Asymptotic Convergence Factors Research Articles

Schwarz methods by domain truncation

Schwarz methods use a decomposition of the computational domain into subdomains and need to impose boundary conditions on the subdomain boundaries. In domain truncation one restricts the unbounded domain to a bounded computational domain and must also put boundary conditions on the computational domain boundaries. In both fields there are vast bodies of literature and research is very active and ongoing. It turns out to be fruitful to think of the domain decomposition in Schwarz methods as a truncation of the domain onto subdomains. Seminal precursors of this fundamental idea are papers by Hagstrom, Tewarson and Jazcilevich (1988), Després (1990) and Lions (1990). The first truly optimal Schwarz method that converges in a finite number of steps was proposed by Nataf (1993), and used precisely transparent boundary conditions as transmission conditions between subdomains. Approximating these transparent boundary conditions for fast convergence of Schwarz methods led to the development of optimized Schwarz methods – a name that has become common for Schwarz methods based on domain truncation. Compared to classical Schwarz methods, which use simple Dirichlet transmission conditions and have been successfully used in a wide range of applications, optimized Schwarz methods are much less well understood, mainly due to their more sophisticated transmission conditions.A key application of Schwarz methods with such sophisticated transmission conditions turned out to be time-harmonic wave propagation problems, because classical Schwarz methods simply do not work in this case. The past decade has given us many new Schwarz methods based on domain truncation. One review from an algorithmic perspective (Gander and Zhang 2019) showed the equivalence of many of these new methods to optimized Schwarz methods. The analysis of optimized Schwarz methods, however, is lagging behind their algorithmic development. The general abstract Schwarz framework cannot be used for the analysis of these methods, and thus there are many open theoretical questions about their convergence. Just as for practical multigrid methods, Fourier analysis has been instrumental for understanding the convergence of optimized Schwarz methods and for tuning their transmission conditions. Similar to local Fourier mode analysis in multigrid, the unbounded two-subdomain case is used as a model for Fourier analysis of optimized Schwarz methods due to its simplicity. Many aspects of the actual situation,e.g.boundary conditions of the original problem and the number of subdomains, were thus neglected in the unbounded two-subdomain analysis. While this gave important insight, new phenomena beyond the unbounded two-subdomain models were discovered.This present situation is the motivation for our survey: to give a comprehensive review and precise exploration of convergence behaviours of optimized Schwarz methods based on Fourier analysis, taking into account the original boundary conditions, many-subdomain decompositions and layered media. We consider as our model problem the operator$-\Delta + \eta $in the diffusive case$\eta>0$(screened Laplace equation) or the oscillatory case$\eta <0$(Helmholtz equation), in order to show the fundamental difference in behaviour of Schwarz solvers for these problems. The transmission conditions we study include the lowest-order absorbing conditions (Robin), and also more advanced perfectly matched layers (PMLs), both developed first for domain truncation. Our intensive work over the last two years on this review has led to several new results presented here for the first time: in the bounded two-subdomain analysis for the Helmholtz equation, we see strong influence of the original boundary conditions imposed on the global problem on the convergence factor of the Schwarz methods, and the asymptotic convergence factors with small overlap can differ from the unbounded two-subdomain analysis. In the many-subdomain analysis, we find the scaling with the number of subdomains,e.g.when the subdomain size is fixed, robust convergence of the double-sweep Schwarz method for the free-space wave problem, either with fixed overlap and zeroth-order Taylor conditions or with a logarithmically growing PML, and we find that Schwarz methods with PMLs work like smoothers that converge faster for higher Fourier frequencies; in particular, for the free-space wave problem, plane waves (in the error) passing through interfaces at a right angle converge more slowly. In addition to our main focus on analysis in Sections 2 and 3, we start in Section 1 with an expository historical introduction to Schwarz methods, and in Section 4 we give a brief interpretation of the recently proposed optimal Schwarz methods for decompositions with cross-points from the viewpoint of transmission conditions. We conclude in Section 5 with a summary of open research problems. In Appendix A we provide a Matlab program for a block LU form of an optimal Schwarz method with cross-points, and in Appendix B we give the Maple program for the two-subdomain Fourier analysis.

Modulus‐based multigrid methods for linear complementarity problems

SummaryBy employing modulus‐based matrix splitting iteration methods as smoothers, we establish modulus‐based multigrid methods for solving large sparse linear complementarity problems. The local Fourier analysis is used to quantitatively predict the asymptotic convergence factor of this class of multigrid methods. Numerical results indicate that the modulus‐based multigrid methods of the W‐cycle can achieve optimality in terms of both convergence factor and computing time, and their asymptotic convergence factors can be predicted perfectly by the local Fourier analysis of the corresponding modulus‐based two‐grid methods.

On a local Fourier analysis for overlapping block smoothers on triangular grids

A general local Fourier analysis for overlapping block smoothers on triangular grids is presented. This analysis is explained in a general form for its application to problems with different discretizations. This tool is demonstrated for two different problems: a stabilized linear finite element discretization of Stokes equations and an edge-based discretization of the curl–curl operator by lowest-order Nédélec finite element method. In this latter, special Fourier modes have to be considered in order to perform the analysis. Numerical results comparing two- and three-grid convergence factors predicted by the local Fourier analysis to real asymptotic convergence factors are presented to confirm the predictions of the analysis and show their usefulness.

On Block Preconditioners for Pde-Constrained Optimization Problems

Recently, Bai proposed a block-counter-diagonal and a block-counter-triangular preconditioning matrices to precondition the GMRES method for solving the structured system of linear equations arising from the Galerkin finite-element discretizations of the distributed control problems in (Computing 91 (2011) 379-395). He analyzed the spectral properties and derived explicit expressions of the eigenvalues and eigenvectors of the preconditioned matrices. By applying the special structures and properties of the eigenvector matrices of the preconditioned matrices, we derive upper bounds for the 2-norm condition numbers of the eigenvector matrices and give asymptotic convergence factors of the preconditioned GMRES methods with the block-counter-diagonal and the block-counter-triangular preconditioners. Experimental results show that the convergence analyses match well with the numerical results.

Multicolor Fourier analysis of the multigrid method for quadratic FEM discretizations

Quadratic finite element methods offer some advantages for the numerical solution of partial differential equations (PDEs), due to their improved approximation properties in comparison to linear approaches. The algebraic linear systems arising from the discretization of PDEs by this kind of methods require an efficient resolution, and multigrid methods provide a good way to solve this problem. To design geometric multigrid methods, local Fourier analysis (LFA) is a very useful tool. However, LFA for quadratic finite element discretizations can not be performed in a standard way, since the discrete operator is defined by different stencils depending on the location of the points in the grid. In this work, a multicolor local Fourier analysis is presented to analyze multigrid solvers for this type of discretizations. With the help of this analysis, some point-wise and line-wise smoothers are analyzed. Some results showing the good correspondence between the two-grid convergence factors predicted by the analysis and the experimentally computed asymptotic convergence factors are presented. Finally, this analysis is applied to design a very efficient multigrid solver for semi-structured triangular grids, based on a hybrid-smoother.

Consensus of second-order and high-order discrete-time multi-agent systems with random networks

This paper studies the convergence and convergence speed for the second-order and the high-order discrete-time multi-agent systems with random networks and arbitrary weights. Random networks mean that the existence of any edge is probabilistic and independent of any other edge. By introducing the agreement set, velocity control gain and high-order state control gain, some consensus protocols are provided for the discrete-time random networks. Moreover, the per-step and asymptotic convergence factors are proposed to measure the convergence and convergence speed. Some examples and simulation results are given to illustrate the effectiveness of the obtained theoretical results.

Convergence speed in distributed consensus over dynamically switching random networks

Characterizing convergence speed is one of the most important research challenges in the design of distributed consensus algorithms for networked multi-agent systems. In this paper, we consider a group of agents that communicate via a dynamically switching random information network. Each link in the network, which represents the directed/undirected information flow between any ordered/unordered pair of agents, could be subject to failure with a certain probability. Hence we model the information flow using dynamically switching random graphs. We characterize the convergence speed for the distributed discrete-time consensus algorithm over a variety of random networks with arbitrary weights. In particular, we propose the asymptotic and per-step (mean square) convergence factors as measures of the convergence speed and derive the exact value for the per-step (mean square) convergence factor. Numerical examples are also given to illustrate our theoretical results.

A note on the convergence of the secant method for simple and multiple roots

The secant method is one of the most popular methods for root finding. Standard text books in numerical analysis state that the secant method is superlinear: the rate of convergence is set by the gold number. Nevertheless, this property holds only for simple roots. If the multiplicity of the root is larger than one, the convergence of the secant method becomes linear. This communication includes a detailed analysis of the secant method when it is used to approximate multiple roots. Thus, a proof of the linear convergence is shown. Moreover, the values of the corresponding asymptotic convergence factors are determined and are found to be also related with the golden ratio.

Analysis of algebraic systems arising from fourth‐order compact discretizations of convection‐diffusion equations

AbstractWe study the properties of coefficient matrices arising from high‐order compact discretizations of convection‐diffusion problems. Asymptotic convergence factors of the convex hull of the spectrum and the field of values of the coefficient matrix for a one‐dimensional problem are derived, and the convergence factor of the convex hull of the spectrum is shown to be inadequate for predicting the convergence rate of GMRES. For a two‐dimensional constant‐coefficient problem, we derive the eigenvalues of the nine‐point matrix, and we show that the matrix is positive definite for all values of the cell‐Reynolds number. Using a recent technique for deriving analytic expressions for discrete solutions produced by the fourth‐order scheme, we show by analyzing the terms in the discrete solutions that they are oscillation‐free for all values of the cell Reynolds number. Our theoretical results support observations made through numerical experiments by other researchers on the non‐oscillatory nature of the discrete solution produced by fourth‐order compact approximations to the convection‐diffusion equation. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 155–178, 2002; DOI 10.1002/num.1041

Asynchronous multisplitting two-stage iterations for systems of weakly nonlinear equations

For the large sparse systems of weakly nonlinear equations arising in the discretizations of many classical differential and integral equations, this paper presents a class of asynchronous parallel multisplitting two-stage iteration methods for getting their solutions by the high-speed multiprocessor systems. Under suitable assumptions, we study the global convergence properties of these asynchronous multisplitting two-stage iteration methods. Moreover, for this class of new methods, we establish their local convergence theories, and precisely estimate their asymptotic convergence factors under some reasonable assumptions when the involved nonlinear mapping is only assumed to be directionally differentiable. Numerical computations show that our new methods are feasible and efficient for parallely solving the system of weakly nonlinear equations.

A class of two‐stage iterative methods for systems of weakly nonlinear equations

The discretizations of many differential equations by the finite difference or the finite element methods can often result in a class of system of weakly nonlinear equations. In this paper, by applying the two-tage iteration technique and in accordance with the special properties of this weakly nonlinear system, we first propose a general two-tage iterative method through the two-tage splitting of the system matrix. Then, by applying the accelerated overrelaxation (AOR) technique of the linear iterative methods, we present a two-tage AOR method, which particularly uses the AOR iteration as the inner iteration and is substantially a relaxed variant of the afore-presented method. For these two classes of methods, we establish their local convergence theories, and precisely estimate their asymptotic convergence factors under some suitable assumptions when the involved nonlinear mapping is only B-differentiable. When the system matrix is either a monotone matrix or an H-matrix, and the nonlinear mapping is a P-bounded mapping, we thoroughly set up the global convergence theories of these new methods. Moreover, under the assumptions that the system matrix is monotone and the nonlinear mapping is isotone, we discuss the monotone convergence properties of the new two-tage iteration methods, and investigate the influence of the matrix splittings as well as the relaxation parameters on the convergence behaviours of these methods. Numerical computations show that our new methods are feasible and efficient for solving the system of weakly nonlinear equations.

Parallel multisplitting two-stage iterative methods for large sparse systems of weakly nonlinear equations

The finite difference or the finite element discretizations of many differential or integral equations often result in a class of systems of weakly nonlinear equations. In this paper, by reasonably applying both the multisplitting and the two-stage iteration techniques, and in accordance with the special properties of this system of weakly nonlinear equations, we first propose a general multisplitting two-stage iteration method through the two-stage multiple splittings of the system matrix. Then, by applying the accelerated overrelaxation (AOR) technique of the linear iterative methods, we present a multisplitting two-stage AOR method, which particularly uses the AOR-like iteration as inner iteration and is substantially a relaxed variant of the afore-presented method. These two methods have a forceful parallel computing function and are much more suitable to the high-speed multiprocessor systems. For these two classes of methods, we establish their local convergence theories, and precisely estimate their asymptotic convergence factors under some suitable assumptions when the involved nonlinear mapping is only directionally differentiable. When the system matrix is either an H-matrix or a monotone matrix, and the nonlinear mapping is a P-bounded mapping, we thoroughly set up the global convergence theories of these new methods. Moreover, under the assumptions that the system matrix is monotone and the nonlinear mapping is isotone, we discuss the monotone convergence properties of the new multisplitting two-stage iteration methods, and investigate the influence of the multiple splittings as well as the relaxation parameters upon the convergence behaviours of these methods. Numerical computations show that our new methods are feasible and efficient for parallel solving of the system of weakly nonlinear equations.

On polynomial preconditioning and asymptotic convergence factors for indefinite Hermitian matrices

We are concerned with the minimal residual method combined with polynomial preconditioning for solving large linear systems Ax= b with indefinite Hermitian coefficient matrices A. The standard approach for choosing the polynomial preconditioner leads to preconditioned systems which are positive definite. Here, we investigate a different strategy which leaves the preconditioned coefficient matrix indefinite. More precisely, the polynomial preconditioner is designed to cluster the positive (negative) eigenvalues of A around 1 (around some negative constant). In particular, it is shown that such indefinite polynomial preconditioners can be obtained as the optimal solutions of a certain two-parameter family of Chebyshev approximation problems. The problem of selecting the parameters so that the resulting indefinite polynomial preconditioner speeds up the convergence of the minimal residual method optimally is also addressed. For this task, we propose an approach based on the concept of asymptotic convergence factors. Finally, some numerical examples of indefinite polynomial preconditioners are given.

Asynchronous multilevel adaptive methods for solving partial differential equations on multiprocessors: Basic ideas

Several mesh refinement methods exists for solving partial differential equations that make efficient use of local grids on scalar computers. On distributed memory multiprocessors, such methods benefit from their tendency to create multiple refinement regions, yet they suffer from the sequential way that the levels of refinement are treated. The asynchronous fast adaptive composite grid method (AFAC) is developed here as a method that can process refinement levels in parallel while maintaining full multilevel convergence speeds. In the present paper, we develop a simple two-level AFAC theory and provide estimates of its asymptotic convergence factors as it applies to very large scale examples. In a companion paper, we report on extensive timing results for AFAC, implemented on an Intel iPSC hypercube.

Optimum first and second order extrapolations of Successive Overrelaxation type methods for certain classes of matrices

This paper deals with the iterative solution of the linear systemx=Bx+c when its Jacobi matrixB is weakly 2-cyclic consistently ordered and has a complex eigenvalue spectrum which lies on a straight-line segment. The optimization problem of the following three methods is considered and solved: i) The extrapolation of the optimum Successive Overrelaxation (SOR) ii) The second order extrapolation of a “good” SOR and iii) The second order extrapolation of the Gauss-Seidel method. In addition a variant of the second order methods considered, suitable for the solution of the system even ifB isnot necessarily weakly 2-cyclic consistently ordered, is proposed. Finally a reference to a theoretical comparison of the various optimum methods in the paper is made and their asymptotic convergence factors for selected eigenvalue spectra are illustrated in a Table in support of the theory developed.