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.

Fast Maximnurm Likelihood Joint Estimation of Frequency and Frequency Rate

Abstract: A fast maximum likelihood algorithm is presented that jointly estimates the frequency and frequency rate of a sinusoid corrupted by additive Gaussian white noise It consists of a coarse search and a fine search First the two-dimensional frequency-frequency rate plane is subdivided into parallelograms whose size depends on the region of convergence of Newton's method used in maximizing the log-likelihood function (LLF) The size of the parallelogram is explicitly computed and is optimal for the method used The coarse search consists of maximizing the LLF over the vertices of the parallelograms Then starting at the vertex where the LLF attained its maximum, a two-dimensional Newton's method to find the absolute maximum of the LLF is implemented This last step consists of the fine search The rate of convergence of Newton's method is cubic, and is extremely fast Furthermore Newton's method will converge after two iterations when the starting point used in the method lies within 75 percent of the distances defined by the parallelogram of convergence whose center coincides with the true values of frequency and frequency rate In this case, the root mean square error (RMSEs) for frequency and frequency rate are practically equal to the Cramer-Rao bound at all signal-to-noise ratio (SNR)?15 dB The frequency-frequency rate ambiguity function is shown to be even and its periodicities are extracted

Real-time frequency and harmonic evaluation using artificial neural networks

Abstract: With increasing harmonic pollution in the power system, real-time monitoring and analysis of harmonic variations have become important. Because of limitations associated with conventional algorithms, particularly under supply-frequency drift and transient situations, a new approach based on nonlinear least-squares parameter estimation has been proposed as an alternative solution for high-accuracy evaluation. However, the computational demand of the algorithm is very high and it is more appropriate to use Hopfield type feedback neural networks for real-time harmonic evaluation. The proposed neural network implementation determines simultaneously the supply-frequency variation, the fundamental-amplitude/phase variation as well as the harmonics-amplitude/phase variation. The distinctive feature is that the supply-frequency variation is handled separately from the amplitude/phase variations, thus ensuring high computational speed and high convergence rate. Examples by computer simulation are used to demonstrate the effectiveness of the implementation. A set of data taken on site was used as a real application of the system.

Local Linear Convergence for Alternating and Averaged Nonconvex Projections

Abstract: The idea of a finite collection of closed sets having “linearly regular intersection” at a point is crucial in variational analysis. This central theoretical condition also has striking algorithmic consequences: in the case of two sets, one of which satisfies a further regularity condition (convexity or smoothness, for example), we prove that von Neumann’s method of “alternating projections” converges locally to a point in the intersection, at a linear rate associated with a modulus of regularity. As a consequence, in the case of several arbitrary closed sets having linearly regular intersection at some point, the method of “averaged projections” converges locally at a linear rate to a point in the intersection. Inexact versions of both algorithms also converge linearly.

On the complexity analysis of randomized block-coordinate descent methods

Abstract: In this paper we analyze the randomized block-coordinate descent (RBCD) methods proposed in Nesterov (SIAM J Optim 22(2):341---362, 2012), Richtarik and Takaa? (Math Program 144(1---2):1---38, 2014) for minimizing the sum of a smooth convex function and a block-separable convex function, and derive improved bounds on their convergence rates. In particular, we extend Nesterov's technique developed in Nesterov (SIAM J Optim 22(2):341---362, 2012) for analyzing the RBCD method for minimizing a smooth convex function over a block-separable closed convex set to the aforementioned more general problem and obtain a sharper expected-value type of convergence rate than the one implied in Richtarik and Takaa? (Math Program 144(1---2):1---38, 2014). As a result, we also obtain a better high-probability type of iteration complexity. In addition, for unconstrained smooth convex minimization, we develop a new technique called randomized estimate sequence to analyze the accelerated RBCD method proposed by Nesterov (SIAM J Optim 22(2):341---362, 2012) and establish a sharper expected-value type of convergence rate than the one given in Nesterov (SIAM J Optim 22(2):341---362, 2012).

X-FEM in isogeometric analysis for linear fracture mechanics

Abstract: The extended finite element method (X-FEM) has proven to be an accurate, robust method for solving problems in fracture mechanics. X-FEM has typically been used with elements using linear basis functions, although some work has been performed using quadratics. In the current work, the X-FEM formulation is incorporated into isogeometric analysis to obtain solutions with higher order convergence rates for problems in linear fracture mechanics. In comparison with X-FEM with conventional finite elements of equal degree, the NURBS-based isogeometric analysis gives equal asymptotic convergence rates and equal accuracy with fewer degrees of freedom (DOF). Results for linear through quartic NURBS basis functions are presented for a multiplicity of one or a multiplicity equal the degree.