Topic
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: This paper considers the so-called "inexact Uzawa" algorithm for iteratively solving linear block saddle point problems, and shows that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled.
Abstract: In this paper, we consider the so-called "inexact Uzawa" algorithm for iteratively solving linear block saddle point problems. Such saddle point problems arise, for example, in finite element and finite difference discretizations of Stokes equations, the equations of elasticity, and mixed finite element discretization of second-order problems. We consider both the linear and nonlinear variants of the inexact Uzawa iteration. We show that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled. Bounds for the rate of convergence are provided in terms of the rate of convergence for the preconditioned Uzawa algorithm and the reduction factor corresponding to the preconditioner for the upper left-hand block. In the case of nonlinear iteration, the inexact Uzawa algorithm is shown to converge provided that the nonlinear process approximating the inverse of the upper left-hand block is of sufficient accuracy. Bounds for the nonlinear iteration are given in terms of this accuracy parameter and the rate of convergence of the preconditioned linear Uzawa algorithm. Applications to the Stokes equations and mixed finite element discretization of second-order elliptic problems are discussed and, finally, the results of numerical experiments involving the algorithms are presented.
396 citations
TL;DR: The implementation and development of a new Taylor?Galerkin finite-element scheme within an unstructured/hybrid, parallel solver for unsteady LES is described, which is third-order in space and time and has a low dissipative error.
395 citations
TL;DR: In this paper, the authors focus on minimum contrast estimators on sieves, which are commonly used in practice as D-dimensional linear spaces generated by some basis: piecewise polynomials, wavelets, Fourier, etc.
Abstract: This paper, which we dedicate to Lucien Le Cam for his seventieth birthday, has been written in the spirit of his pioneering works on the relationships between the metric structure of the parameter space and the rate of convergence of optimal estimators. It has been written in his honour as a contribution to his theory. It contains further developments of the theory of minimum contrast estimators elaborated in a previous paper. We focus on minimum contrast estimators on sieves. By a `sieve' we mean some approximating space of the set of parameters. The sieves which are commonly used in practice are D-dimensional linear spaces generated by some basis: piecewise polynomials, wavelets, Fourier, etc. It was recently pointed out that nonlinear sieves should also be considered since they provide better spatial adaptation (think of histograms built from any partition of D subintervals of [0,1] as a typical example). We introduce some metric assumptions which are closely related to the notion of finite-dimensional metric space in the sense of Le Cam. These assumptions are satisfied by the examples of practical interest and allow us to compute sharp rates of convergence for minimum contrast estimators.
394 citations
TL;DR: This parameter-expanded Ei M, PX-EM, algorithm shares the simplicity and stability of ordinary EM, but has a faster rate of convergence since its M step performs a more efficient analysis.
Abstract: SUMMARY The EM algorithm and its extensions are popular tools for modal estimation but ar-e often criticised for their slow convergence. We propose a new method that can often make EM much faster. The intuitive idea is to use a 'covariance adjustment' to correct the analysis of the M step, capitalising on extra information captured in the imputed complete data. The way we accomplish this is by parameter expansion; we expand the complete-data model while preserving the observed-data model and use the expanded complete-data model to generate EM. This parameter-expanded Ei M, PX-EM, algorithm shares the simplicity and stability of ordinary EM, but has a faster rate of convergence since its M step performs a more efficient analysis. The PX-EM algorithm is illustrated for the multivariate t distribution, a random effects model, factor analysis, probit regression and a Poisson imaging model.
393 citations
TL;DR: It is found that the adaptive coefficient μ, which controls the rate of convergence of the algorithm, must be restricted to an interval significantly smaller than the domain commonly stated in the literature.
Abstract: Statistical analysis of the least mean-squares (LMS) adaptive algorithm with uncorrelated Gaussian data is presented. Exact analytical expressions for the steady-state mean-square error (mse) and the performance degradation due to weight vector misadjustment are derived. Necessary and sufficient conditions for the convergence of the algorithm to the optimal (Wiener) solution within a finite variance are derived. It is found that the adaptive coefficient μ, which controls the rate of convergence of the algorithm, must be restricted to an interval significantly smaller than the domain commonly stated in the literature. The outcome of this paper, therefore, places fundamental limitations on the mse performance and rate of convergence of the LMS adaptive scheme.
392 citations