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.

Covariance Structure and Convergence Rate of the Gibbs Sampler with Various Scans

Abstract: This paper presents results on covariance structure and convergence for the Gibbs sampler with both systematic and random scans. It is shown that, under conditions that guarantee the compactness of the Markov forward operator and irreducibility of the corresponding chain, the Gibbs sampling scheme converges geometrically in terms of Pearson χ 2 -distance. In particular, for the random scan, the autocovariance can be expressed as variances of iterative conditional expectations. As a consequence, the autocorrelations are all positive and decrease monotonically

Iteration methods for convexly constrained ill-posed problems in hilbert space

Abstract: Minimization problems in Hilbert space with quadratic objective function and closed convex constraint set C are considered In case the minimum is not unique we are looking for the solution of minimal norm If a problem is ill-posed, ie if the solution does not depend continuously on the data, and if the data are subject to errors then it has to be solved by means of regularization methods The regularizing properties of some gradient projection methods—ie convergence for exact data, order of convergence under additional assumptions on the solution and stability for perturbed data—are the main issues of this paper

The p‐Version of the Finite Element Method

Abstract: In the p-version of the finite element method, the triangulation is fixed and the degree p, of the piecewise polynomial approximation, is progressively increased until some desired level of precision is reached. In this paper, we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation. These properties lead to an a priori estimate of the asymptotic rate of convergence of the p-version. The estimate shows that the p-version gives results which are not worse than those obtained by the conventional finite element method (called the h-version, in which h represents the maximum diameter of the elements), when quasi-uniform triangulations are employed and the basis for comparison is the number of degrees of freedom. Furthermore, in the case of a singularity problem, we show (under conditions which are usually satisfied in practice) that the rate of convergence of the p-version is twice that of the h-version with quasi-uniform mesh. Inverse approximation theorems which determine the smoothness of a function based on the rate at which it is approximated by piecewise polynomials over a fixed triangulation are proved for both singular and nonsingular problems. We present numerical examples which illustrate the effectiveness of the p-version for a simple one-dimensional problem and for two problems in two-dimensional elasticity.We also discuss roundott error and computational costs associated with the p-version. Finally, we describe some important features, such as hierarchic basis functions, which have been utilized in COMET-X, an experimental computer implemen- tation of the p-version.