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.

A linearly conforming point interpolation method (lc-pim) for 2d solid mechanics problems

Abstract: A linearly conforming point interpolation method (LC-PIM) is developed for 2D solid problems. In this method, shape functions are generated using the polynomial basis functions and a scheme for the selection of local supporting nodes based on background cells is suggested, which can always ensure the moment matrix is invertible as long as there are no coincide nodes. Galerkin weak form is adopted for creating discretized system equations, and a nodal integration scheme with strain smoothing operation is used to perform the numerical integration. The present LC-PIM can guarantee linear exactness and monotonic convergence for the numerical results. Numerical examples are used to examine the present method in terms of accuracy, convergence, and efficiency. Compared with the finite element method (FEM) using linear triangle elements and the radial point interpolation method (RPIM) using Gauss integration, the LC-PIM can achieve higher convergence rate and better efficiency.

Convergence of quasi-Newton matrices generated by the symmetric rank one update

Abstract: Quasi-Newton algorithms for unconstrained nonlinear minimization generate a sequence of matrices that can be considered as approximations of the objective function second derivatives This paper gives conditions under which these approximations can be proved to converge globally to the true Hessian matrix, in the case where the Symmetric Rank One update formula is used The rate of convergence is also examined and proven to be improving with the rate of convergence of the underlying iterates The theory is confirmed by some numerical experiments that also show the convergence of the Hessian approximations to be substantially slower for other known quasi-Newton formulae

Large-scale sparse logistic regression

Abstract: Logistic Regression is a well-known classification method that has been used widely in many applications of data mining, machine learning, computer vision, and bioinformatics. Sparse logistic regression embeds feature selection in the classification framework using the l1-norm regularization, and is attractive in many applications involving high-dimensional data. In this paper, we propose Lassplore for solving large-scale sparse logistic regression. Specifically, we formulate the problem as the l1-ball constrained smooth convex optimization, and propose to solve the problem using the Nesterov's method, an optimal first-order black-box method for smooth convex optimization. One of the critical issues in the use of the Nesterov's method is the estimation of the step size at each of the optimization iterations. Previous approaches either applies the constant step size which assumes that the Lipschitz gradient is known in advance, or requires a sequence of decreasing step size which leads to slow convergence in practice. In this paper, we propose an adaptive line search scheme which allows to tune the step size adaptively and meanwhile guarantees the optimal convergence rate. Empirical comparisons with several state-of-the-art algorithms demonstrate the efficiency of the proposed Lassplore algorithm for large-scale problems.

Computational Models of Electromagnetic Resonators: Analysis of Edge Element Approximation

Abstract: The purpose of this paper is to address some difficulties which arise in computing the eigenvalues of Maxwell's system by a finite element method. Depending on the method used, the spectrum may be polluted by spurious modes which are difficult to pick out among the approximations of the physically correct eigenvalues. Here we propose a criterion to establish whether or not a finite element scheme is well suited to approximate the eigensolutions and, in the positive case, we estimate the rate of convergence of the eigensolutions. This criterion involves some properties of the finite element space and of a suitable Fortin operator. The lowest-order edge elements, under some regularity assumptions, give an example of space satisfying the required conditions. The construction of such a Fortin operator in very general geometries and for any order edge elements is still an open problem. Moreover, we give some justification for the spectral pollution which occurs when nodal elements are used. Results of numerical experiments confirming the theory are also reported.