation model typically involve solving systems of linear equations of huge dimensions. Multi-thread methods are therefore often preferred in order to reduce the required computation time. Multi-thread (parallel) execution is typically achieved by domain decomposition methods. In the case of non-homogeneous materials, the problem conditioning can be significantly improved if the computational domain is split efficiently, as each subdomain can cover only a certain material set within some defined parameter range. For linear problems, e.g. heat conduction, it is very easy to split the domain in this way. A problem arises for the nonlinear Richards equation, where the values of the constitutive functions, even over a homogeneous material, can vary within several orders of magnitude, see e.g. Kuraz et al. (Appl Math Comput 2012, in press). If the Rothe method is considered for time integration, a robust algorithm will be obtained if the domain is split adaptively over the time integration levels. The domain decomposition technique considered here is the standard multiplicative Schwarz method with coarse level, see e.g. Toselli and Widlund (Domain decomposition: methods algorithms and theory, Springer, Berlin, 2005). A method of the domain decomposition adaptivity will be studied in this paper.

Greedy and randomized versions of the multiplicative Schwarz method

We consider sequential, i.e., Gauss–Seidel type, subspace correction methods for the iterative solution of symmetric positive definite variational problems, where the order of subspace correction steps is not deterministically fixed as in standard multiplicative Schwarz methods. Here, we greedily choose the subspace with the largest (or at least a relatively large) residual norm for the next update step, which is also known as the Gauss–Southwell method. We prove exponential convergence in the energy norm, with a reduction factor per iteration step directly related to the spectral properties, e.g., the condition number, of the underlying space splitting. To avoid the additional computational cost associated with the greedy pick, we alternatively consider choosing the next subspace randomly, and show similar estimates for the expected error reduction. We give some numerical examples, in particular applications to a Toeplitz system and to multilevel discretizations of an elliptic boundary value problem, which illustrate the theoretical estimates.

Parallel GMRES with a multiplicative Schwarz preconditioner

This paper presents a robust hybrid solver for linear systems that combines a Krylov subspace method as accelerator with a Schwarz-based preconditioner. This preconditioner uses an explicit formulation associated to one iteration of the multiplicative Schwarz method. The Newtonbasis GMRES, which aim at expressing a good data parallelism between subdomains is used as accelerator. In the first part of this paper, we present the pipeline parallelism that is obtained when the multiplicative Schwarz preconditioner is used to build the Krylov basis for the GMRES method. This is referred as the first level of parallelism. In the second part, we introduce a second level of parallelism inside the subdomains. For Schwarz-based preconditioners, the number of subdomains are keeped small to provide a robust solver. Therefore, the linear systems associated to subdomains are solved efficiently with this approach. Numerical experiments are performed on several problems to demonstrate the benefits of using these two levels of parallelism in the solver, mainly in terms of numerical robustness and global efficiency. Cet article présente un solveur hybride robuste pour des systèmes linéaires. Ce solveur parallèle construit un préconditionneur de type Schwarz pour accélerer une méthode basée sur les sous-espaces de Krylov. Le préconditionneur est défini à partir d’une formulation explicite correspondant à une itération de Schwarz multiplicatif. Dans le but de réduire les communications et les dépendences entre les sous-domaines, nous utilisons la version de GMRES qui dissocie la construction de la base de Krylov et son orthogonalisation. Nous présentons dans un premier temps le parallélisme qui est obtenu lorsque ce préconditionneur Schwarz multiplicatif est utilisé dans la construction de la base de Krylov. C’est le premier niveau de parallélisme. Dans la deuxième partie de ce travail, nous introduisons un deuxième niveau de parallélisme à l’intérieur de chaque sous-domaine. Pour des décompositions de domaines avec recouvrement, le nombre de sous-domaines doit rester faible pour fournir un solveur robuste. De ce fait, les systèmes linéaires associés aux sous-domaines sont résolus de manière efficace avec ce deuxième niveau de parallélisme. Plusieurs tests numériques sont présentés à la fin du document pour valider l’efficacité de cette approche.

A multiplicative multisplitting method for solving the linear complementarity problem

The convergence of the multiplicative multisplitting-type method for solving the linear complementarity problem with an H-matrix is discussed using classical and new results from the theory of splitting. This directly results in a sufficient condition for guaranteeing the convergence of the multiplicative multisplitting method. Moreover, the multiplicative multisplitting method is applied to the H-compatible splitting and the multiplicative Schwarz method, separately. Finally, we establish the monotone convergence of the multiplicative multisplitting method under appropriate conditions.

A multiplicative Schwarz method and its application to nonlinear acoustic-structure interaction

A new Schwarz method for nonlinear systems is presented, constituting the multiplicative variant of a straightforward additive scheme. Local convergence can be guaranteed under suitable assumptions. The scheme is applied to nonlinear acoustic-structure interaction problems. Numerical examples validate the theoretical results. Further improvements are discussed by means of introducing overlapping subdomains and employing an inexact strategy for the local solvers.

A multiplicative Schwarz method with active subdomains for transient convection–diffusion problems

AbstractAn efficient algorithm to find the solution of transient convection–diffusion problems with dominant convection is presented. The main idea is to follow the solution front and activate those subdomains where the solution satisfies a given threshold value. We call this novel method ‘the multiplicative Schwarz method with active subdomains’, and it is motivated by the solution of a problem from activated‐carbon filters used in the automotive industry to reduce emissions. Numerical experiments show that this method is more efficient than the preconditioned conjugate gradient method with an incomplete Cholesky factorization. Copyright © 2009 John Wiley & Sons, Ltd.

Schwarz Methods for Quasi-Likelihood in Generalized Linear Models

The topic is maximum likelihood estimation in generalized linear models by an approximation using a sequence of sub-models, referred to here as blocks. More broadly, the topic is the solution of a non-linear system of equations of the type that arises in quasi-likelihood estimation, to be solved for δ ∈ R m . The Schwarz methods use a sequence of sub-models, each sub-model corresponding to a subset of the components of δ, the sub-models being patched together to yield the solution for the full model. The technique might be useful for model comparison, where the fitted values from a sub-model are used as starting values for a larger model. And we explore the use of the multiplicative Schwarz method and the damped additive Schwarz method for generalized linear model with large sample sizes. It is shown that the convergence of the proposed iterative procedures is guaranteed for canonical link functions and uncanonical link functions.

Thermal models for copper-alloyed dies in pressure die casting

A recent innovation in pressure die casting is the use of copper-alloyed dies suitably protected with a thermally sprayed steel layer. The thermal response of copper-alloyed dies is dictated principally by the deposited layer, the cooling arrangement and the casting geometry. This paper is concerned with the development of efficient numerical models for the prediction of both steady-state and transient thermal behaviour of the new die designs. Die temperatures are cyclic but useful information is obtained from a steady-state model, which provides for time-averaged temperatures and energy fluxes. The modelling strategy presented in the paper involves the indirect determination of transient temperatures. A perturbation approach is adopted, where a model for the difference between transient and steady-state temperature is developed. It is shown that this approach can be utilised to determine transient temperature efficiently once steady-state information is available with the transient perturbation model only involving surfaces where a significant variation in temperature occurs. The die-thermal models are founded on the boundary element method as die surface temperatures are of primary importance in pressure die casting. The finite element method is used to model casting solidification, where the formulation adopted provides for accurate energy transport. The dies form a multi-domain environment for thermal predictions making it necessary to utilise a suitably constructed coarse preconditioner to enhance numerical stability and provide for efficient computation. A multiplicative Schwarz method is presented that enables a parameter matrix accelerated GMRES method to be applied on each domain. Numerical experiments are performed to demonstrate the computational effectiveness of the approach. Predicted temperatures are compared with thermocouple readings obtained from a copper–steel die on a commercial die casting machine.

An Iterative domain decomposition algorithm for a nonlinear convection-diffusion problem

This article describes an iterative domain decomposition algorithm for solving nonlinear singularly perturbed convection-diusi on problems where convection is dominant. The domain decomposition algorithm consists of the two iterative processes: outer iterations and inner iterations. One outer iterative step represents computing nonlinear dierence subproblems on overlapping subdomains in serial according to upwind error propagation (the multiplicative Schwarz method). At the level of the inner iterations, each nonlinear subproblem is solved by the monotone additive Schwarz algorithm. The advantages of the algorithm are that the algorithm solves only linear discrete systems at each iterative step, converges monotonically to the exact solution of the system, and is potentially parallelisable. Results of numerical experiments are presented.

Multiplicative Schwarz algorithms for the p-version Galerkin boundary element method in 3D

We study a 2-level multiplicative Schwarz method for the p version Galerkin boundary element method for a weakly singular integral equation of the first kind in 3D. We prove that the rate of convergence of the multiplicative Schwarz operator for the p version grows only logarithmically in p and is independent of h.

Convergence Rate of a Schwarz Multilevel Method for the Constrained Minimization of Nonquadratic Functionals

In [L. Badea, Convergence Rate of a Multiplicative Schwarz Method for Strongly Nonlinear Inequalities, V. Barbu, I. Lasiecka, D. Tiba, and C. Varsan, eds., Kluwer Academic Publishers, Boston, 2003], the convergence of a subspace correction method applied to the constrained minimization of a functional in a general reflexive Banach space has been proved, provided that the convex set verifies a certain assumption. This assumption is weaker than that in which the convex set is decomposed according to the space decomposition as a sum of subsets. In the Sobolev spaces, the proposed method becomes a multiplicative Schwarz method for the solution of the variational inequalities coming from the minimization of nonquadratic functionals. We prove in this paper that this assumption holds for the one-, two- and multilevel multiplicative Schwarz methods in the finite element space, and we explicitly write the constants in the error estimations depending on the overlapping and mesh parameters. Our error estimates are similar with those obtained for the minimization of quadratic functionals in [L. Badea, X.-C. Tai, and J. Wang, SIAM J. Numer. Anal., (2003), pp. 1052-1073], or with those obtained for the one-obstacle problem in [X.-C. Tai, Numer. Math., 93 (2003), pp. 755-786].

SchurRAS: A Restricted Version of the Overlapping Schur Complement Preconditioner

This paper presents a preconditioner based on solving approximate Schur complement systems with the overlapping restricted additive Schwarz (RAS) methods. The construction of the preconditioner, called SchurRAS, is as simple as in the standard RAS. The communication requirements for each application of the preconditioning operation are also similar to those of the standard RAS approach. In the particular case when the degree of overlap is one, then SchurRAS and RAS involve exactly the same communication volume per step. In addition, SchurRAS has the same degree of parallelism as RAS. In some numerical experiments with a model problem, the convergence rate of the method was found to be similar to that of the multiplicative Schwarz (MS) method. The Schur-based RAS usually outperforms the standard RAS both in terms of iteration count and CPU time. For a few two-dimensional scaled problems, SchurRAS was about twice as fast as the stardard RAS on 64 processors.