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Summary: This paper presents some recent advances for parallel‐in‐time methods applied to linear elasticity. With recent computer architecture changes leading to stagnant clock speeds, but ever increasing numbers of cores, future speedups will be available through increased concurrency. Thus, sequential algorithms, such as time stepping, will suffer a bottleneck. This paper explores multigrid reduction in time (MGRIT) for an important application area, linear elasticity. Previously, efforts at parallel‐in‐time for elasticity have experienced difficulties, for example, the beating phenomenon. As a result, practical parallel‐in‐time algorithms for this application area currently do not exist. This paper proposes some solutions made possible by MGRIT (e.g., slow temporal coarsening and FCF‐relaxation) and, more importantly, a different formulation of the problem that is more amenable to parallel‐in‐time methods. Using a recently developed convergence theory for MGRIT and Parareal, we show that the changed formulation of the problem avoids the instability issues and allows the reduction of the error using two temporal grids. We then extend our approach to the multilevel case, where we demonstrate how slow temporal coarsening improves convergence. The paper ends with supporting numerical results showing a practical algorithm enjoying speedup benefits over the sequential algorithm. [ABSTRACT FROM AUTHOR]

Additive Schwarz preconditioned GMRES is a powerful method for solving large sparse linear systems of equations on parallel computers. The algorithm is often implemented in the Euclidean norm, or the discrete l2 norm, however, the optimal convergence result is available only in the energy norm (or the equivalent Sobolev H1 norm). Very little progress has been made in the theoretical understanding of the l2 behaviour of this very successful algorithm. To add to the difficulty in developing a full l2 theory, in this note, we construct explicit examples and show that the optimal convergence of additive Schwarz preconditioned GMRES in l2 cannot be obtained using the existing GMRES theory. More precisely speaking, we show that the symmetric part of the preconditioned matrix, which plays a role in the Eisenstat–Elman–Schultz theory, has at least one negative eigenvalue, and we show that the condition number of the best possible eigenmatrix that diagonalizes the preconditioned matrix, key to the Saad–Schultz theory, is bounded from both above and below by constants multiplied by h-1/2. Here h is the finite element mesh size. The results presented in this paper are mostly negative, but we believe that the techniques used in our proofs may have wide applications in the further development of the l2 convergence theory and in other areas of domain decomposition methods. Copyright © 2002 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The ChebFilterCG algorithm, proposed by Golub, Ruiz, and Touhami [SIAM J. Matrix Anal. Appl. 29 (2007), pp. 774-795] is an iterative method that combines Chebyshev filter and conjugate gradient for solving symmetric positive definite linear systems with multiple right-hand sides. The Chebyshev filter is used to produce initial residuals rich in eigenvectors corresponding to the smallest eigenvalues, which are then used in the initial phase of the conjugate gradient. This paper presents a convergence analysis of ChebFilterCG. In particular, it is shown theoretically and numerically that the algorithm yields an approximation of the invariant subspace associated with the smallest eigenvalues that can be recycled for solving several linear systems with the same matrix and different right-hand sides. A refined error bound when solving these systems is also given. The choice and influence of the Chebyshev filtering steps is discussed. Numerical experiments are described to illustrate that the Chebyshev filter does not degrade the distribution of the smallest eigenvalues and highlight the effect of rounding errors when large outlying eigenvalues are present. Finally, it is shown that the method may become more effective when an additional Chebyshev filtering step is used in the initialization phase of ChebFilterCG. [ABSTRACT FROM AUTHOR]

SUMMARY An iterative method for finding the largest eigenvalue of a nonnegative tensor was proposed by Ng, Qi, and Zhou in 2009. In this paper, we establish an explicit linear convergence rate of the Ng-Qi-Zhou method for essentially positive tensors. Numerical results are given to demonstrate linear convergence of the Ng-Qi-Zhou algorithm for essentially positive tensors. Copyright © 2011 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY In this paper, a practical two-term acceleration algorithm is proposed, the interval of the parameter which guarantees the convergence of the acceleration algorithm is analyzed in detail. Further, the acceleration ratio of the new acceleration algorithm is obtained in advance. The new acceleration algorithm is less sensitive to the parameter than the Chebyshev semi-iterative method. Finally, some numerical examples show that the accelerated algorithm is effective. Copyright © 2011 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper presents some variants of the inexact Newton method for solving systems of nonlinear equations defined by locally Lipschitzian functions. These methods use variants of Newton's iteration in association with Krylov subspace methods for solving the Jacobian linear systems. Global convergence of the proposed algorithms is established under a nonmonotonic backtracking strategy. The local convergence based on the assumptions of semismoothness and BD-regularity at the solution is characterized, and the way to choose an inexact forcing sequence that preserves the rapid convergence of the proposed methods is also indicated. Numerical examples are given to show the practical viability of these approaches. Copyright © 2009 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In 1979, Gay proved that Broyden's methods, when used for n-square linear systems, terminate in at most 2n iterations (SIAM J. Numer. Anal. 1979; 16:623–630). Also, the ABS methods were introduced in 1984 (Numer. Math. 1984; 45:1361–1376). In this paper we show another (handy) proof of Gay's theorem by these algorithms. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper considers the inexact Barzilai–Borwein (BB) algorithm applied to saddle point problems. To this aim, we study the convergence properties of the inexact BB algorithm for symmetric positive definite linear systems. Suppose that gk and &gtilde;k are the exact residual and its approximation of the linear system at the kth iteration, respectively. We prove the R-linear convergence of the algorithm if ∥&gtilde;k – gk∥⩽η∥&gtilde;k∥ for some small η>0 and all k. To adapt the algorithm for solving saddle point problems, we also extend the R-linear convergence result to the case when the right-hand term ∥&gtilde;k∥ is replaced by ∥&gtilde;k–1∥. Although our theoretical analyses cannot provide a good estimate to the parameter η, in practice, we find that η can be as large as the one in the inexact Uzawa algorithm. Further numerical experiments show that the inexact BB algorithm performs well for the tested saddle point problems. Copyright © 2007 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The DEFLATED-GMRES(m, k) method is one of the major iterative solvers for the large sparse linear systems of equations, x = b. This algorithm assembles a preconditioner adaptively for the GMRES(m) method based on eigencomponents gathered from the Arnoldi process during iterations. It is usually known that if a restarted GMRES(m) method is used to solve linear systems of equations, the information of the smallest eigencomponents is lost at each restart and the super-linear convergence may also be lost. In this paper, we propose an adaptive procedure that combines the DEFLATED-GMRES(m, k) algorithm and the determination of a restart frequency automatically. It is shown that a new algorithm combining elements of both will reduce the negative effects of the restarted procedure. The numerical experiments are presented on three test problems by using the MIMD parallel machine AP3000. From these numerical results, we show that the proposed algorithm leads to faster convergence than the conventional DEFLATED-GMRES(m, k) method. Copyright © 2000 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The solution of nonsymmetric systems of linear equations continues to be a difficult problem. A main algorithm for solving nonsymmetric problems is restarted GMRES. The algorithm is based on restarting full GMRES every s iterations, for some integer s > 0. This paper considers the impact of the restart frequency s on the convergence and work requirements of the method. It is shown that a good choice of this parameter can lead to reduced solution time, while an improper choice may hinder or preclude convergence. An adaptive procedure is also presented for determining automatically when to restart. The results of numerical experiments are presented. [ABSTRACT FROM AUTHOR]

We focus on the interplay between the choice of partition (problem decomposition) and the corresponding rate of convergence of parallel numerical algorithms. Using a specific algorithm, for which the numerics depend upon the partition, we demonstrate that the rate of convergence can depend strongly on the choice of the partition. This dependence is shown to be a function of the algorithm and of the choice of problem. Information gleaned from tests using various 2-way partitions leads to new partitions for which some degree of convergence robustness is exhibited. The incorporation of a known correction for approximate Schur complements into the original algorithm yields a modified parallel algorithm which numerical experiments indicate achieves robust convergence behaviour with respect to the choice of partition. We conclude that tests of a parallel algorithm which vary the method of partitioning can provide constructive information regarding the robustness of the algorithm and guidance for modifying the algorithm or the choice of partitioning algorithm to make the overall computations more robust. Copyright © 2003 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

GMRES(k) is widely used for solving non-symmetric linear systems. However, it is inadequate either when it converges only for k close to the problem size or when numerical error in the modified Gram–Schmidt process used in the GMRES orthogonalization phase dramatically affects the algorithm performance. An adaptive version of GMRES(k) which tunes the restart value k based on criteria estimating the GMRES convergence rate for the given problem is proposed here. This adaptive GMRES(k) procedure outperforms standard GMRES(k), several other GMRES-like methods, and QMR on actual large scale sparse structural mechanics postbuckling and analog circuit simulation problems. There are some applications, such as homotopy methods for high Reynolds number viscous flows, solid mechanics postbuckling analysis, and analog circuit simulation, where very high accuracy in the linear system solutions is essential. In this context, the modified Gram–Schmidt process in GMRES, can fail causing the entire GMRES iteration to fail. It is shown that the adaptive GMRES(k) with the orthogonalization performed by Householder transformations succeeds whenever GMRES(k) with the orthogonalization performed by the modified Gram–Schmidt process fails, and the extra cost of computing Householder transformations is justified for these applications. © 1998 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]