Summary. Weighted max-norm bounds are obtained for Algebraic Additive Schwarz Iterations with overlapping blocks for the solution of Ax = b, when the coefficient matrix A is an M -matrix. The case of inexact local solvers is also covered. These bounds are analogous to those that exist using A-norms when the matrix A is symmetric positive definite. A new theorem concerningP -regular splittings is presented which provides a useful tool for the A-norm bounds. Furthermore, a theory of splittings is developed to represent Algebraic Additive Schwarz Iterations. This representation makes a connection with multisplitting methods. With this representation, and using a comparison theorem, it is shown that a coarse grid correction improves the convergence of Additive Schwarz Iterations when measured in weighted max norm.

https://www.math.temple.edu/~szyld/papers/wnorm_nummath.pdf

Weighted max norms, splittings, and overlapping additive Schwarz iterations

Summary. The convergence of multiplicative Schwarz-type methods for solving linear systems when the coefficient matrix is either a nonsingular M -matrix or a symmetric positive definite matrix is studied using classical and new results from the theory of splittings. The effect on convergence of algorithmic parameters such as the number of subdomains, the amount of overlap, the result of inexact local solves and of “coarse grid” corrections (global coarse solves) is analyzed in an algebraic setting. Results on algebraic additive Schwarz are also included.

https://www.math.temple.edu/~szyld/papers/multShw_NM.pdf

Algebraic theory of multiplicative Schwarz methods

A class of asynchronous Schwarz methods for the parallel solution of nonsingular linear systems of the form is investigated. This class includes, in particular, an asynchronous algebraic Schwarz method as well as asynchronous multisplitting. Theorems are obtained demonstrating c onvergence for the cases when 1 is nonnegative and when is an -matrix. The results shown are for both the situations with or without overlap between the domains in which an underlying mesh is divided, if such a mesh exists. Numerical experiments on systems of up to over ten million variables on up to 256 processors are presented. They illustrate the c onvergence properties of the method, as well as the fact that when the domains are not all of the same size, the asynchronous method can be up to 50% faster than the corresponding synchronous one. 1. Introduction. Consider the discretization of a linear second order elliptic boundary value problem (b.v.p.) on a domain Ω, and a fixed number of open subdomainsΩ Ω, 1 , such that Ω Ω. In general, one may assume that contiguous subdomains have a nonempty intersection and, in fact, this intersection may be larger than just the boundary of the subdomains, i.e., there may be some overlap between contiguous subdomains; see Section 4 for specific examples of these subdomains. The general idea of the Schwarz Alternating Procedure (SAP) is to solve the b.v.p. restricted to each subdomain, using as boundary conditions the function values of the (ap- proximate) solution of neighboring subdomains. This process is repeated and, under certain conditions, the process converges to the solution of the b.v.p. on Ω. In the additive Schwarz approach, which is the one considered in this paper, the approximate solution on Ω is obtained by adding up the solutions on all subdomains. When there is overlap, some weights are introduced to keep the resulting approximation consistent with the original b.v.p. These ideas go back to Schwarz (21), and have been revived in the last decade as the basis for domain decomposition methods, where these procedures are used as a preconditioner for the solution of the discretized nonsingular linear system

Asynchronous weighted additive schwarz methods

Lösung linearer Gleichungssysteme auf Parallelrechnern

Linear multisplitting methods are known as parallel iterative methods for solving a linear systemAx=b. We extend the idea of multisplittings to the problem of solving a nonlinear system of equationsF(x)=0. Our nonlinear multisplittings are based on several nonlinear splittings of the functionF. In a parallel computing environment, each processor would have to calculate the exact solution of an individual nonlinear system belonging to `his' nonlinear multisplitting and these solutions are combined to yield the next iterate. Although the individual systems are usually much less involved than the original system, the exact solutions will in general not be available. Therefore, we consider important variants where the exact solutions of the individual systems are approximated by some standard method such as Newton's method. Several methods proposed in literature may be regarded as special nonlinear multisplitting methods. As an application of our systematic approach we present a local convergence analysis of the nonlinear multisplitting methods and their variants. One result is that the local convergence of these methods is determined by an induced linear multisplitting of the Jacobian ofF.

Parallel nonlinear multisplitting methods

This is the fourth of a series of papers about the convergence rate of the Schwarz Alternating Procedure (SAP) for solving Mathematical Physics problems. In this paper, some typical values of the optimal overrelaxation factor, the maximum convergence factor and the convergence range of SAP for various values of ρ (see Evans et al. [1]) are given.

The convergence rate of the Schwarz alternating procedure (IV): with pseudo-boundary relaxation factor

This is the second of a series of papers about the convergence rate of SAP (Schwarz Alternating Procedure) for solving Mathematical Physics problems. In this paper, the convergence rate is obtained for two-dimensional model problems.

The convergence rate of the schwarz alternating procedure (II)—for two-dimensional problems

In this paper we propose an asynchronous parallel mixed algorithm for solving linear and nonlinear equations. This algorithm can be used not only on serial and parallel computers, but also on MIMD multiprocessor systems. The convergence of the algorithm has been proved under certain conditions. This paper gives some special cases of the algorithm which are known to us as efficient iterative methods. Numerical experiments are given to illustrate the method.

An asynchronous parallel mixed algorithm for linear and nonlinear equations

After introducing the parallel Schwarz overrelaxation method for linear systems, we analyse the convergence factor of the method in detail. The optimal overrelaxation parameter ω of the method is discussed in this paper. Some examples are also shown.

The convergence factor of the parallel Schwarz overrelaxation method for linear systems

This is the sixth paper about the rate of convergence of the SAP (Schwarz Alternating Procedure) for solving Mathematical Physics problems. In this paper, the convergence rate is obtained for solving the problem: by SAP with the third boundary value conditions on the pseudo-boundaries.

Shao Jian-Ping

Papers

The convergence rate of the Schwarz alternating procedure (IV): with pseudo-boundary relaxation factor

The convergence rate of the schwarz alternating procedure (II)—for two-dimensional problems

An asynchronous parallel mixed algorithm for linear and nonlinear equations

The convergence factor of the parallel Schwarz overrelaxation method for linear systems

The convergence rate of the Schwarz alternating procedure(VI) -for unsymmetric problems