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Rate of convergence

About: Rate of convergence is a research topic. Over the lifetime, 31257 publications have been published within this topic receiving 795334 citations. The topic is also known as: convergence rate.


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TL;DR: Numerical results show that the HSS iteration method and its inexact variant are efficient and robust solvers for this class of continuous Sylvester equations.
Abstract: We present a Hermitian and skew-Hermitian splitting (HSS) iteration method for solving large sparse continuous Sylvester equations with non-Hermitian and positive definite/semidefinite matrices. The unconditional convergence of the HSS iteration method is proved and an upper bound on the convergence rate is derived. Moreover, to reduce the computing cost, we establish an inexact variant of the HSS iteration method and analyze its convergence property in detail. Numerical results show that the HSS iteration method and its inexact variant are efficient and robust solvers for this class of continuous Sylvester equations.

137 citations

Journal ArticleDOI
TL;DR: In this article, a general presentation of non-overlapping domain decomposition methods for harmonic wave propagation models is given, which leads to concise convergence proofs and contains some recent developments about the use of non local transmission conditions.

137 citations

Journal ArticleDOI
TL;DR: In this paper, a second-order adjoint model was proposed for a shallow-water equation model on a limited-area domain, and the sensitivity of the cost function with respect to distributed observations was analyzed.
Abstract: The adjoint method application in variational data assimilation provides a way of obtaining the exact gradient of the cost functionj with respect to the control variables. Additional information may be obtained by using second order information. This paper presents a second order adjoint model (SOA) for a shallow-water equation model on a limited-area domain. One integration of such a model yields a value of the Hessian (the matrix of second partial derivatives, ∇2 J) multiplied by a vector or a column of the Hessian of the cost function with respect to the initial conditions. The SOA model was then used to conduct a sensitivity analysis of the cost function with respect to distributed observations and to study the evolution of the condition number (the ratio of the largest to smallest eigenvalues) of the Hessian during the course of the minimization. The condition number is strongly related to the convergence rate of the minimization. It is proved that the Hessian is positive definite during the process of the minimization, which in turn proves the uniqueness of the optimal solution for the test problem. Numerical results show that the sensitivity of the response increases with time and that the sensitivity to the geopotential field is larger by an order of magnitude than that to theu andv components of the velocity field. Experiments using data from an ECMWF analysis of the First Global Geophysical Experiment (FGGE) show that the cost functionJ is more sensitive to observations at points where meteorologically intensive events occur. Using the second order adjoint shows that most changes in the value of the condition number of the Hessian occur during the first few iterations of the minimization and are strongly correlated to major large-scale changes in the reconstructed initial conditions fields.

137 citations

Journal ArticleDOI
TL;DR: It is demonstrated that the solution errors of PDEs due to quadrature inaccuracy can be significantly reduced when the variationally inconsistent methods are corrected with the proposed method, and consequently the optimal convergence rate can be either partially or fully restored.
Abstract: Author(s): Hillman, Michael Charles | Advisor(s): Chen, Jiun-Shyan | Abstract: The rate of convergence in Galerkin methods for solving boundary value problems is determined by the order of completeness in the trial space and order of accuracy in the domain integration. If insufficiently accurate domain integration is employed, the optimal convergence rate cannot be attained. For meshfree methods accurate domain integration is difficult to achieve without costly high order quadrature, and the lack of accuracy in the domain integration may lead to sub-optimal convergence, or even solutions that diverge with refinement. The difficulty in domain integration is due to the overlap of the shape function supports and the rational nature of the shape functions themselves. This dissertation introduces a general framework to achieve the optimal order of convergence consistent with the order of trial space without high order quadrature.First, the conditions for achieving arbitrary order exactness in a boundary value problem using the Galerkin approximation with quadrature are derived. The conditions are derived in a general form and are applicable to all types of problems: the test function gradients in the Galerkin approximation should be consistent with the chosen numerical integration, and this is termed variational consistency. Specifically, integration by parts of the inner product between the test function and differential operator acting on the desired exact solution should hold when evaluated with the chosen quadrature. Specific problems are then considered with the conditions explicitly stated, including elasticity, the Euler-Bernoulli beam, the Kirchhoff-Love plate, and the non-linear formulation of solid mechanics.Treating the type of numerical integration as a given, test function gradients are then constructed to satisfy this condition. The resulting method is arbitrarily high order exact and applicable to all types of integration. The method is then used as a correction to several commonly used numerical integration methods and applied to the various boundary value problems. It is demonstrated that the error induced by numerical integration is greatly reduced, and optimal convergence is either partially or fully restored. Further, it is shown that the variationally consistent integration methods are more effective than their standard counterparts in terms of computing time and solution error.

137 citations

Journal ArticleDOI
TL;DR: The method is shown to be globally linearly convergent following the methodology established by Burke and Xu and a local acceleration step is added to the method so that it is also locally quadratically convergent under suitable assumptions.
Abstract: A noninterior continuation method is proposed for nonlinear complementarity problems. It improves the noninterior continuation methods recently studied by Burke and Xu [Math. Oper. Res., 23 (1998), pp. 719--734] and Xu [The Global Linear Convergence of an Infeasible Non-Interior Path-following Algorithm for Complementarity Problems with Uniform P-functions, Preprint, Department of Mathematics, University of Washington, Seattle, 1996]; the interior point neighborhood technique is extended to a broader class of smoothing functions introduced by Chen and Mangasarian [Comput. Optim. Appl., 5 (1996), pp. 97--138]. The method is shown to be globally linearly convergent following the methodology established by Burke and Xu. In addition, a local acceleration step is added to the method so that it is also locally quadratically convergent under suitable assumptions.

137 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20241
2023693
20221,530
20212,129
20202,036
20191,995